Algebraic Tools for Modal Logic
Mai Gehrke
Yde Venema
ESSLLI’01
August 13 – 17, 2001
Helsinki, Finland
Algebraic Tools for Modal Logic
Mai Gehrke
Yde Venemay
General aim
There is a long and strong tradition in logic research of applying algebraic techniques in order
to deepen our understanding of logic. Such applications are possible because many logics
correspond to classes of algebras; typically, the consequence relation of the logic translates
into the (quasi-)equational theory of the corresponding class of algebras.
This correspondence between logic and algebra allows one, on a rst level, to study the
algebras in order to understand the deductive system. But often, metalogical properties also
end up having algebraic counterparts. In modal logic, a striking example of this phenomenon
can be found using the duality theory between Kripke structures and Boolean Algebras with
Operators. For instance, a modal logic is complete if and only if its corresponding algebraic
variety is generated by the class of algebras that are dual to the Kripke frames of the logic.
A central tool in proving completeness for modal logics is the notion of canonicity, which
happens to have an equally important and interesting algebraic expression. For a long time
the algebraic and the logical strand of research on these issues have been carried out in relative
isolation, and one of the aims of this course is to strengthen the ties between the two research
lines.
More concretely, the purpose of the course is twofold. Apart from giving a general introduction to the fundamental ideas and methods of applying algebra in logic, we also intend to
present recent developments from algebra as well as modal logic, in an integrated format. Our
intention is to illuminate and generalize existing results concerning the issues of completeness,
canonicity and correspondence for modal and generalized modal logics.
Course outline
In the rst part of the course (consisting of the rst two lectures) we give a general introduction
to the algebraic perspective on logic. As our running examples we take classical propositional
logic and modal logic. We show how the technical notion of a modal logic corresponds to the
algebraic one of a variety of Boolean algebras with operators, and we discuss the connection
between the logical and the algebraic angle on properties such as completeness and canonicity.
Department of
Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA. E-mail:
[email protected].
y Institute
of Logic, Language and Computation, Univ. of Amsterdam, Plantage Muidergracht 24, 1018 TV
Amsterdam. E-mail:
[email protected].
1
In the last three lectures we concentrate on the notion of canonicity, which plays an equally
fundamental role in the theory of modal logic as in the algebraic theory of Boolean algebras
with operators. The aim of this part of the course is to to present some recent results in the
eld that generalize and illuminate the classical results.
Intended audience & prerequisites
The intended audience for this course consists of two groups of people, and of each group we
presuppose some background knowledge.
First and foremost, we have tailored our course towards students that have encountered
modal logic before and want to study its general theory in more detail. We presuppose that
these students are familiar with the canonical model method for proving completeness of a
modal logic, and with the notion of a boolean algebra. Furthermore, some experience with
mathematical argumentation is required. It would be useful if the student had encountered
some lattice theory as well, but this is not strictly needed.
And second, the course could be of value for mathematics students that have an interest in
universal algebra, and more particular, in the theory of lattices and lattices with additional
operators. For these students, some familiarity with duality theory would be useful, but
knowledge of modal logic is not strictly required.
About these notes
These Notes consist of three parts.
1.
Modal Logic and Their Algebras,
by Patrick Blackburn, Maarten de Rijke and Yde
Venema. This is an excerpt from the recently published textbook Modal Logic by the
same three authors (Cambridge University Press, Cambridge, 2001). There are two
chapters, of which the rst starts with a recollection of some basic facts concerning
modal logic and then gives a fairly detailed description of the canonical model method
for proving modal completeness results. The second chapter introduces the algebraic
approach towards modal logic, and ends with a discussion of the importance of the
notion of canonicity in logic and algebra.
2. The article On the Canonicity of Sahlqvist Identities by Bjarni Jonsson, (Studia Logica,
1994) provides an algebraic description of the notion of canonicity. It modernizes and
further develops the classical algebraic methodology for proving canonicity and leads
to an algebraic proof of an important result in modal logic stating that all so-called
Sahlqvist formulas are canonical.
3.
Monotone Bounded Distributive Lattice Expansions, by Mai Gehrke and Bjarni J
onsson,
(Mathematica Japonica, 2000). This paper treats canonical extensions of distributive
lattice based algebras from an algebraic point of view. It includes the abstract algebraic
characterization of canonical extensions, the algebraic denition of extensions of maps,
and algebraic canonicity results mainly of semantic nature.
2
Modal Logic and Their Algebras
Patrick Blackburn, Maarten de Rijke and Yde Venema
1
Modal Logic: Basic Concepts
In this chapter we recollect some basic facts concerning modal logic, concentrating
on completeness theory.
1.1 Syntax
Definition 1.1 The basic modal language is defined using a set of proposition letters (or proposition symbols or propositional variables) whose elements are usually denoted p, q , r , and so on, and a unary modal operator 3 (‘diamond’). The
well-formed formulas of the basic modal language are given by the rule
::= p j ? j : j
_ j 3;
where p ranges over elements of .
Just as the familiar first-order existential and universal quantifiers are duals to
each other (in the sense that 8x $ :9x :), we have a dual operator (‘box’) for
our diamond which is defined by := :3:. We also make use of the classical abbreviations for conjunction, implication, bi-implication and the constant true
(‘top’): ^ := :(: _: ), ! := : _ , $ := ( ! )^( ! )
and > := : ?.
a
Although we generally assume that the set of proposition letters is a countably
infinite set fp0 ; p1 ; : : :g, occasionally we need to make other assumptions. This is
why we take as an explicit parameter when defining the set of modal formulas.
That is the basic modal language. Let us now generalize it. There are two obvious
ways to do so. First, there seems no good reason to restrict ourselves to languages
with only one diamond. Second, there seems no good reason to restrict ourselves
to modalities that take only a single formula as argument. Thus the general modal
languages we will now define may contain many modalities, of arbitrary arities.
1
2
1 Modal Logic: Basic Concepts
Definition 1.2 A modal similarity type is a pair = (O; ) where O is a nonempty set, and is a function O ! N . The elements of O are called modal operators; we use M (‘triangle’), M0 , M1 , . . . , to denote elements of O . The function assigns to each operator M 2 O a finite arity, indicating the number of arguments
M can be applied to.
In line with Definition 1.1, we often refer to unary triangles as diamonds, and
denote them by 3a or hai, where a is taken from some index set. We often assume
that the arity of operators is known, and do not distinguish between and O .
a
Definition 1.3 A modal language ML(; ) is built up using a modal similarity
type = (O; ) and a set of proposition letters . The set Form (; ) of modal
formulas over and is given by the rule
:= p j ? j : j 1 _ 2 j M(1 ; : : : ; (M) );
a
where p ranges over elements of .
Definition 1.4 We now define dual operators for non-nullary triangles. For each
M 2 O the dual O of M is defined as O(1 ; : : : ; n ) := :M(:1 ; : : : ; :n ). The
dual of a triangle of arity at least 2 is called a nabla. As in the basic modal language,
the dual of a diamond is called a box, and is written 2a or [a].
a
Example 1.5 (An Arrow Language) The type ! of arrow logic is a similarity
type with modal operators other than diamonds. The language of arrow logic is
designed to talk about the objects in arrow structures (entities which can be pictured
as arrows). The well-formed formulas of the arrow language are given by the rule
:= p j ? j : j _
j Æ j j 1’:
That is, 1’ (‘identity’) is a nullary modality (a modal constant), the ‘converse’ operator is a diamond, and the ‘composition’ operator Æ is a dyadic operator. Possible
readings of these operators are:
1’
Æ
identity
‘skip’;
converse
‘ conversely’ ;
composition ‘first , then ’:
a
1.2 Semantics
Basic modal language
We start by defining frames, models, and the satisfaction relation for the basic
modal language.
1.2 Semantics
3
Definition 1.6 A frame for the basic modal language is a pair F = (W; R) such
that W is a non-empty set and R is a binary relation on W .
That is, a frame for the basic modal language is simply a relational structure
bearing a single binary relation. Elements of W will be referred to by various
names such as nodes, worlds, or points.
A model for the basic modal language is a pair M = (F; V ), where F is a frame
for the basic modal language, and V is a function assigning to each proposition
letter p in a subset V (p) of W . Formally, V is a map: ! P(W ), where
P(W ) denotes the power set of W . Informally we think of V (p) as the set of
points in our model where p is true. The function V is called a valuation. Given a
model M = (F; V ), we say that M is based on the frame F, or that F is the frame
underlying M.
a
Definition 1.7 Suppose w is a state in a model M = (W; R; V ). Then we inductively define the notion of a formula being satisfied (or true) in M at state w as
follows:
M; w p
M; w ?
M; w :
w 2 V (p); where p 2 ;
never;
iff not M; w ;
M; w _ iff M; w or M; w ;
M; w 3 iff for some v 2 W with Rwv we have M; v : (1.1)
iff
It follows from this definition that M; w 2 if and only if for all v 2 W such
that Rwv , we have M; v . Finally, we say that a set of formulas is true at a
state w of a model M, notation: M; w , if all members of are true at w. a
It is convenient to extend the valuation V from proposition letters to arbitrary
formulas so that V () always denotes the set of states at which is true:
V ()
:= fw j M; w g:
Definition 1.8 A formula is globally or universally true in a model M (notation:
M ) if it is satisfied at all points in M (that is, if M; w , for all w 2 W ).
A formula is satisfiable in a model M if there is some state in M at which is
true; a formula is falsifiable or refutable in a model if its negation is satisfiable.
A set of formulas is globally true (satisfiable, respectively) in a model M if
M; w for all states w in M (some state w in M, respectively).
a
Arbitrary modal languages
We now define frames, models and satisfaction for modal languages of arbitrary
similarity type.
4
1 Modal Logic: Basic Concepts
Definition 1.9 Let be a modal similarity type. A -frame is a tuple F consisting
of the following ingredients:
(i) a non-empty set W ,
(ii) for each n 0, and each n-ary modal operator
an (n + 1)-ary relation RM .
M in the similarity type ,
We turn such a frame into a model in exactly the same way that we did for
the basic modal language: by adding a valuation. That is, a -model is a pair
M = (F; V ) where F is a -frame, and V is a valuation with domain and range
P(W ), where W is the universe of F.
The notion of a formula being satisfied (or true) at a state w in a model M =
(W; fRM j M 2 g; V ) (notation: M; w ) is defined inductively. The clauses
for the atomic and boolean cases are the same as for the basic modal language (see
Definition 1.7). As for the modal case, when (M) > 0 we define
M; w M(1 ; : : : ; n )
2 W with RMwv : : : vn
we have, for each i, M; vi i :
This is an obvious generalization of the way 3 is handled in the basic modal laniff for some v1 , . . . , vn
1
guage. Before going any further, the reader should formulate the satisfaction clause
for O(1 ; : : : ; n ).
As before, we often write w for M; w where M is clear from the
context. The concept of global truth (or universal truth) in a model is defined
as for the basic modal language: it simply means truth at all states in the model.
And, as before, we sometimes extend the valuation V supplied by M to arbitrary
formulas.
a
Example 1.10 (Arrow Models) Consider the similarity type ! of arrow logic,
cf. Example 1.5. Arrow frames are structures of the form F = (W; C; R; I ), where
C is a ternary relation interpreting Æ, R a binary relation for and I a unary relation for 1’. An arrow model is a structure M = (F; V ) such that F = (W; C; R; I )
is an arrow frame and V is a valuation. Then:
M; a 1’
M; a M; a Æ
Ia;
iff M; b for some b with Rab;
iff M; b and M; c for some b and c with Cabc:
iff
Interesting examples of such structures are the two-dimensional ones arising as
follows. Let U be some set, and let U U be the set of pairs over U ; suppose now
that the relations C , R and I on U U are defined as follows:
Cabc iff a0 = b0 ; a1 = c1 and b1 = c0 ;
Rab iff a0 = b1 and a1 = b0 ;
1.3 Completeness
5
Ia iff a0 = a1 :
Then we call the structure (U U; C; R; I ) the two-dimensional arrow frame over
U , notation: SU . In such a structure, the semantics works out as follows. V now
maps propositional variables to sets of pairs over U ; that is, to binary relations.
The truth definition can be rephrased as follows:
M; (a0 ; a1 ) 1’
M; (a0 ; a1 ) M; (a0 ; a1 ) Æ
a0 = a1 ;
iff M; (a1 ; a0 ) ;
iff M; (a0 ; u) and M; (u; a1 ) for some u 2 U:
iff
a
Validity
Definition 1.11 A formula is valid at a state w in a frame F (notation: F; w )
if is true at w in every model (F; V ) based on F; is valid in a frame F (notation:
F ) if it is valid at every state in F. A formula is valid on a class of frames
F (notation: F ) if it is valid on every frame F in F; and it is valid (notation:
) if it is valid on the class of all frames. The set of all formulas that are valid in
a class of frames F is called the logic of F (notation: F ).
a
Our definition of the logic of a frame class F (as the set of ‘all’ formulas that are
valid on F) is underspecified: we did not say which collection of proposition letters
should be used to build formulas. But usually the precise form of this collection
is irrelevant for our purposes. Readers that are worried about this may define, given
a set of proposition letters, F () to be the set f 2 Form (; ) j F g.)
Note that the above is a fairly restricted viewpoint on what a logic is: in particular, we do not take the semantics consequence relation into consideration.
1.3 Completeness
Normal Modal Logics
Suppose we are interested in a certain class of frames F: are there syntactic mechanisms capable of generating F , the formulas valid on F? One response to such
questions is embodied in the concept of a normal modal logic. A normal modal
logic is simply a set of formulas satisfying certain syntactic closure conditions. In
this section we are working in a fixed countable set of proposition letters, and we
first confine ourselves to the basic modal language.
Definition 1.12 (Normal Modal Logics) A normal modal logic in the basic modal
language is a set of modal formulas that contains, besides all propositional tautologies, the axioms
6
(K)
(Dual)
1 Modal Logic: Basic Concepts
2(p ! q) ! (2p ! 2q),
3p $ :2:p.
while it is closed under the rules
(modus ponens)
(generalization)
(uniform substitution)
if 2 and ! 2 then 2 ,
if 2 then 2 2 , and
if belongs to then so do all of its substitution instances.
If 2 we say that is a theorem of and write ` ; if not, we write 6` . If
1 and 2 are modal logics such that 1 2 , we say that 2 is an extension of
1 .
In what follows, we usually drop the words ‘normal modal’ and talk simply of
‘logics.’
a
Remark 1.13 Apart from the inclusion of the Dual axiom – needed because we
took the diamond as the primitive operator in our language – the above definition of
a normal modal logic is probably the most popular way of stipulating what normal
logics are. But it is not the only way. Here, for example, is a simple diamond-based
formulation of the concept, which will be useful in our algebraic work: a logic is
normal if it contains (besides the propositional tautologies) the axioms 3? $ ?
and 3(p _ q ) $ 3p _ 3q , and is closed under the rules of modus ponens, universal
substitution and: ` ! implies ` 3 ! 3 . This formulation is equivalent
to Definition 1.12, as the interested reader may verify.
a
Example 1.14
(i) The collection of all formulas is a logic, the inconsistent
logic.
(ii) The set of formulas valid in a given frame class is a logic.
T
(iii) If fi j i 2 I g is a collection of logics, then i2I i is a logic.
(iv) If M is a class of models, then the set of formulas that are true throughout
any model in M need not be a logic since it may not be closed under uniform
substitution.
a
Example 1.14(iii) guarantees that there is a smallest normal modal logic containing
any set of formulas.
Definition 1.15 The normal modal logic generated or axiomatized by , notation:
K , is the smallest logic containing ; the formulas in are called the axioms of
this logic. The logic generated by the empty set is the minimal normal modal logic
K.
a
1.3 Completeness
7
This generative perspective, which is essentially syntactic, can be phrased in the
terminology of Hilbert style axiomatizations, but since we are not interested in the
details of proof systems we will refrain from doing so.
Defining a logic by stating which formulas generate it (that is, extending the
minimal normal logic K with certain axioms of interest) is the usual way of syntactically specifying normal logics. Here are some of the better known axioms,
together with their traditional names:
(4)
(T)
(B)
(D)
(.3)
33p ! 3p
p ! 3p
p ! 23p
2p ! 3p
3p ^ 3q ! 3(p ^ 3q) _ 3(p ^ q) _ 3(q ^ 3p)
There is a convention for talking about the logics generated by such axioms: if
A1 ; : : : ; An are axioms then KA1 : : : An is the normal logic generated by A1 , . . . ,
An . But irregularities abound. Many historical names are firmly entrenched, thus
modal logicians talk of T, S4, B, and S5 instead of KT, KT4, KB and KT4B
respectively.
Soundness and Completeness
Now that we know what normal modal logics are, we are ready to introduce the two
fundamental concepts linking the syntactic and semantic perspectives: soundness
and completeness.
Definition 1.16 (Soundness and Completeness) Let F be a class of frames (or
models, or general frames). A normal modal logic is sound with respect to F if
F . (Equivalently: is sound with respect to F if for all formulas , and all
structures F 2 F, ` implies F .) If is sound with respect to F we say that
F is a class of frames (or models, or general frames) for .
Conversely, is complete with respect to F if for any formula , if F ; that
is, if F then ` .
a
These notions, which only refer to the theorems of the logic and the set of validities of a frame class, are often referred to as the weak versions of soundness
and completeness. The corresponding strong notions, which relate the derivability
of formulas from assumptions to the semantic consequence relation over a frame
class, will not be discussed in these notes.
Table 1.1 lists a number of well-known logics together with classes of frames for
which they are sound. Recall that a right-unboundedness frame (W; R) is a frame
such that 8x9y Rxy . Also, a frame (W; R) satisfying 8x8y 8z (Rxy ^ Rxz !
(Ryz _ y = z _ Rzy)) is said to have no branching to the right.
8
1 Modal Logic: Basic Concepts
These completeness results are among the best known in modal logic, and we
will soon be able to prove them. Together with their soundness counterparts, they
constitute perspicuous semantic characterizations of important logics. K4, for example, is not just the logic obtained by enriching K with some particular axiom:
it is precisely the set of formulas valid on all transitive frames. There is always
something arbitrary about syntactic presentations; it is pleasant (and useful) to have
these semantic characterizations at our disposal.
K
K4
T
B
KD
S4
S5
K4.3
S4.3
KL
the class of all frames
the class of transitive frames
the class of reflexive frames
the class of symmetric frames
the class of right-unbounded frames
the class of reflexive, transitive frames
the class of frames whose relation is an equivalence relation
the class of transitive frames with no branching to the right
the class of reflexive, transitive frames with no branching to the right
the class of finite transitive trees (weak completeness only)
Table 1.1. Some Soundness and Completeness Results
These completeness results are among the best known in modal logic, and we will
soon be able to prove them. Together with their soundness counterparts, they constitute perspicuous semantic characterizations of important logics. K4, for example, is not just the logic obtained by enriching K with some particular axiom: it is
precisely the set of formulas valid on all transitive frames. There is always something arbitrary about syntactic presentations; it is pleasant (and useful) to have
these semantic characterizations at our disposal. Not all normal modal logics can
be semantically characterized as the set of formulas valid in a certain frame class.
This fact makes the following definition meaningful.
Definition 1.17 A normal modal logic is complete is it is the logic of some frame
class F, i.e., if = F .
a
Arbitrary Modal Languages
To conclude this section, we extend the definition of normal modal logics to arbitrary similarity types.
Definition 1.18 Assume we are working with a modal language of similarity type
. A modal logic in this language is a set of formulas containing all tautologies that
is closed under modus ponens and uniform substitution. A modal logic is normal
1.4 Canonical Models
9
if for every operator O it contains the axiom KiO (for all i such that 1 i (O))
and the axiom DualO , and is closed under the generalization rules described below.
The required axioms are obvious polyadic analogs of the earlier K and Dual
axioms:
(KiO )
O(r ; : : : ; p ! q; : : : ; r O ) !
! O(r ; : : : ; p; : : : ; r O ) ! O(r ; : : : ; q; : : : ; r O ) :
M(r ; : : : ; r O ) $ :O(:r ; : : : ; :r O ):
1
( )
1
(DualO )
1
( )
( )
1
1
( )
( )
(Here p; q; r1 ; : : : ; r(O) are distinct propositional variables, and the occurrences
KiO of p and q occur in the i-th argument place of O.) Finally, for a polyadic
operator O, generalization takes the following form:
` implies ` O(?; : : : ; ; : : : ; ?):
That is, an n-place operator O is associated with n generalization rules, one for
each of its n argument positions.
Note that these axioms and rules do not apply to nullary modalities. Nullary
modalities are rather like propositional variables and – as far as the minimal logic
is concerned – they do not give rise to any axioms or rules.
a
Definition 1.19 Let be a modal similarity type. Given a set of -formulas ,
we define K , the normal modal logic axiomatized or generated by , to be the
smallest normal modal -logic containing all formulas in . Formulas in are
called axioms of this logic, and may be called an axiomatization of K . The
normal modal logic generated by the empty set is denoted by K .
a
1.4 Canonical Models
Soundness proofs are usually routine, but completeness of a logic is often hard to
prove. The standard approach to show that a logic is complete with respect to a
frame class F uses contraposition: one proves that if a formula does not belong
to a logic, it is satisfiable in some model based on a frame in F. Thus completeness
theorems are essentially model existence theorems. In this section we discuss, in a
fair bit of detail, the most important method for proving completeness; it is called
the canonical model method because it constructs a single model in which all nontheorems of the logic can be refuted.
The idea behind the construction of this canonical model is of an elegant simplicity: let the points of the model form a coherently related collection of sets of
formulas, and make sure that any modal formula is true at such a point if and only
if it is a member of it. In a slogan:
truth = membership.
10
1 Modal Logic: Basic Concepts
Maximal Consistent Sets
First we need some auxiliary technical definitions. Fix a normal modal logic
The points of the canonical model for will be maximal -consistent sets.
.
Definition 1.20 If [ fg is a set of formulas then is deducible in from
(or: is -deducible from ) if there are formulas 1 ,. . . , n 2 such that
` ( ^ ^ n) ! :
If this is the case we write
` , if not, 6` . A set of formulas is consistent if 6` ?, and -inconsistent otherwise. A formula is -consistent if
fg is -consistent; otherwise is -inconsistent. A set of formulas is maximal
1
-consistent, or: a -MCS, if is -consistent, and any set of formulas properly
containing is -inconsistent. It is easy to see that is -consistent iff : 2 .
a
Why use MCSs in completeness proofs? To see this, note that every point w in every
model M for a logic is associated with a set of formulas, namely f j M; w g. It is easy to check (and the reader should do so) that this set of formulas is
actually a -MCS. That is: if is true in some model for , then belongs to a
-MCS. The idea of the canonical model construction is to turn this observation
around.
We need to learn a little more about MCSs.
Proposition 1.21 (Properties of MCSs) If is a logic and
(i)
is closed under modus ponens: if , !
(ii) ;
(iii) for all formulas : 2 or : 2 ;
(iv) for all formulas , : _ 2 iff 2 or
2
is a -MCS then:
, then
2
2
;
.
Proof. Left to the reader.
a
As MCSs are to be our building blocks, it is vital that we have enough of them. In
fact, any consistent set of formulas can be extended to a maximal consistent one.
Lemma 1.22 (Lindenbaum’s Lemma) If is a -consistent set of formulas then
there is a -MCS + such that + .
Proof. Let 0 , 1 , 2 ; : : : be an enumeration of the formulas of our language. We
define the set + as the union of a chain of -consistent sets as follows:
0
n+1
=
=
;
n [ fn g; if this is -consistent
n [ f:n g; otherwise
1.4 Canonical Models
+
=
S
11
n0 n :
The proof of the following properties of + is left as an exercise to the reader:
(i) n is -consistent, for all n; (ii) exactly one of and : is in + , for every
formula ; (iii) if + ` , then 2 + ; and finally (iv) + is a -MCS.
a
The Canonical Model
In particular, Lindenbaum’s Lemma guarantees that every -consistent formula
belongs to some -MCS. Thus, if we take as the universe of our canonical model
the collection of all -MCSs, we are sure that every -consistent formula is an
element of some point in the model. Hence, if we can indeed prove a Truth Lemma
stating that ‘truth = membership’, we have shown that every -consistent formula
is satisfiable in the canonical model for .
Obviously, to make this work it is crucial to find the right definition of the canonical valuation and the canonical accessibility relations. The first is straightforward;
for the latter, let us have look at how maximal consistent sets are related in a concrete model. If w is related to w0 in some model M, then it is clear that the information embodied in the MCSs associated with w and w0 is ‘coherently related’.
The idea behind the canonical model construction is to try and turn this observation
around: that is, to work backwards from collections of coherently related MCSs to
the desired model.
We are now ready to build models out of MCSs, and in particular, to build the
very special models known as canonical models. With the help of these structures
we will be able to prove the Canonical Model Theorem, a universal completeness
result for normal logics. We first define canonical models and prove this result for
the basic modal language; at the end of the section we generalize our discussion to
languages of arbitrary similarity type.
Definition 1.23 The canonical model M for a normal modal logic (in the basic
language) is the triple (W ; R ; V ) where:
W is the set of all -MCSs;
R is the binary relation on W defined by Rwu if for all formulas ,
2 u implies 3 2 w. R is called the canonical relation;
(iii) V is the valuation defined by V (p) = fw 2 W j p 2 wg. V is called
(i)
(ii)
the canonical (or natural) valuation.
The pair F
= (W ; R ) is called the canonical frame for .
a
All three clauses deserve comment. First, the canonical valuation equates the truth
of a propositional symbol at w with its membership in w. Our ultimate goal is to
12
1 Modal Logic: Basic Concepts
prove a Truth Lemma which will lift this ‘truth = membership’ equation to arbitrary
formulas.
Second, note that the states of M consist of all -consistent MCSs. The significance of this is that, by Lindenbaum’s Lemma, any -consistent set of formulas
is a subset of some point in M – hence, by the Truth Lemma proved below, any
-consistent set of formulas is true at some point in this model. In short, the single structure M is a ‘universal model’ for the logic , which is why it is called
‘canonical.’
Finally, consider the canonical relation: a state w is related to a state u precisely
when for each formula in u, w contains the information 3 . Intuitively, this
captures what we mean by MCSs being ‘coherently related.’ A dual formulation in
terms of boxes instead of diamonds is ac The following lemma, stating that an alternative formulation in terms of boxes instead of diamonds is actually equivalent,
shows that we are getting things right:
Lemma 1.24 For any normal logic
implies 2 v .
, R wv iff for all formulas ,
2 2w
Proof. For the left to right direction, suppose R wv . Further suppose 62 v . As v
is an MCS, by Proposition 1.21 : 2 v . As R wv , 3: 2 w. As w is consistent,
:3: 62 w. That is, 2 62 w and we have established the contrapositive. We
leave the right to left direction to the reader.
a
In fact, the definition of R is exactly what we require; all that remains to be
checked is that enough ‘coherently related’ MCSs exist for our purposes.
Lemma 1.25 (Existence Lemma) For any normal modal logic and any state
w 2 W , if 3 2 w then there is a state v 2 W such that R wv and 2 v.
Proof. Suppose 3 2 w. We will construct a state v such that R wv and 2 v .
Let v be fg [ f j 2 2 wg. Then v is consistent. For suppose not. Then
there are 1 , . . . , n such that ` ( 1 ^ ^ n ) ! :, and it follows by an
easy argument that ` 2( 1 ^ ^ n ) ! 2:: As the reader should check, the
formula (2 1 ^ ^ 2 n ) ! 2( 1 ^ ^ n ) is a theorem of every normal
modal logic, hence by propositional calculus, ` (2 1 ^ ^2 n ) ! 2:. Now,
2 1 ^ ^ 2 n 2 w (for 2 1, . . . , 2 n 2 w, and w is an MCS) thus it follows
that 2: 2 w. Using Dual, it follows that :3 2 w. But this is impossible: w is
an MCS containing 3. We conclude that v is consistent after all.
Let v be any MCS extending v ; such extensions exist by Lindenbaum’s Lemma.
By construction 2 v . Furthermore, for all formulas , 2 2 w implies 2 v .
Hence by Lemma 1.24, R wv .
a
With this established, the rest is easy. First we lift the ‘truth = membership’ equation to arbitrary formulas:
1.4 Canonical Models
13
Lemma 1.26 (Truth Lemma) For any normal modal logic and any formula ,
M ; w iff 2 w.
Proof. By induction on the degree of . The base case follows from the definition
of V . The boolean cases follow from Proposition 1.21. It remains to deal with the
modalities. The left to right direction is more or less immediate from the definition
of R :
M ; w 3
9v (Rwv ^ M ; v )
9v (Rwv ^ 2 v)
(Induction Hypothesis)
3 2 w
(Definition R ):
For the right to left direction, suppose 3 2 w. By the equivalences above, it
suffices to find an MCS v such that R wv and 2 v – and this is precisely what
the Existence Lemma guarantees.
a
iff
iff
only if
The Canonical Model Theorem
We now have all the material to prove the Canonical Model Theorem. Although it
is not strictly necessary here, it is often useful to think of completeness proofs in
the following way.
Proposition 1.27 A logic is complete with respect to a class of structures (models or frames) S iff every -consistent formula is satisfiable on some S 2 S.
Proof. To prove the right to left implication we argue by contraposition. Suppose
is not complete with respect to S. Thus there is a formula such that S but
62 . But then : is -consistent, but not satisfiable on any structure in S. The
left to right direction is left to the reader.
a
The following Theorem is the main result concerning canonical models. It states
that every normal modal logic is identical to the set of formulas that are through
at every state of its canonical model. (Comparing this to the remark in Example 1.14(iv), we see that the canonical model is rather special indeed.)
Theorem 1.28 (Canonical Model Theorem) Any normal modal logic is complete
with respect to its canonical model. That is, for any formula , we have
M iff 2 :
Proof. Soundness is left to the reader. For completeness, suppose is a consistent
formula of the normal modal logic . By Lindenbaum’s Lemma there is a -MCS
to which belongs. By the Truth Lemma, M ; , whence is satisfiable in
the canonical model. Thus by the previous proposition, is complete with respect
to its canonical model.
a
14
1 Modal Logic: Basic Concepts
At first glance, the Canonical Model Theorem may seem rather abstract. It is a
completeness result with respect to a single model, not a class of frames, and a
rather abstract model at that. (That K4 is complete with respect to the class of transitive frames is interesting; that it is complete with respect to the singleton class
containing only its canonical model seems rather dull.) But appearances are misleading: canonical models are by far the most important tool used in completeness
proofs. For a start, the Canonical Model Theorem immediately yields the following
result:
Theorem 1.29 K is complete with respect to the class of all frames.
Proof. By Proposition 1.27, to prove this result it suffices to find, for any Kconsistent formula , a model M (based on any frame whatsoever) and a state
w in M such that M; w . This is easy: simply choose M to be (FK; V K), the
canonical model for K, and let w be any K-MCS containing .
a
More importantly, it is often easy to get useful information about the structure of
canonical frames. For example, as we will learn in the next section, the canonical
frame for K4 is transitive – and this immediately yields the (more interesting) result that K4 is complete with respect to the class of transitive frames. Even when
a canonical model is not as cleanly structured as we would like, it still embodies a vast amount of information about its associated logic; one of the important
themes pursued later in the chapter is how to make use of this information indirectly. Furthermore, canonical models are mathematically natural. As we will
learn in Chapter 2, from an algebraic perspective canonical models are not abstract
oddities: indeed, they are precisely the structures one is lead to by considering the
ideas underlying the Stone Representation Theorem.
1.4.1 Arbitrary Modal Languages
To conclude this section we sketch the generalizations required to extend the results
obtained so far to languages of arbitrary similarity types.
Definition 1.30 Let be a modal similarity type, and a normal modal logic in
; V )M2 for has
the language over . The canonical model M = (W ; RM
W and V as defined in Definition 1.23, while for an n-ary operator M 2 the
(W )n+1 is defined by R wu1 : : : un if for all formulas 1 2 u1 ,
relation RM
M
. . . , n 2 un we have M( 1 ; : : : ; n ) 2 w.
a
There is an analog of Lemma 1.24.
Lemma 1.31 For any normal modal logic
, RM wu1 : : : un iff for all formulas
1.5 Applications
; : : : ; n,
i 2 ui .
1
O(
1
; : : : ; n)
2
15
1i
w implies that for some i such that
n,
a
Proof. Left to the reader.
Now for the crucial lemma – we must show that enough coherently related MCSs
exist. This requires a more delicate approach than was needed for Lemma 1.25.
Lemma 1.32 (Existence Lemma) Suppose M( 1 ; : : : ; n ) 2 w. Then there are
u1 , . . . , un such that 1 2 u1 , . . . , n 2 un and R wu1 : : : un .
Proof. The proof of Lemma 1.25 establishes the result for any unary operators in
the language, so it only remains to prove the (trickier) case for modalities of higher
arity. To keep matters simple, assume that M is binary; this illustrates the key new
idea needed.
So, suppose M( 1 ; 2 ) 2 w. Let 0 , 1 , . . . enumerate all formulas. We construct two sequences of sets of formulas
f g = 1
0
1
and
f g = 2
0
1
such that all i and i are finite and consistent, i+1 is either i [ fi g or
V
V
i [f:i g, and similarly for i+1 . Moreover, putting i := i and i := i ,
we will have that M(i ; i ) 2 w.
The key step in the inductive construction is
M(i ; i) 2 w ) M (i ^ (i _ :i); i ^ (i _ :i)) 2 w
) M ((i ^ i) _ (i ^ :i); (i ^ i) _ (i ^ :i)) 2 w
) one of the formulas M(i ^ [:]i ; i ^ [:]i ) is in w.
If, for example, M(i ^ i ; i ^ :i ) 2 w, we take i := i [ fi g, i :=
i [ f:i g. Under
this definition, all i and i have the required properties.
S
S
Finally, let u = i i and u = i i . It is easy to see that u , u are -MCSs
wu u , as required.
and RM
a
+1
1
1
2
+1
1
2
2
With this lemma established, the real work has been done. The Truth Lemma
and the Canonical Model Theorem for general modal languages are now obvious
analogs of Lemma 1.26 and Theorem 1.28. The precise statements and proofs are
left to the reader.
1.5 Applications
In this section we put canonical models to work. First we show how to prove some
of the frame completeness results noted in Table 1.1 using a simple and uniform
method of argument. This leads us to isolate one of the most important concepts
of modal completeness theory: canonicity.
16
1 Modal Logic: Basic Concepts
Suppose we suspect that a normal modal logic is complete with respect to a
class of frames F; how should we go about proving it? Actually, there is no infallible strategy. (Indeed, many normal modal logics are not complete with respect
to any class of frames whatsoever.) Nonetheless, a very simple technique works
in a large number of interesting cases: simply show that the canonical frame for
belongs to F. We call such proofs completeness-via-canonicity arguments, for
reasons which will soon become clear. Let us consider some examples.
Theorem 1.33 The logic K4 is complete with respect to the class of transitive
frames.
Proof. Given a K4-consistent formula , it suffices to find a model (F; V ) and a
state w in this model such that (1) (F; V ); w , and (2) F is transitive. Let
MK4 be the canonical model for K4 and let be any K4-MCS containing . By
Lemma 1.26, MK4 ; so step (1) is established. It remains to show that FK4
is transitive. So suppose w, v and u are points in this frame such that RK4 wv and
RK4vu. We wish to show that RK4wu. Suppose 2 u. As RK4vu, 3 2 v, so
as RK4 wv , 33 2 w. But w is a K4-MCS, hence it contains 33 ! 3 , thus
by modus ponens it contains 3 . Thus RK4 wu.
a
In spite of its simplicity, the preceding result is well worth reflecting on. Two
important observations should be made.
First, the proof actually establishes something more general than the theorem
claims: namely, that the canonical frame of any normal logic containing 33p !
3p is transitive. The proof works because all MCSs in the canonical frame contain
the 4 axiom; it follows that the canonical frame of any extension of K4 is transitive,
for all such extensions contain the 4 axiom.
Second, the result suggests that there may be a connection between the structure
of canonical frames and the frame correspondences studied in modal correspondence theory. It is well-known that the 4 axiom defines transitivity (in the sense
that a frame F is transitive iff F 4 – and now we know that it imposes this
property on canonical frames as well.
Theorem 1.34 T, KB and KD are complete with respect to the classes of reflexive
frames, of symmetric frames, and of right-unbounded frames, respectively.
Proof. For the first claim, it suffices to show that the canonical model for T is
reflexive. Let w be a point in this model, and suppose 2 w. As w is a T-MCS,
! 3 2 w, thus by modus ponens, 3 2 w. Thus RTww.
For the second claim, it suffices to show that the canonical model for KB is
symmetric. Let w and v be points in this model such that RKB wv , and suppose
that 2 w. As w is a KB-MCS, ! 23 2 w, thus by modus ponens 23 2 w.
Hence by Lemma 1.24, 3 2 v . But this means that RKB vw, as required.
1.6 Completeness and canonicity
17
For the third claim, it suffices to show that the canonical model for KD is rightunbounded. (This is slightly less obvious than the previous claims since it requires
an existence proof.) Let w be any point in the canonical model for KD. We
must show that there exists a v in this model such that RKD wv . As w is a KDMCS it contains 2p ! 3p, thus by closure under uniform substitution it contains
2> ! 3>. Moreover, as > belongs to all normal modal logics, by generalization
2> does too; so 2> belongs to KD, hence by modus ponens 3> 2 w. Hence,
by the Existence Lemma, w has an RKD successor v .
a
Once again, these result hint at a link between definability and the structure of
canonical frames: after all, T defines reflexivity, B defines symmetry, and D right
unboundedness. And yet again, the proofs actually establish something more general than the theorem states: the canonical frame of any normal logic containing
T is reflexive, the canonical frame of any normal logic containing B is symmetric,
and the canonical frame of any normal logic containing D is right unbounded. This
allows us to ‘add together’ our results. Here are two examples:
Theorem 1.35 S4 is complete with respect to the class of reflexive, transitive frames.
S5 is complete with respect to the class of frames whose relation is an equivalence
relation.
Proof. The proof of Theorem 1.33 shows that the canonical frame of any normal
logic containing the 4 axiom is transitive, while the proof of the first clause of
Theorem 1.34 shows that the canonical frame of any normal logic containing the
T axiom is reflexive. As S4 contains both axioms, its canonical frame has both
properties, thus the completeness result for S4 follows.
As S5 contains both the 4 and the T axioms, it also has a reflexive, transitive
canonical frame. As it also contains the B axiom (which by the proof of the second
clause of Theorem 1.34 means that its canonical frame is symmetric), its canonical
relation is an equivalence relation. The desired completeness result follows.
a
1.6 Completeness and canonicity
As the above examples suggest, canonical models are an important tool for proving frame completeness results. Moreover, their utility evidently hinges on some
sort of connection between the properties of canonical frames and the frame correspondences studied earlier. Let us introduce some terminology to describe this
important phenomenon.
Definition 1.36 (Canonicity) A formula is canonical if, for any normal logic ,
2 implies that is valid on the canonical frame for . A normal logic is
canonical if its canonical frame is a frame for . (That is, is canonical if for all
such that ` , is valid on the canonical frame for .)
a
18
1 Modal Logic: Basic Concepts
Clearly 4, T, B and D axioms are all canonical formulas. For example, any normal
logic containing the 4 axiom has a transitive canonical frame, and the 4 axiom is
valid on transitive frames. Similarly, any modal logic containing the B axiom has
a symmetric canonical frame, and the B axiom is valid on symmetric frames.
Moreover K4, T, KB, KD, S4 and S5 are all canonical logics. Our previous
work has established that all the axioms involved are valid on the relevant canonical frames. But modus ponens, uniform substitution, and generalization preserve
frame validity. It follows that every formula in each of these logics is valid on that
logic’s canonical frame. In general, to show that KA1 : : : An is a canonical logic
it suffices to show that A1 ; : : : ; An are canonical formulas.
The general importance of the notion of canonicity is given by the following Theorem.
Theorem 1.37 Every canonical logic is complete.
Proof. Let be a canonical normal modal logic, and let F be the class of frames
for . Clearly then is sound with respect to F . For completeness, take a consistent formula . It follows from the canonical model theorem that is satisfiable in F , and since is canonical, this frame actually belongs to F . Thus the
theorem follows from Proposition 1.27.
a
Of the various questions that naturally suggest themselves, we mention two clusters. First, which formulas are canonical? For instance, can we decide on the basis
of the syntactic shape alone whether a formula is canonical? Unfortunately, the answer to this question is negative, but then perhaps we can find sufficient syntactic
criteria for canonicity? There are some interesting positive answers here, and this
issue will keep us occupied for a large part of the course. For instance, there is an
important collection of modal formulas, the so-called Sahlqvist formulas that are
all canonical. The precise definition will be given later on, but all of the formulas
discussed above belong to the Sahlqvist fragment of modal logic.
The second cluster of questions circles around the relation between the notions
of correspondence and canonicity. All the above examples concern canonical
modal formulas that are elementary, that is, they correspond to first order frame
properties like transitivity or reflexivity; in fact, all Sahlqvist formulas have this
property. Not all canonical modal formulas are elementary, but all modal formulas
that are known to be canonical, do correspond to a first order property when we
confine our attention to the canonical frames.
When it comes to proving the canonicity of a given modal formula, once we
know that the formula is elementary, the standard approach is to show that the
canonical frame satisfies the first order condition that corresponds to the formula.
This was the approach that we took in the previous section. The aim of these
1.6 Completeness and canonicity
19
Notes however is to provide tools that are tailored towards more direct proofs.
This approach is based on algebraic methods that we will develop next.
2
Algebraizing Modal Logic
In this chapter we develop an algebraic semantics for modal logic. The basic idea
is to extend the algebraic treatment of classical propositional logic (which uses
boolean algebras) to modal logic. The algebras employed to do this are called
boolean algebras with operators (BAOs). The boolean part handles the underlying
propositional logic, the additional operators handle the modalities.
But why algebraize modal logic? There are two main reasons. First, the algebraic perspective allows us to bring powerful new techniques to bear on modallogical problems. Second, the algebraic semantics turns out to be better-behaved
than frame-based semantics: we will be able to prove an algebraic completeness
result for every normal modal logic. No analogous result holds for frames.
In order to make the reader familiar with the basic ideas underlying algebraic
logic, we first discuss the algebraic approach towards propositional logic and only
then turn to the algebraization of modal logic. In the third section we prove two
results that underpin the whole approach, namely the Stone theorem for boolean
algebras and its exension, the Jónsson-Tarksi theorem for boolean algebras with
operators.
2.1 Logic as Algebra
What do algebra and logic have in common? And why bring algebra into the study
of logic? This section provides some preliminary answers: we show that algebra
and logic share key ideas, and analyze classical propositional logic algebraically.
Along the way we will meet a number of important concepts (notably formula
algebras, the algebra of truth values, set algebras, abstract boolean algebras, and
Lindenbaum-Tarski algebras) and results (notably the Stone Representation Theorem), but far more important is the overall picture. Algebraic logic offers a natural
way of re-thinking many basic logical issues, but it is important not to miss the
wood for the trees. The bird’s eye view offered here should help guide the reader
through the more detailed modal investigations that follow.
20
2.1 Logic as Algebra
21
Algebra as logic
Most school children learn how to manipulate simple algebraic equations. Given
the expression (x + 3)(x + 1), they learn how to multiply these factors to form
x2 + 4x + 3, and (somewhat later) study methods for doing the reverse (that is, for
decomposing quadratics into factors).
Such algebraic manipulations are essentially logical. For a start, we have a welldefined syntax: we manipulate equations between terms. This syntax is rarely
explicitly stated, but most students (building on the analogy with basic arithmetic)
swiftly learn how to build legitimate terms using numerals, variables such as x, y
and z , and +, and . Moreover, they learn the rules which govern this symbol
manipulation process: replacing equals by equals, doing the same thing to both
sides of an equation, appealing to commutativity, associativity and distributivity to
simplify and rearrange expressions. High-school algebra is a form of proof theory.
But there is also a semantic perspective on basic algebra, though this usually
only becomes clear later. As students learn more about mathematics, they realize
that the familiar ‘laws’ do not hold for all mathematical objects: for example, matrix multiplication is not commutative. Gradually the student grasps that variables
need not be viewed as standing for numbers: they can be viewed as standing for
other objects as well. Eventually the semantic perspective comes into focus: there
are various kinds of algebras (that is, sets equipped with collections of functions,
or operations, which satisfy certain properties), and terms denote elements in algebras. Moreover, an equation such as x y = y x is not a sacrosanct law: it is
simply a property that holds for some algebras and not for others.
So algebra has a syntactic dimension (terms and equations) and a semantic dimension (sets equipped with a collection of operations). And in fact there is a
tight connection between the proof theory algebra offers and its semantics. In
Appendix A we give a standard derivation system for equational logic (that is, a
standard set of rules for manipulating equations) and state a fundamental result due
to Birkhoff: the system is strongly sound and complete with respect to the standard
algebraic semantics. Algebra really can be viewed as logic.
But logic can also be viewed as algebra. We will now illustrate this by examining classical propositional logic algebraically. Our discussion is based around three
main ideas: the algebraization of propositional semantics in the class of set algebras; the algebraization of propositional axiomatics in the class of abstract boolean
algebras; and how the Stone Representation Theorem links these approaches.
Algebraizing propositional semantics
Consider any propositional formula, say (p _ q ) ^ (p _ r ). The most striking thing
about propositional formulas (as opposed to first-order formulas) is their syntactic
simplicity. In particular, there is no variable binding – all we have is a collection of
22
2 Algebraizing Modal Logic
atomic symbols (p, q , r , and so on) that are combined into more complex expressions using the symbols ?, >, :, _ and ^. Recall that we take ?, : and _ as the
primitive symbols, treating the others as abbreviations.
Now, as the terminology ‘proposition letters’ suggests, we think of p, q , and r
as symbols denoting entities called propositions, abstract bearers of information.
So what do ?, >, :, _ and ^ denote? Fairly obviously: ways of combining
propositions, or operations on propositions. More precisely, _ and ^ must denote
binary operations on propositions (let us call these operations + and respectively),
: must denote a unary operation on propositions (let us call it ), while ? and >
denote special nullary operations on propositions (that is, they are the names of
two special propositions: let us call them 0 and 1 respectively). In short, we have
worked our way towards the idea that formulas can be seen as terms denoting
propositions.
But which kinds of algebras are relevant? Here is a first step towards an answer.
Definition 2.1 Let Bool be the algebraic similarity type having one constant (or
nullary function symbol) ?, one unary function symbol :, and one binary function
symbol _. Given a set of propositional variables , Form () is the set of Bool terms in ; this set is identical to the collection of propositional formulas in .
Algebras of type Bool are usually presented as 4-tuples A = (A; +; ; 0). We
make heavy use of the standard abbreviations and 1. That is, a b is shorthand for
( a + b), and 1 is shorthand for 0.
a
But this only takes us part of the way. There are many different algebras of this
similarity type – and we are only interested in algebras which can plausibly be
viewed as algebras of propositions. So let us design such an algebra. Propositional
logic is about truth and falsehood, so let us take the set 2 = f0; 1g as the set A
underlying the algebra; we think of ‘0’ as the truth value false, and ‘1’ as the value
true. But we also need to define suitable operations over these truth values, and we
want these operations to provide a natural interpretation for the logical connectives.
Which operations are appropriate?
Well, the terms we are working with are just propositional formulas. So how
would we go about evaluating a formula in the truth value algebra? Obviously
we would have to know whether the proposition letters in are true or false, but let
us suppose that this has been taken care of by a function : ! 2 mapping the set
of proposition letters to the set 2 of truth values. Given such a (logicians will
call a valuation, algebraists will call it an assignment) it is clear what we have to
2.1 Logic as Algebra
23
do: compute ~() according to the following rules:
~(p)
~(?)
~(:)
~( _ )
=
=
=
=
(p); for all p 2 ;
0;
1 ~();
max(~(); ~( )):
(2.1)
Clearly the operations used here are the relevant ones; they simply restate the familiar truth table definitions. This motivates the following definition:
Definition 2.2 The algebra of truth values is 2 = (f0; 1g; +; ; 0)), where
a = 1 a and a + b = max(a; b), respectively.
+ are defined by
and
a
Let us sum up our discussion so far. The crucial observations are that formulas can
be viewed as terms, that valuations can be identified with algebraic assignments
in the algebra 2, and that evaluating the truth of a formula under such a valuation/assignment is exactly the same as determining the meaning of the term in the
algebra 2 under the assignment/valuation.
So let us move on. We have viewed meaning as a map ~ from the set Form ()
to the set f0; 1g – but it is useful to consider this meaning function in more mathematical detail. Note the ‘shape’ of the conditions on ~ in (2.1): the resemblance
to the defining condition of a homomorphism is too blatant to miss. But since homomorphisms are the fundamental maps between algebras (see Appendix A) why
not try and impose algebraic structure on the domain of such meaning functions
(that is, on the set of formulas/terms) so that meaning functions really are homomorphisms? This is exactly what we are about to do. We first define the needed
algebraic structure on the set of formulas.
Definition 2.3 Let be a set of proposition letters. The propositional formula
algebra over is the algebra
Form() = (Form (); +; ; ?);
where is the collection of propositional formulas over , and and
operations defined by := : and + := _ , respectively.
+ are the
a
In other words, the carrier of this algebra is the collection of propositional formulas
over the set of proposition letters , and the operations and + give us a simple
mathematical picture of the dynamics of formula construction.
Proposition 2.4 Let be some set of proposition letters. Given any assignment
: ! 2, the function ~ : Form () ! 2 assigning to each formula its meaning
under this valuation, is a homomorphism from Form() to 2.
24
2 Algebraizing Modal Logic
Proof. A precise definition of homomorphism is given in Appendix A. Essentially,
homomorphisms between algebras map elements in the source algebra to elements
in the target algebra in an operation preserving way – and this is precisely what the
conditions in ~ in (2.1) express.
a
The idea of viewing formulas as terms, and meaning as a homomorphism, is fundamental to algebraic logic.
Another point is worth stressing. As the reader will have noticed, sometimes we
call a sequence of symbols like p _ q a formula, and sometimes we call it a term.
This is intentional. Any propositional formula can be viewed as – simply is – an
algebraic term. The one-to-one correspondence involved is so obvious that it is not
worth talking about ‘translating’ formulas to terms or vice-versa; they are simply
two ways of looking at the same thing. We simply choose whichever terminology
seems most appropriate to the issue under discussion.
But let us move on. As is clear from high-school algebra, algebraic reasoning is
essentially equational. So a genuinely algebraic logic of propositions should give
us a way of determining when two propositions are equal. For example, such a
logic should be capable of determining that the formulas p _ (q ^ p) and p denote
the same proposition. How does the algebraic approach to propositional semantics
handle this? As follows: an equation s t is valid in an algebra A if for every
assignment to the variables occurring in the terms, s and t have the same meaning
in A (see Appendix A for further details). Hence, an algebraic way of saying that
a formula is a classical tautology (notation: j=C ) is to say that the equation
> is valid in the algebra of truth values.
Now, an attractive feature of propositional logic (a feature which extends to
modal logic) is that not only terms, but equations correspond to formulas. There
is nothing mysterious about this: we can define the bi-implication connective $ in
classical propositional logic, and viewed as an operation on propositions, $ asserts
that both terms have the same meaning:
~( $ ) =
1
0
if ~() = ~(
otherwise :
);
So to speak, propositional logic is intrinsically equational.
Theorem 2.5 neatly summarizes our discussion so far: it shows how easily we
can move from a logical to an algebraic perspective and back again.
Theorem 2.5 (2 Algebraizes Classical Validity) Let
formulas/terms. Then
and
j=C iff 2 j= >:
2 j= iff j=C $ :
j=C $ ( $ >).
be propositional
(2.2)
(2.3)
(2.4)
2.1 Logic as Algebra
Proof. Immediate from the definitions.
25
a
Remark 2.6 The reader may wonder about the presence of (2.3) and in particular,
of (2.4) in the Theorem. The point is that for a proper, ‘full’, algebraization of a
logic, one has to establish not only that the membership of some formula in the
logic can be rendered algebraically as the validity of some equation in some
(class of) algebra(s). One also has to show that conversely, there is a translation
of equations to formulas such that the equation holds in the class of algebras if
and only if its translation belongs to the logic. And finally, one has to prove that
translating a formula to an equation , and then translating this equation back
to a formula, one obtains a formula 0 that is equivalent to the original formula .
The fact that our particular translations satisfy these requirements is stated by (2.3)
and (2.4), respectively.
Since we will not go far enough into the theory of algebraic logic to use these
‘full’ algebraizations, in the sequel we will only mention the first kind of equivalence when we algebraize a logic. Nevertheless, in all the cases that we consider,
the second and third requirements are met as well.
a
Set algebras
Propositional formulas/terms and equations may be interpreted in any algebra of
type Bool . Most algebras of this type are uninteresting as far as the semantics
of propositional logic is concerned – but other algebras besides 2 are relevant. A
particularly important example is the class of set algebras. As we will now see,
set algebras provide us with a second algebraic perspective on the semantics of
propositional logic. And as we will see in the following section, the perspective
they provide extends neatly to modal logic.
Definition 2.7 (Set Algebras) Let A be a set. As usual, we denote the power set
of A (the set of all subsets of A) by P(A). The power set algebra P(A) is the
structure
P(A) = (P(A); [; ; ?);
where ? denotes the empty set,
is the operation of taking the complement of
a set relative to A, and [ that of taking the union of two sets. From these basic
operations we define in the standard way the operation \ of taking the intersection
of two sets, and the special element A, the top set of the algebra.
A set algebra or field of sets is a subalgebra of a power set algebra. That is, a set
algebra (on A) is a collection of subsets of A that contains ? and is closed under
[ and (so any set algebra contains A and is closed under \ as well). The class
of all set algebras is called Set.
a
26
2 Algebraizing Modal Logic
Set algebras provide us with a simple concrete picture of propositions and the way
they are combined – moreover, it is a picture that even at this stage contains a
number of traditional modal ideas. Think of A as a set of worlds (or situations,
or states) and think of a proposition as a subset of A. And think of a proposition
as a set of worlds – the worlds that make it true. So viewed, ? is a very special
proposition: it is the proposition that is false in every situation, which is clearly a
good way of thinking about the meaning of ?. Similarly, A is the proposition true
in all situations, which is a suitable meaning for >. It should also be clear that [
is a way of combining propositions that mirrors the role of _. After all, in what
worlds is p _ q true? In precisely those worlds that make p true or q true. Finally,
complementation mirrors negation, for :p is true in precisely those worlds where
p is not true.
As we will now show, set algebras and the algebra 2 make precisely the same
equations true. We will prove this algebraically by showing that the class of set
algebras coincides (modulo isomorphism) to the class of subalgebras of powers of
2. The crucial result needed is the following:
Proposition 2.8 Every power set algebra is isomorphic to a power of 2, and conversely.
Proof. Let A be an arbitrary set, and consider the following function
elements of P(A) to 2-valued maps on A:
(X )(a) =
1
0
mapping
if a 2 X;
otherwise:
In other words, (X ) is the characteristic function of X . The reader should verify
that is an isomorphism between P(A) and 2A , where the latter algebra is as
defined in Definition A.6.
Conversely, to show that every power of 2 is isomorphic to some power set
algebra, let 2I be some power of 2. Consider the map : 2I ! P(I ) defined by
(f ) = fi 2 I j f (i) = 1g:
Again, we leave it for the reader to verify that
tween 2I and P(I ).
is the required isomorphism be-
Theorem 2.9 (Set algebraizes classical validity) Let
formulas/terms. Then
j=C iff
Set j= >:
a
and
be propositional
(2.5)
Proof. It is not difficult to show from first principles that the validity of equations is
preserved under taking direct products (and hence powers) and subalgebras. Thus,
with the aid of Theorem 2.5 and Proposition 2.8, the result follows.
a
2.1 Logic as Algebra
27
Algebraizing propositional axiomatics
We now have two equational perspectives on the semantics of propositional logic:
one via the algebra 2, the other via set algebras. But what about the syntactic
aspects of propositional logic? It is time to see how the equational perspective
handles such notions as theoremhood and provable equivalence.
Assume we are working in some fixed (sound and complete) proof system for
classical propositional logic. Let `C mean that is a theorem of this system, and
call two propositional formulas and provably equivalent (notation: C )
if the formula $ is a theorem. Theorem 2.11 is a syntactic analog of Theorem 2.9: it is the fundamental result concerning the algebraization of propositional
axiomatics. Its statement and proof makes use of boolean algebras, so let us define
these important entities right away.
Definition 2.10 (Boolean Algebras) Let A = (A; +; ; 0) be an algebra of the
boolean similarity type. Then A is called a boolean algebra iff it satisfies the
following identities (recall that x y and 1 are shorthand for ( x + y ) and 0,
respectively):
(B 0)
(B 1)
(B 2)
(B 3)
(B 4)
x+y =y+x
x + (y + z ) = (x + y) + z
x+0 =x
x + ( x) = 1
x + (y z ) = (x + y) (x + z )
xy =yx
x (y z ) = (x y) z
x1=1
x ( x) = 0
x (y + z ) = (x y) + (x z )
The operations + and are called join and meet, respectively, and the elements 1
and 0 are referred to as the top and bottom elements. We order the elements of a
boolean algebra by defining a b if a + b = b (or equivalently, if a b = a). Given
a boolean algebra A = (A; +; ; 0), the set A is called its carrier set. We call the
class of boolean algebras BA.
a
By a famous result of Birkhoff (discussed in Appendix A) a class of algebras defined by a collection of equations can be structurally characterized as a variety.
Thus in what follows we sometimes speak of the variety of boolean algebras, rather
than the class of boolean algebras.
If you have not encountered boolean algebras before, you should check that the
algebra 2 and the set algebras defined earlier are both examples of boolean algebras
(that is, check that these algebras satisfy the listed identities). In fact, set algebras
are what are known as concrete boolean algebras. As we will see when we discuss the Stone Representation Theorem, the relationship between abstract boolean
algebras (that is, any algebraic structure satisfying the previous definition) and set
algebras lies at the heart of the algebraic perspective on propositional soundness
and completeness.
28
2 Algebraizing Modal Logic
But this is jumping ahead: our immediate task is to state the syntactic analog of
Theorem 2.9 promised above.
Theorem 2.11 (BA Algebraizes Classical Theoremhood) Let and
sitional formulas/terms. Then
`C iff
BA j= >:
be propo(2.6)
Proof. Soundness (the direction from left to right in (2.6) can be proved by a
straightforward inductive argument on the length of propositional proofs. Completeness will follow from the Propositions 2.14 and 2.15 below.
a
How are we to prove this completeness result? Obviously we have to show that every non-theorem of classical propositional logic can be falsified on some boolean
algebra (falsified in the sense that there is some assignment under which the formula does not evaluate to the top element of the algebra). So the key question is:
how do we build falsifying algebras? Our earlier work on relational completeness
suggests an answer. In our completeness proof we made use of canonical models:
that is, we manufactured models out of syntactical ingredients (sets of formulas)
taking care to hardwire in all the crucial facts about the logic. So the obvious
question is: can we construct algebras from (sets of) formulas in a way that builds
in all the propositional logic we require? Yes, we can. Such algebras are called
Lindenbaum-Tarski algebras. In essence, they are ‘canonical algebras.’
First, some preliminary work. The observation underpinning what follows is
that the relation of provable equivalence is a congruence on the formula algebra. A
congruence on an algebra is essentially an equivalence relation on the algebra that
respects the operations (a precise definition is given in Appendix A) and it is not
hard to see that provable equivalence is such a relation.
Proposition 2.12 The relation
algebra.
C is a congruence on the propositional formula
Proof. We have to prove that C is an equivalence relation satisfying
C
only if : C
:
(2.7)
and
0 C
0
and 1
C
1
only if (0 _ 1 ) C
( _ ):
0
1
(2.8)
In order to prove that C is reflexive, we have to show that for any formula , the
formula $ is a theorem of the proof system. The reader is invited to prove
this in his or her favorite proof system for proposition calculus. The properties of
symmetry and transitivity are also left to the reader.
2.1 Logic as Algebra
29
But we want to prove that C is not merely an equivalence relation but a congruence. We deal with the case for negation, leaving (2.8) to the reader. Suppose that
C , that is, `C $ . Again, given that we are working with a sound and
complete proof system for propositional calculus, this implies that `C : $ : .
Given this, (2.7) is immediate.
a
The equivalence classes under C are the building blocks for what follows. As any
such class is a maximal set of mutually equivalent formulas, we can think of such
classes as propositions. And as C is a congruence, we can define a natural algebraic structure on these propositions. Doing so gives rise to Lindenbaum-Tarski
algebras.
Definition 2.13 (Lindenbaum-Tarski Algebra) Given a set of proposition letters
, let Form ()=C be the set of equivalence classes that C induces on the set
of formulas, and for any formula let [] denote the equivalence class containing
. Then the Lindenbaum-Tarski algebra (for this language) is the structure
LC () := (Form ()=C ; +; ; 0);
where +, and 0 are defined by: [] + [ ] := [ _ ], [] := [:] and 0 :=
[?]. Strictly speaking, we should write [] instead of [], for ’s congruence
class depends on the set of proposition letters. But unless there is potential for
confusion, we usually will not bother to do so.
a
Lindenbaum-Tarski algebras are easy to work with. For instance, it is easy to see
that the meet operation in such an algebra is given by [] [ ] := [ ^ ], while
the top element 1 is [>]. As another example, we show that a + ( a) = 1 for
all elements a of LC (). The first observation is that a, just like any element of
LC (), is of the form [] for some formula . But then we have
a + ( a) = [] + ( []) = [] + [:] = [ _ (:)] = [>] = 1;
(2.9)
where the fourth equality holds because `C ( _ :) $ >.
It is fairly obvious that the structure of a Lindenbaum-Tarski algebra only depends on the cardinality of the set of proposition letters; the reader is asked to
prove this in Exercise 2.1.4.
We need two results concerning Lindenbaum-Tarski algebras. First, we have to
show that they are indeed an ‘algebraic canonical model’ – that is, that they give us
a counterexample for every non-theorem of propositional logic. Second, we have
to show that they are counterexamples of the right kind: that is, we need to prove
that any Lindenbaum-Tarski algebra is a boolean algebra.
Proposition 2.14 Let be some propositional formula, and a set of proposition
30
2 Algebraizing Modal Logic
letters of size not smaller than the number of proposition letters occurring in .
Then
`C iff LC () j= >:
(2.10)
Proof. We may and will assume that actually contains all variables occurring in
, cf. Exercise 2.1.4. We first prove the easy direction from right to left. Assume
that is not a theorem of classical propositional logic. This implies that and
> are not provably equivalent, whence we have [] 6= [>]. We have to find an
assignment on LC () that forms a counterexample to the validity of . There is
one obvious candidate, namely the assignment given by (p) = [p]. It can easily
be verified (by a straightforward formula induction) that with this definition we
obtain ~( ) = [ ] for all formulas that use variables from the set . But then by
our assumption on we find that
~() = [] 6= [>] = 1;
as required.
For the other direction we have to work a bit harder. If `C then it is obvious
that ~() = [] = [>] = 1, but only looking at is not sufficient now. We have to
show that ~() = [>] for all assignments .
So let be an arbitrary assignment. That is, assigns an equivalence class
(under C ) to each proposition letter. For each variable p, take a representing
formula (p) in the equivalence class (p); that is, we have (p) = [(p)]. We
may view as a function mapping proposition letters to formulas; in other words,
is a substitution. Let ( ) denote the effect of performing this substitution on the
formula . It can be proved by an easy formula induction that, for any formula ,
we have
~(
) = [( )]:
(2.11)
Now, the collection of propositional theorems is closed under uniform substitution
(depending on the formulation of your favorite sound and complete proof system,
this is either something that is hardwired in or can be shown to hold). This closure
property implies that the formula () is a theorem, and hence that () C >, or
equivalently, [()] = [>]. But then it follows from (2.11) that
~() = [>];
which is precisely what we need to show that LC () j= .
a
Thus it only remains to check that LC () is the right kind of algebra.
Proposition 2.15 For any set of proposition letters, LC () is a boolean algebra.
2.1 Logic as Algebra
31
Proof. Fix a set . The proof of this Proposition boils down to proving that all the
identities B0–4 hold in LC (). In (2.9) we proved that the first part of B3 holds;
we leave the reader to verify that the other identities hold as well.
a
Summarizing, we have seen that the axiomatics of propositional logic can be algebraized in a class of algebras, namely the variety of boolean algebras. We have
also seen that Lindenbaum-Tarski algebras act as canonical representatives of the
class of boolean algebras. (For readers with some background in universal algebra,
we remark that Lindenbaum-Tarski algebras are in fact the free boolean algebras.)
Weak completeness via Stone
It is time to put our findings together, and to take one final step. This step is more
important than any taken so far.
Theorem 2.9 captured tautologies as equations valid in set algebras:
j=C iff Set j= >:
On the other hand, in Theorem 2.11 we found an algebraic semantics for the notion
of classical theoremhood:
`C iff BA j= >:
But there is a fundamental logical connection between j=C and `C : the soundness
and completeness theorem for propositional logic tells us that they are identical.
Does this crucial connection show up algebraically? That is, is there an algebraic
analog of the soundness and completeness result for classical propositional logic?
There is: it is called the Stone Representation Theorem.
Theorem 2.16 (Stone Representation Theorem) Any boolean algebra is isomorphic to a set algebra.
Proof. We will make a more detailed statement of this result, and prove it, in Section 2.3.
a
(Incidentally, this immediately tells us that any boolean algebra is isomorphic to a
subalgebra of a power of 2 – for Proposition 2.8 tells us that any power set algebra
is isomorphic to a subalgebra of a power of 2.) But what really interests us here
is the logical content of Stone’s Theorem. In essence, it is the key to the weak
completeness of classical propositional logic.
Corollary 2.17 (Soundness and Weak Completeness) For any formula ,
valid iff it is a theorem.
is
Proof. Immediate from the equations above, since by the Stone Representation
Theorem, the equations valid in Set must coincide with those valid in BA.
a
32
2 Algebraizing Modal Logic
The relation between Theorem 2.11 and Corollary 2.17 is the key to much of our
later work. Note that from a logical perspective, Corollary 2.17 is the interesting result: it establishes the soundness and completeness of classical propositional logic
with respect to the standard semantics. So why is Theorem 2.11 important? After
all, as it proves completeness with respect to an abstractly defined class of boolean
algebras, it does not have the same independent logical interest. This is true, but
given that the abstract algebraic counterexamples it provides can be represented as
standard counterexamples – and this is precisely what Stone’s Theorem guarantees
– it enables us to prove the standard completeness result for propositional logic.
To put it another way, the algebraic approach to completeness factors the algebra
building process into two steps. We first prove completeness with respect to an
abstract algebraic semantics by building an abstract algebraic model. It is easy to
do this – we just use Lindenbaum-Tarski algebras. We then try and represent the
abstract algebras in the concrete form required by the standard semantics; that is,
in terms of set algebras or of the algebra 2.
In the next two sections we extend this approach to modal logic. Algebraizing
modal logic is more demanding than algebraizing propositional logic. For a start,
there is not just one logic to deal with – we want to be able to handle any normal
modal logic whatsoever. Moreover, the standard semantics for modal logic is given
in terms of frame-based models – so we are going to need a representation result
that tells us how to represent algebras as relational structures.
But all this can be done. In the following section we will generalize boolean
algebras to boolean algebras with operators; these are the abstract algebras we will
be dealing with throughout the chapter. We also generalize set algebras to complex algebras; these are the concrete algebras which model the idea of set-based
algebras of propositions for modal languages. We then define the LindenbaumTarski algebras we need – and every normal modal logic will give rise to its own
Lindenbaum-Tarski algebra. This is all a fairly straightforward extension of ideas
we have just discussed. We then turn, in Section 2.3, to the crucial representation
result: the Jónsson-Tarski Theorem. This is an extension of Stone’s Representation Theorem that tells us how to represent a boolean algebra with operators as an
ordinary modal model. It is an elegant result in its own right, but for our purposes
its importance is the bridge it provides between completeness in the universe of
algebras and completeness in the universe of relational structures.
Exercises for Section 2.1
2.1.1 Let A and B be two sets, and f : A ! B some map. Show that f 1 : P (B ) !
P (A) given by f 1 (Y ) = fa 2 A j f (a) 2 Y g is a homomorphism from the power set
algebra of B to that of A.
2.1.2 Prove that every power set algebra is isomorphic to a power of the algebra 2, and
2.2 Algebraizing Modal Logic
33
that conversely, every power of 2 is isomorphic to a power set algebra. That is, fill in the
details of the proof of Theorem 2.8.
2.1.3 Here is a standard set of axioms for propositional calculus: p ! (q ! p), (p !
(q ! r )) ! ((p ! q ) ! (p ! r )), and (:p ! :q ) ! (q ! p). Show that all three
axioms are valid on any set algebra. That is, show that whatever subset is used to interpret
the proposition letters, these formulas are true in all worlds. Furthermore, show that modus
ponens and uniform substitution preserve validity.
2.1.4 Let and be two sets of proposition letters.
(a)
(b)
(c)
(d)
Prove that Form() is a subalgebra of Form( ) iff .
Prove that LC () can be embedded in LC ( ) iff jj j j.
Prove that LC () and LC ( ) are isomorphic iff jj = j j.
Does imply that LC () is a subalgebra of LC ( )?
2.2 Algebraizing Modal Logic
Let us adapt the ideas introduced in the previous section to modal logic. The most
basic principle of algebraic logic is that formulas of a logical language can be
viewed as terms of an algebraic language, so let us first get clear about the algebraic
languages we will use in the remainder of this chapter:
Definition 2.18 Let be a modal similarity type. The corresponding algebraic
similarity type F contains as function symbols all modal operators, together with
the boolean symbols _ (binary), : (unary), and ? (constant). For a set of variables, we let Ter () denote the collection of F -terms over .
a
The algebraic similarity type F can be seen as the union of the modal similarity
type and the boolean type Bool . In practice we often identify and F , speaking
of -terms instead of F -terms. The previous definition takes the formulas-asterms paradigm quite literally: by our definitions
Form (; ) = Ter ():
Just as boolean algebras were the key to the algebraization of classical propositional logic, in modal logic we are interested in boolean algebras with operators or
BAO s. Let us first define BAO s abstractly; we will discuss concrete BAO s shortly.
Definition 2.19 (Boolean Algebras with Operators) Let be a modal similarity
type. A boolean algebra with -operators is an algebra
A = (A; +; ; 0; fM )M2
such that (A; +; ; 0) is a boolean algebra and every fM is an operator of arity
(M); that is, fM is an operation satisfying
34
2 Algebraizing Modal Logic
(normality) fM (a1 ; : : : ; a(M) ) = 0 whenever ai
(additivity) for all i (such that 0 < i (M)),
= 0 for some i (0 < i (M)).
fM(a1 ; : : : ; ai + a0i ; : : : ; a(M) ) =
fM(a1 ; : : : ; ai ; : : : ; a(M) ) + fM(a1 ; : : : ; a0i ; : : : ; a(M) ):
If we abstract from the particular modal similarity type , or if is known from
context, we simply speak of boolean algebras with operators, or BAOs.
a
Now, the boolean structure is obviously there to handle the propositional connectives, but what is the meaning of the normality and additivity conditions on the fM ?
Consider a unary operator f . In this case these conditions boil down to:
f (0)
f (x + y)
= 0;
= fx + fy:
But these equations correspond to the following modal formulas:
3? $ ?;
3(p _ q) $ 3p _ 3q;
both of which formulas are modal validities. Indeed (as we noted in Remark 1.13)
they can be even be used to axiomatize the minimal normal logic K. Thus, even
at this stage, it should be clear that our algebraic operators are well named: their
defining properties are modally crucial.
Furthermore, note that all operators have the property of monotonicity. An operation g on a boolean algebra is monotonic if a b implies ga gb. (Here refers
to the ordering on boolean algebra given in Definition 2.10: a b iff a b = a
iff a + b = b.) Operators are monotonic, because if a b, then a + b = b, so
fa + fb = f (a + b) = fb, and so fa fb. Once again there is an obvious modal
analog, namely the rule of proof mentioned in Remark 1.13: if ` p ! q then
` 3p ! 3q.
Example 2.20 Consider the collection of binary relations over a given set U . This
collection forms a set algebra on which we can define the operations j (composition), () 1 (inverse) and Id (the identity relation); these are binary, unary and
nullary operations respectively. It is easy to verify that these operations are actually operators; to give a taste of the kind of argumentation required, we show that
composition is additive in its second argument:
(x; y) 2 R j (S [ T )
iff
iff
iff
there is a z with (x; z ) 2 R and (z; y ) 2 S [ T
there is a z with (x; z ) 2 R and (z; y ) 2 S or (z; y ) 2 T
there is a z with (x; z ) 2 R and (z; y ) 2 S ,
2.2 Algebraizing Modal Logic
35
or there is a z with (x; z ) 2 R and (z; y ) 2 T
iff
iff
(x; y) 2 R j S or (x; y) 2 R j T
(x; y) 2 R j S [ R j T:
a
The reader should check the remaining cases.
Algebraizing modal semantics
However it is the next type of BAO that is destined to play the leading role: complex
algebras. These structures make crucial use of operations mR that we define next.
Let us first consider the basic modal similarity type with one diamond. Given a
frame F = (W; R), let mR be the following operation on the power set of W :
mR (X ) = fw 2 W j Rwx for some x 2 X g:
Think of mR (X ) as the set of states that ‘see’ a state in X . This operation corresponds to the diamond in the sense that for any valuation V and any formula we
have V (3) = mR (V ()).
Moving to the general case, we obtain the following definition.
Definition 2.21 Given an (n +1)-ary relation R on a set W , we define the following n-ary operation mR on the power set P(W ) of W :
mR (X1 ; : : : ; Xn ) =
fw 2 W j Rww1 : : : wn for some w1 2 X1 ; : : : ; wn 2 Xn g:
a
Example 2.22 Let be the converse operator of arrow logic, and recall that we
use the letter R to denote the accessibility relation for . Thus on a square frame
SU , by the rather special nature of R, we have that, for any subset X of U 2 :
mR (X )
= f(a ; a ) 2 U j a = x and a = x for some (x ; x ) 2 X g
= f(x ; x ) 2 U j (x ; x ) 2 X g:
In other words, mR (X ) is nothing but the converse of the binary relation X .
a
0
1
1
0
2
2
0
1
0
1
0
0
1
1
Definition 2.23 (Complex Algebras) Let be a modal similarity type, and F =
(W; RM)M2 a -frame. The (full) complex algebra of F (notation: F+), is the
expansion of the power set algebra P(W ) with operations mRM for every operator
M in . A complex algebra is a subalgebra of a full complex algebra. If K is a
class of frames, then we denote the class of full complex algebras of frames in K
by CmK.
a
36
2 Algebraizing Modal Logic
It is important that you fully understand this definition. For a start, note that complex algebras are set algebras (that is, concrete propositional algebras) to which
mR operations have been added. Recall that for a binary relation R, the unary
operation mR yields the set of all states which ‘see’ a state in a given subset X of
the universe:
mR (X ) = fy 2 W j there is an x 2 X such that Ryxg:
For a relation of arity n + 1, the n-ary operation mR maps an n-tuple of subsets of
the universe to the set of all points which ‘see’ an n-tuple of states each of which
belongs to the corresponding subset. It easily follows that if we have some model
in mind and denote with V~ () the set of states where is true, then
V~ (M(1 ; : : : ; n ))
=
mRM (V~ (1 ; : : : ; n )):
Thus it should be clear that complex algebras are intrinsically modal. In the previous section we said that set algebras model propositions as sets of possible worlds.
By adding the mR operations, we have modeled the idea that one world may be
able to access the information in another. In short, we have defined a class of concrete algebras which capture the modal notion of access between states in a natural
way.
How are complex algebras connected with abstract BAOs? One link is obvious:
Proposition 2.24 Let be a modal similarity type, and
frame. Then F+ is a boolean algebra with -operators.
F = (W; RM )M2 a -
Proof. We have to show that operations of the form mR are normal and additive.
This rather easy proof is left to the reader; see Exercise 2.2.2.
a
The other link is deeper. As we will learn in the following section (Theorem 2.45), complex algebras are to BAOs what set algebras are to boolean algebras:
every abstract boolean algebra with operators has a concrete set theoretic representation, for every boolean algebra with operators is isomorphic to a complex
algebra.
But we have a lot to do before we are ready to prove this – let us continue our
algebraization of the semantics of modal logic. We will now define the interpretation of -terms and equations in arbitrary boolean algebras with -operators. As
we saw for propositional logic, the basic idea is very simple: given an assignment
that tells us what the variables stand for, we can inductively define the meaning of
any term.
Definition 2.25 Assume that is a modal similarity type and that is a set of
variables. Assume further that A = (A; +; ; 0; fM )M2 is a boolean algebra with
2.2 Algebraizing Modal Logic
37
-operators. An assignment for is a function : ! A. We can extend uniquely to a meaning function ~: Ter () ! A satisfying:
~(p)
~(?)
~(:s)
~(s _ t)
~(M(s1 ; : : : ; sn ))
= (p); for all p 2 ;
= 0;
= ~(s);
= ~(s) + ~(t);
= fM(~(s ); : : : ; ~(sn)):
Now let s t be a -equation. We say that s t is true in A (notation: A j= s t)
if for every assignment : ~(s) = ~(t).
a
1
But now consider what happens when A is a complex algebra F+ . Since elements
of F+ are subsets of the power set P(W ) of the universe W of F, assignments are
simply ordinary modal valuations! The ramifications of this observation are listed
in the following proposition:
Proposition 2.26 Let be a modal similarity type, a -formula,
an assignment (or valuation) and w a point in F. Then
(F; ); w F
F j= +
F a -frame, w 2 ~();
iff F+ j= >;
iff F $ :
iff
(2.12)
(2.13)
(2.14)
Proof. We will only prove the first part of the proposition (for the basic modal
similarity type); the second and third part follow immediately from this and the
definitions.
Let , F and be as in the statement of the theorem. We will prove (2.12) (for
all w) by induction on the complexity of . The only interesting part is the modal
case of the inductive step. Assume that is of the form 3 . The key observation
is that
~(3
) = mR3 (~( )):
(2.15)
We now have:
(F; ); w 3
iff
iff
iff
iff
there is a v such that R3 wv and (F; ); v there is a v such that R3 wv and v 2 ~( )
w 2 mR3 (~(
w 2 ~(3 ):
))
Here the second equivalence is by the inductive hypothesis, and the last one by
(2.15). This proves (2.12).
a
38
2 Algebraizing Modal Logic
The previous proposition is easily lifted to the level of classes of frames and complex algebras. The resulting theorem is a fundamental one: it tells us that classes
of complex algebras algebraize modal semantics. It is the modal analog of Theorem 2.9.
Theorem 2.27 Let be a modal similarity type,
class of -frames. Then
K
CmK j= iff
iff
and
-formulas, and K a
CmK j= >;
K$ :
Proof. Immediate by Proposition 2.26.
(2.16)
(2.17)
a
This proposition allows us to identify the modal logic K of a class of frames K
(that is, the set of formulas that are valid in each F 2 Kg) with the equational
theory of the class CmK of complex algebras of frames in K (that is, the set of
equations fs t j F+ j= s t, for all F 2 Kg).
Let us summarize what we have learned so far. We have developed an algebraic
approach to the semantics of modal logic in terms of complex algebras. These
complex algebras, concrete boolean algebras with operators, generalize to modal
languages the idea of algebras of propositions provided by set algebras. And most
important of all, we have learned that complex algebras embody all the information
about normal modal logics that frames do. Thus, mathematically speaking, we can
dispense with frames and instead work with complex algebras.
Algebraizing modal axiomatics
Turning to the algebraization of modal axiomatics, we encounter a situation similar
to that of the previous section. Once again, we will see that the algebraic counterpart of a logic is an equational class of algebras. To give a precise formulation we
need the following definition.
Definition 2.28 Given a formula , let be the equation >. Now let be a
modal similarity type. For a set of -formulas, we define V to be the class of
those boolean algebras with -operators in which the set = f j 2 g is
valid.
a
We now state the algebraic completeness theorem for modal logic. It is the obvious
analog of Theorem 2.11.
Theorem 2.29 (Algebraic Completeness) Let
be a modal similarity type, and
2.2 Algebraizing Modal Logic
39
a set of -formulas. Then K (the normal modal -logic axiomatized by ) is
sound and complete with respect to V . That is, for all formulas we have
`K iff V j= :
Proof. We leave the soundness direction as an exercise to the reader. Completeness
is an immediate corollary of Theorems 2.34 and 2.35.
a
As a corollary to the soundness direction of Theorem 2.29, we have that VK =
V , for any set of formulas. In the sequel this will allow us to forget about
axiom sets and work with logics instead.
To prove the completeness direction of Theorem 2.29, we need a modal version
of the basic tool used to prove algebraic completeness results: Lindenbaum-Tarski
algebras. As in the the case of propositional languages, we will build an algebra on
top of the set of formulas in such a way that the relation of provable equivalence
between two formulas is a congruence relation. The key difference is that we do
not have just one relation of provable equivalence, but many: we want to define the
notion of Lindenbaum-Tarski algebras for arbitrary normal modal logics.
Definition 2.30 Let be an algebraic similarity type, and a set of proposition
letters. The formula algebra of over is the algebra Form(; ) = (Form (; );
+; ; ?; fM)M2 where +, and ? are given as in Definition 2.3, while for each
modal operator M, the operation fM is given by
fM(t1 ; : : : ; tn ) = M(t1 ; : : : ; tn ):
a
Notice the double role of M in this definition: on the right-hand side of the equation,
M is a ‘static’ part of the term M(t1; : : : ; tn), whereas in the-left hand side we have
a more ‘dynamic’ perspective on the interpretation fM of the operation symbol M.
Definition 2.31 Let be a modal similarity type, a set of propositional variables,
and a normal modal -logic. We define as a binary relation between formulas (in ) by
If , we say that and
iff
` $
:
are equivalent modulo .
a
Proposition 2.32 Let be a modal similarity type, a set of proposition letters
and a normal modal -logic. Then is a congruence relation on Form(; ).
Proof. We confine ourselves to proving the proposition for the basic modal similarity type. First, we have to show that is an equivalence relation; this is easy,
and we leave the details to the reader. Next, we must show that is a congruence
40
2 Algebraizing Modal Logic
relation on the formula algebra; that is, we have to demonstrate that
following properties:
0 0
and 1
1
imply
implies
implies
0 _ 1 0 _
: : ;
3 3 :
1
We leave the proofs as exercises to the reader.
has the
;
(2.18)
a
Proposition 2.32 tells us that the following are correct definitions of functions on
the set Form (; )= of equivalence classes under :
[] + [ ] := [ _ ];
[] := [:];
fM([ ]; : : : ; [n ]) := [M( ; : : : ; n )]:
For unary diamonds, the last clause boils down to: f3[] := [3].
1
(2.19)
1
Given Proposition 2.32, the way is open to define the Lindenbaum-Tarski algebra
for any normal modal logic : we simply define it to be the quotient algebra of the
formula algebra over the congruence relation .
Definition 2.33 (Lindenbaum-Tarski Algebras) Let be a modal similarity
type, a set of proposition letters, and a normal modal -logic in this language.
The Lindenbaum-Tarski algebra of over the set of generators is the structure
L () := (Form (; )= ; +; ; fM );
where the operations +,
and fM are defined as in (2.19).
a
As with propositional logic, we need two results about Lindenbaum-Tarski algebras. First, we must show that modal Lindenbaum-Tarski algebras are boolean
algebras with operators; indeed, we need to show that the Lindenbaum-Tarski algebra of any normal modal logic belongs to V . Second, we need to prove that
Lindenbaum-Tarski algebras provide canonical counterexamples to the validity of
non-theorems of in V . The second point is easily dealt with:
Theorem 2.34 Let be a modal similarity type, and a normal modal -logic.
Let be some propositional formula, and a set of proposition letters of size not
smaller than the number of proposition letters occurring in . Then
` iff L() j= :
(2.20)
Proof. This proof is completely analogous to that of Proposition 2.14 and is left to
the reader.
a
So let us verify that Lindenbaum-Tarski algebras are canonical algebraic models of
the right kind:
2.2 Algebraizing Modal Logic
41
Theorem 2.35 Let be a modal similarity type, and be a normal modal -logic.
Then for any set of proposition letters, L () belongs to V .
Proof. Once we have shown that L () is a boolean algebra with -operators, the
theorem immediately follows from Theorem 2.34. Now, that L () is a boolean
algebra is clear, so the only thing that remains to be done is to show that the modalities really give rise to -operators.
As an example, assume that contains a diamond 3; let us prove additivity of
f3. We have to show that
f3(a + b) = f3a + f3b;
for arbitrary elements a and b of L (). Let a and b be such elements; by definition
there are formulas and such that a = [] and b = [ ]. Then
f3(a + b) = f3([] + [
]) = f3([ _ ]) = [3( _ )]
while
f3a + f3b = f3([]) + f3([
]) = [3] + [3 ] = [3 _ 3 ]:
It is easy to check that
` 3( _ ) $ (3 _ 3 );
whence it follows that [3( _ )] = [3 _ 3 ]. We leave it for the reader to fill
in the remaining details of this proof as Exercise 2.2.4.
a
As an immediate corollary we have the following result: modal logics are always
complete with respect to the variety of boolean algebras with operators where their
axioms are valid. This is in sharp contrast to the situation in relational semantics,
where modal logics need not be complete with respect to the class of frames that
they define.
This is an interesting result, but it is not what we really want, for it proves completeness with respect to abstract BAOs rather than complex algebras. Not only are
complex algebras concrete algebras of propositions, we also know (recall Proposition 2.26) that complex algebras embody all the information of relevance to frame
validity – so we really should be aiming for completeness results with respect to
classes of complex algebras.
And that is why the long-promised Jónsson-Tarski Theorem, which we state
and prove in the following section, is so important. This tells us that every boolean
algebra with operators is isomorphic to a complex algebra, and thus guarantees that
we can represent the Lindenbaum-Tarski algebras of any normal modal logics as a complex algebra. In effect, it will convert Theorem 2.34 into a completeness
result with respect to complex algebras. Moreover, because of the link between
42
2 Algebraizing Modal Logic
complex algebras and relational semantics, it will open the door to exploring frame
completeness algebraically.
Exercises for Section 2.2
2.2.1 Let A be a boolean algebra. Prove that is an operator. How about +?
2.2.2 Show that every complex algebra is a boolean algebra with operators (that is, prove
Proposition 2.24).
2.2.3 Let A be the collection of finite and co-finite subsets of N . Define f
f (X )
Prove that (A; [;
=
fy 2 N j y + 1 2 X g
N
if X is finite;
if X is co-finite:
:
A ! A by
; ?; f ) is a boolean algebra with operators.
2.2.4 Let be a normal modal logic. Prove that the Lindenbaum-Tarski algebra L is a
boolean algebra with -operators (that is, fill in the missing proof details in Theorem 2.35).
2.2.5 Let be a set of -formulas. Prove that for any formula , `K . That is, prove the soundness direction of Theorem 2.29.
implies V j=
2.2.6 Call a variety V of BAOs complete if it is generated by a class of complex algebras,
i.e., if V = HSPCmK for some frame class K. Prove that a logic is complete iff the
variety V is complete.
2.2.7 Let A be a boolean algebra. In this exercise we assume familiarity with the notion
of an infinite sum (supremum). An operation f : A ! A is called completely additive if it
distributes over infinite sums (in each of its arguments).
(a) Show that every operation of the form mR is completely additive.
(b) Give an example of an operation that is additive, but not completely additive. (Hint:
as the boolean algebra, take the set of finite and co-finite subsets of some frame.)
2.3 The Jónsson-Tarski Theorem
We already know how to construct a BAO from a frame: simply form the frame’s
complex algebra. We will now learn how to construct a frame from a BAO by forming the ultrafilter frame of the algebra. As we will see, this operation generalizes
two constructions that we have met before: taking the ultrafilter extension of a
model, and forming the canonical frame associated with a normal modal logic.
Our new construction will lead us to the desired representation theorem: by taking the complex algebra of the ultrafilter frame of a BAO, we obtain the canonical
embedding algebra of the original BAO. The fundamental result of this section
(and, indeed, of the entire chapter) is that every boolean algebra with operators
can be isomorphically embedded in its canonical embedding algebra. We will
prove this result and along the way discuss a number of other important issues,
2.3 The Jónsson-Tarski Theorem
43
such as the algebraic status of canonical models and ultrafilter extensions, and the
importance of canonical varieties of BAOs for modal completeness theory.
Let us consider the problem of (isomorphically) embedding an arbitrary BAO
A in a complex algebra. Obviously, the first question to ask is: what should be
the underlying frame of the complex algebra? To keep our notation simple, let
us assume for the moment that we are working in a similarity type with just one
unary modality, and that A = (A; +; ; 0; f ) is a boolean algebra with one unary
operator f . Thus we have to find a universe W and a binary relation R on W
such that A can be embedded in the complex algebra of the frame (W; R). Stone’s
Representation Theorem 2.16 gives us half the answer, for it tells us how to embed
the boolean part of A in the power set algebra of the set Uf A of ultrafilters of A.
Let us take a closer look at this fundamental result.
Stone’s Representation Theorem
A central role in the representation theory of boolean algebras is played by filters
and ultrafilters.
Definition 2.36 A filter of a boolean algebra A = (A; +;
satisfying
(F1)
(F2)
(F3)
; 0) is a subset F
A
1 2 F,
F is closed under taking meets; that is, if a; b 2 F then a b 2 F ,
F is upward closed; that is, if a 2 F and a b then b 2 F .
A filter is proper if it does not contain the smallest element 0, or, equivalently, if
F 6= A. An ultrafilter is a proper filter satisfying
(F4) For every a 2 A, either a or
a belongs to F .
a
The collection of ultrafilters of A is called Uf A.
Note the difference in terminology: an (ultra)filter over the set W is an (ultra)filter
of the power set algebra P(W ).
Example 2.37 For any element a of a boolean algebra A, the set a" = fb 2 A j
a bg is a filter. In the field of finite and co-finite subsets of a countable set W ,
the collection of co-finite subsets of W forms an ultrafilter.
a
Example 2.38 Since the collection of filters of a boolean algebra is closed under
taking intersections, we may speak of the smallest filter FD containing a given set
D A. This filter can also be defined as the following set:
fa 2 A j
there are d0 ; : : : ; dn
2 D such that d : : : dn ag
1
(2.21)
44
2 Algebraizing Modal Logic
which explains why we will also refer to FD as the filter generated by D . This
filter is proper if D has the so-called finite meet property; that is, if there is no finite
subset fd0 ; : : : ; dn g of D such that d1 : : : dn = 0.
a
For future reference, we gather some properties of ultrafilters; the proof of the next
proposition is left to the reader.
Proposition 2.39 Let A = (A; +;
; 0) be a boolean algebra. Then
(i) For any ultrafilter u of A and for every pair of elements a; b 2 A we have
that a + b 2 u iff a 2 u or b 2 u.
(ii) Uf A coincides with the set of maximal proper filters on A (‘maximal’ is
understood with respect to set inclusion).
The main result that we need in the proof of Stone’s Theorem is the Ultrafilter
Theorem: this guarantees that there are enough ultrafilters for our purposes.
Proposition 2.40 (Ultrafilter Theorem) Let A be a boolean algebra, a an element
of A, and F a proper filter of A that does not contain a. Then there is an ultrafilter
extending F that does not contain a.
Proof. We first prove that every proper filter can be extended to an ultrafilter. Let
G be a proper filter of A, and consider the set X of all proper filters H extending
G. Suppose that Y is a chain in X ; that is, Y is a nonempty subset of X of which
the elements are pairwise ordered by set inclusion. We leave it to the reader to
S
S
S
verify that Y is a proper filter; obviously, Y extends G; so Y belongs to X
itself. This shows that X is closed under taking unions of chains, whence it follows
from Zorn’s Lemma that X contains a maximal element u. We claim that u is an
ultrafilter.
For suppose otherwise. Then there is a b 2 A such that neither b nor b belongs
to u. Consider the filters H and H 0 generated by u [fbg and u [f bg, respectively.
Since neither of these can belong to X , both must be improper; that is, 0 2 H and
0 2 H 0. But then by definition there are elements u1; : : : ; un, u01; : : : ; u0m in u such
that
u1 : : : un b 0 and u01 : : : u0m b 0:
From this it easily follows that
u1 : : : un u01 ::: u0m = 0;
contradicting the fact that u is a proper filter.
Now suppose that a and F are as in the statement of the proposition. It is not
hard to show that F [ f ag is a set with the finite meet property. In Example 2.38
we saw that there is a proper filter G extending F and containing a. Now we use
2.3 The Jónsson-Tarski Theorem
45
the first part of the proof to find an ultrafilter u extending G. But if u extends G it
also extends F , and if it contains a it cannot contain a.
a
It follows from Proposition 2.40 and the facts mentioned in Example 2.38 that any
subset of a boolean algebra can be extended to an ultrafilter provided that it has
the finite meet property. We now have all the necessary material to prove Stone’s
Theorem.
Theorem 2.16 (Stone Representation Theorem) Any boolean algebra is isomorphic to a field of sets, and hence, to a subalgebra of a power of 2. As a
consequence, the variety of boolean algebras is generated by the algebra 2:
BA = V(f2g):
Proof. Fix a boolean algebra A = (A; +; ; 0). We will embed A in the power set
of Uf A. Consider the map : A ! P(Uf A) defined as follows:
(a)
= fu 2 Uf A j a 2 ug:
We first show that is a homomorphism. As an example we treat the join operation:
(a + b)
=
=
=
=
fu 2 Uf A j a + b 2 ug
fu 2 Uf A j a 2 u or b 2 ug
fu 2 Uf A j a 2 ug [ fu 2 Uf A j b 2 ug
(a) [ (b):
Note that the crucial second equality follows from Proposition 2.39.
It remains to prove that is injective. Suppose that a and b are distinct elements
of A. We may derive from this that either a 6 b or b 6 a. Without loss of
generality we may assume the second. But if b 6 a then a does not belong to the
filter b" generated by fbg, so by Proposition 2.40 there is some ultrafilter u such
that b" u and a 62 u. Obviously, b" u implies that b 2 u. But then we have
that u 2 (b) and u 62 (a).
This shows that A is isomorphic to a field of sets; it then follows by Proposition 2.8 that A is isomorphic to a subalgebra of a power of 2. From this it is
immediate that BA is the variety generated by the algebra 2.
a
Remark 2.41 That every boolean algebra is isomorphic to a subalgebra of a power
of the algebra 2 can be proved more directly by observing that there is a one-toone correspondence between ultrafilters of A and homomorphisms from A onto 2.
Given an ultrafilter u of A, define u : A ! 2 by
u (a) =
1
0
if a 2 u;
otherwise :
46
2 Algebraizing Modal Logic
And conversely, given a homomorphism : A ! 2, define the ultrafilter u by
u
=
1
(1) (= fa 2 A j (a) = 1g):
We leave further details to the reader.
a
Ultrafilter frames
Now that we have a candidate for the universe of the ultrafilter frame of a given
BAO A, let us see how to define a relation R on ultrafilters such that we can embed
A in the algebra (Uf A; R)+ . To motivate the definition of R, we will view the
elements of the algebra as propositions, and imagine that r (a) (the representation
map r applied to proposition a) yields the set of states where a is true according
to some valuation. Hence, reading fa as 3a, it seems natural that a state u should
be in r (fa) if and only if there is a v with Ruv and v 2 r (a). So, in order to
decide whether Ruv should hold for two arbitrary states (ultrafilters) u and v , we
should look at all the propositions a holding at v (that is, all elements a 2 v ) and
check whether fa holds at u (that is, whether fa 2 u). Putting it more formally,
the natural, ‘canonical’ choice for R seems to be the relation Qf given by
Qf uv iff fa 2 u for all a 2 v:
The reader should compare this definition with the definition of the canonical relation given in Definition 1.23. Although one is couched in terms of ultrafilters,
and the other in terms of maximal consistent sets (MCSs), both clearly trade on the
same idea. As we will shortly learn (and as the above identification of ‘ultrafilters’
and ‘maximal sets of propositions’ already suggests), this is no accident.
In the general case, we use the following definition (an obvious analog of Definition 1.30).
Definition 2.42 Given an n-ary operator f on a boolean algebra (A; +; ; 0), we
define the (n + 1)-ary relation Qf on the set of ultrafilters of the algebra by
Qf uu1 : : : un iff f (a1 ; : : : ; an ) 2 u for all a1 2 u1 , . . . , an 2 un :
Let A = (A; +; ; 0; fM )M2 be a boolean algebra with operators. The ultrafilter
frame of A, notation: A+ , is the structure (Uf A; QfM )M2 . The complex algebra
(A+)+ is called the (canonical) embedding algebra of A (notation: EmA).
a
We leave it to the reader to verify that the ultrafilter extension ue F of a frame
F is nothing but the ultrafilter frame of the complex algebra of F, in symbols:
ue F = (F+ )+ .
For later reference, we state the following proposition (an obvious analog of
Lemma 1.31) which shows that we could have given an alternative but equivalent
definition of the relation Qf .
2.3 The Jónsson-Tarski Theorem
Proposition 2.43 Let f be an n-ary operator on the boolean algebra
u1 , . . . , un an (n + 1)-tuple of ultrafilters of A. Then
47
A, and u,
Qf uu1 : : : un iff f ( a1 ; : : : ; an ) 2 u implies that for some i, ai 2 ui .
Proof. We only prove the direction from left to right. Suppose that Qf uu1 : : : un ,
and that f ( a1 ; : : : ; an ) 2 u. To arrive at a contradiction, suppose that there is
no i such that ai 2 ui . But as Qf uu1 : : : un , it follows that f ( a1 ; : : : ; an ) 2 u.
But this contradicts the fact that f ( a1 ; : : : ; an ) 2 u.
a
As the above sequence of analogous definitions and results suggest, we have already encountered a kind of frame which is very much like an ultrafilter frame,
namely the canonical frame of a normal modal logic (see Definition 1.23). The
basic idea should be clear now: the states of the canonical frame are the MCSs of
the logic, and an ultrafilter is nothing but an abstract version of an MCS. But this is
no mere analogy: the canonical frame of a logic is actually isomorphic to the ultrafilter frame of its Lindenbaum-Tarski algebra, and the mapping involved is simple
and intuitive. When making this connection, the reader should keep in mind that
when we defined ‘the’ canonical frame in the previous Chapter, we always had a
fixed, countable set of proposition letters in mind.
Theorem 2.44 Let be a modal similarity type, a normal modal -logic, and the set of proposition letters used to define the canonical frame F . Then
F = (L())+ :
Proof. We leave it to the reader to show that the function defined by
(
) = f[] j 2 g;
mapping a maximal -consistent set to the set of equivalence classes of its members, is the required isomorphism between F and (L ())+ .
a
The Jónsson-Tarski Theorem
We are ready to prove the Jónsson-Tarski Theorem: every boolean algebra with
operators is embeddable in the full complex algebra of its ultrafilter frame.
Theorem 2.45 (Jónsson-Tarski Theorem) Let be a modal similarity type, and
A = (A; +; ; 0; fM )M2 be a boolean algebra with -operators. Then the representation function r : A ! P(Uf A)) given by
r(a) = fu 2 Uf A j a 2 ug
is an embedding of A into EmA.
48
2 Algebraizing Modal Logic
Proof. To simplify our notation a bit, we work in a similarity type with a single
n-ary modal operator, assuming that A = (A; +; ; 0; f ) is a boolean algebra with
a single n-ary operator f . By Stone’s Representation Theorem, the map r : A !
P(Uf (A)) given by
r(x) = fu 2 Uf (A) j x 2 ug
is a boolean embedding. So, it suffices to show that
phism; that is, that
r is also a modal homomor-
r(f (a1 ; : : : ; an )) = mQf (r(a1 ); : : : ; r(an )):
(2.22)
We will first prove (2.22) for unary f . In other words, we have to prove that
r(fa) = mQf (r(a)):
We start with the inclusion from right to left: assume u 2 mQf (r (a)). Then by
definition of mQf , there is an ultrafilter u1 with u1 2 r (a) (that is, a 2 u1 ) and
Qf uu1 . By definition of Qf this implies fa 2 u, or u 2 r(fa).
For the other inclusion, let u be an ultrafilter in r (fa), that is, fa 2 u. To
prove that u 2 mQf (r (a)), it suffices to find an ultrafilter u1 such that Qf uu1 and
u1 2 r(a), or a 2 u1 . The basic idea of the proof is that we first pick out those
elements of A (other than a) that we cannot avoid putting in u1 . These elements are
given by the condition Qf uu1 . By Proposition 2.43 we have that for every element
of the form f ( y ) in u, y has to be in u1 ; therefore, we define
F
:= fy 2 A j
f ( y) 2 ug:
We will now show that there is an ultrafilter u1 F containing a. First, an easy
proof (using the additivity of f ), shows that F is closed under taking meets. Second, we prove that
F 0 := fa y j y 2 F g
has the finite meet property. As F is closed under taking meets, it is sufficient to
show that a y 6= 0 whenever y 2 F . To arrive at a contradiction, suppose that
a y = 0. Then a y, so by the monotonicity of f , fa f ( y); therefore,
f ( y) 2 u, contradicting y 2 F .
By Proposition 2.40 there is an ultrafilter u1 F 0 . Note that a 2 u1 , as 1 2 F .
Finally, Qf uu1 holds by definition of F : if f ( y ) 2 u then y 2 F u1 .
We now prove (2.22) for arbitrary n 1 by induction on the arity n of f . We
have just proved the base case. So, assume that the induction hypothesis holds for
n. We only treat the direction from left to right, since the other direction can be
proved as in the base case. Let f be a normal and additive function of rank n + 1,
and suppose that a1 ; : : : ; an+1 are elements of A such that f (a1 ; : : : ; an+1 ) 2 u.
2.3 The Jónsson-Tarski Theorem
49
We have to find ultrafilters u1 ; : : : ; un+1 of A such that (i) ai 2 ui for all i with
1 i n + 1, and (ii) Qf uu1 : : : un+1. Our strategy will be to let the induction
hypothesis take care of u1 ; : : : ; un and then to search for un+1 .
Let f 0 : An ! A be the function given by
f 0(x1 ; : : : ; xn ) = f (x1 ; : : : ; xn ; an+1 ):
That is, for the time being we fix an+1 . It is easy to see that f 0 is normal and
additive, so we may apply the induction hypothesis. Since f 0 (a1 ; : : : ; an ) 2 u, this
yields ultrafilters u1 ; : : : ; un such that ai 2 ui for all i with 1 i n, and
f (x1 ; : : : ; xn ; an+1 ) 2 u; whenever xi 2 ui (1 i n).
(2.23)
Now we will define an ultrafilter un+1 such that an+1 2 un+1 and Qf uu1 : : : un+1 .
This second condition can be rewritten as follows (we abbreviate ‘x1 2 u1 , . . . ,
xn 2 un ’ by ‘~x 2 ~u’):
Qf uu1 : : : un+1
iff for all ~x; y : if ~x 2 ~u, then y 2 un+1 implies f (~x; y ) 2 u
iff for all ~x; y : if ~x 2 ~u, then f (~x; y ) 62 u implies y 62 un+1
iff for all ~x; y : if ~x 2 ~u, then f (~x; y ) 2 u implies y 2 un+1
iff for all ~x; z : if ~x 2 ~u, then f (~x; z ) 2 u implies z 2 un+1 .
This provides us with a minimal set of elements that un+1 should contain; put
F
:= fz 2 A j 9~x 2 ~u ( f (~x; z) 2 u)g:
If f (~x; z ) 2 u, we say that ~x drives z into F . We now take the first condition
into account as well, defining F 0 := fan+1 g [ F .
Our aim is to prove the existence of an ultrafilter un+1 containing F 0 . It will be
clear that this is sufficient to prove the theorem (note that an+1 2 F 0 as 1 2 F ). To
be able to apply the Ultrafilter Theorem 2.40, we will show that F 0 has the finite
meet property. We first need the following fact:
F is closed under taking meets.
(2.24)
Let z 0 ; z 00 be in F ; assume that z 0 and z 00 are driven into F by ~x0 and ~x00 , respectively. We will now see that ~x := (x01 x001 ; : : : ; x0n x00n ) drives z := z 0 z 00 into F ,
that is, that f (~x; z ) 2 u.
Since f is monotonic, we have f (~x; z 0 ) f (~x0 ; z 0 ), and hence we find that
f (~x0; z 0 ) f (~x; z 0 ). As u is upward closed and f (~x0; z 0 ) 2 u by
our ‘driving assumption’, this gives f (~x; z 0 ) 2 u. In the same way we find
f (~x; z 00 ) 2 u. Now
f (~x; z ) = f (~x; (z 0 z 00 )) = f (~x; ( z 0 ) + ( z 00 )) = f (~x; z 0 ) + f (~x; z 00 );
50
2 Algebraizing Modal Logic
whence
f (~x; z ) = [ f (~x; z 0 )] [ f (~x; z 00 )]:
Therefore, f (~x; z ) 2 u, since u is closed under taking meets. This proves
(2.24).
We can now finish the proof and show that indeed
F 0 has the finite meet property.
(2.25)
By (2.24) it suffices to show that an+1 z 6= 0 for all z 2 F . To prove this, we reason
by contraposition: suppose that z 2 F and an+1 z = 0. Let ~x 2 ~u be a sequence
that drives z into F , that is, f (~x; z ) 2 u. From an+1 z = 0 it follows that
an+1 z , so by monotonicity of f we get f (~x; z ) f (~x; an+1 ). But then
f (~x; an+1 ) 2 u, which contradicts (2.23). This proves that indeed an+1 z 6= 0
and hence we have shown (2.25) and thus, Theorem 2.45.
a
Exercises for Section 2.3
2.3.1 Let be a normal modal logic. Give a detailed proof that the canonical frame F is
isomorphic to the ultrafilter frame of L (over a countable set of proposition letters).
2.3.2 Let A denote the collection of sets X of integers satisfying one of the following
four conditions: (i) X is finite, (ii) X is co-finite, (iii) X E is finite, (iv) X E is
co-finite. Here E denotes the set of all even integers, and denotes symmetric difference:
X E = (X n E ) [ (E n X ). Consider the following algebra A = (A; [; ; ?; f ) where
the operation f is given by
f (X ) =
fx
Z
1
j x 2 Xg
if X is of type (i) or (iii);
if X is of type (ii) or (iv):
(a) Show that A is a boolean algebra with operators.
(b) Describe A+ .
W be the set Z [ f 1; 1g and let S be the successor relation on Z, that is,
S = f(z; z + 1) j z 2 Zg.
(a) Give a BAO whose ultrafilter frame is isomorphic to the frame F = (W; R) with
R = S [ f( 1; 1); (1; 1)g.
(b) Give a BAO whose ultrafilter frame is isomorphic to the frame F = (W; R) with
R = S [ (W f 1; 1g).
(c) Give a BAO whose ultrafilter frame is isomorphic to the frame F = (W; R) with
R = S [ f( 1; 1)g [ (W f1g).
2.3.3 Let
2.3.4 An operation on a boolean algebra is called 2-additive if it satisfies
f (x + y + z ) = f (x + y) + f (x + z ) + f (y + z ):
Prove an analog of the Jónsson-Tarski Theorem for boolean algebras augmented with 2additive operations.
2.4 Canonicity: the algebraic perspective
51
2.4 Canonicity: the algebraic perspective
To conclude this chapter, we discuss the algebraic perspective on canonicity. Clearly
the Jónsson-Tarski Theorem guarantees that we can represent the LindenbaumTarski algebras of normal modal logics as complex algebras, so it immediately
converts Theorem 2.34 into a completeness result with respect to complex algebras.
But modal logicians want more: because of the link between complex algebras
and relational semantics, it seems to offer a plausible algebraic handle on frame
completeness. And in fact it does – but we need to be careful. As should be clear
from our work in the previous Chapter, even with the Jónsson-Tarski Theorem at
our disposal, one more hurdle remains to be cleared. In Exercise 2.2.6 we defined
the notion of a complete variety of BAOs: a variety V is complete if there is a frame
class K that generates V in the sense that V = HSPCmK. The exercise asked
the reader to show that any logic is complete if and only if V is a complete
variety. Now does the Jónsson-Tarski Theorem establish such a thing? Not really
– it does show that every algebra A is a complex algebra over some frame, thus
proving that for any logic we have that V CmK for some frame class K. So,
this certainly gives V HSPCmK. However, in order to prove completeness,
we have to establish an equality instead of an inclusion. One way to prove this is
to show that the complex algebras that we have found form a subclass of V . By
Proposition 2.26 it would suffice to show that for any algebra A in the variety V ,
the frame A+ is a frame for the logic . This requirement gives us an algebraic
handle on the notion of canonicity.
Let us examine a concrete example. Recall that K4 is the normal logic generated by the 4 axiom, 33p ! 3p. We know from Theorem 1.33 that K4 is
complete with respect to the class of transitive frames. How can we prove this
result algebraically?
A little thought reveals that the following is required: we have to show that the
Lindenbaum-Tarski algebras for K4 are embeddable in full complex algebras of
transitive frames. Recall that the 4 axiom characterizes the transitive frames, thus
in our proposed completeness proof, we would have to show that 4 is valid in the
ultrafilter frame (LK4 ())+ of LK4 (), or equivalently, that ((LK4 ())+ )+ belongs to the variety V4 . Note that by Theorem 2.34 we already know that LK4 ()
belongs to V4 .
As this example suggests, proving frame completeness results for extensions of
K algebraically leads directly to the following question: which varieties of BAOs
are closed under taking canonical embedding algebras? In fact, this is the required
algebraic handle on canonicity and motivates the following definition.
Definition 2.46 Let be a modal similarity type, and C a class of boolean algebras
with -operators. C is canonical if it is closed under taking canonical embedding
52
2 Algebraizing Modal Logic
algebras; that is, if for all algebras A, EmA is in C whenever A is in C. Likewise,
an equation is canonical if its validity is preserved when moving from a BAO to its
canonical embedding algebra.
a
Thus we now have two notions of canonicity, namely the logical one of Definition 1.36 and the algebraic one just defined. Using Theorem 2.34, we show that
these two concepts are closely related.
Proposition 2.47 Let be a modal similarity type, and
V is a canonical variety, then is canonical.
a set of -formulas. If
Proof. Assume that the variety V is canonical, and let be the fixed countable set
of proposition letters that we use to define canonical frames. By Theorem 2.34, the
Lindenbaum-Tarski algebra LK () is in V ; then, by assumption, its canonical
embedding algebra EmLK is in V . However, from Theorem 2.44 it follows that
this algebra is isomorphic to the complex algebra of the canonical frame of K:
EmLK() = ((LK())+ )+ = (FK)+:
Now the fact that (FK )+ is in V means that FK by Proposition 2.26. But
this implies that is canonical.
a
An obvious question is whether the converse of Proposition 2.47 holds as well;
that is, whether a variety V is canonical if is a canonical set of modal formulas. However, note that canonicity of only implies that one particular boolean
algebra with operators has its embedding algebra in V , namely the LindenbaumTarski algebra over a countably infinite number of generators. This is because
throughout the completeness chapter we were working in a fixed, countable set of
proposition letters. In fact, we are facing an open problem here:
Open Problem 1 Let be a modal similarity type, and a canonical set of formulas. Is V a canonical variety?
Equivalently, suppose that E is a set of equations such that for all countable
boolean algebras with -operators we have the following implication
if A j= E then EmA j= E:
(2.26)
Is VE a canonical variety? In other words, does (2.26) hold for all boolean algebras with -operators?
This is an interesting problem. However, arguably the restriction of the notion of
canonicity to countable languages that we adopted in the previous Chapter was not
mathematically natural. Thus, let us redefine the logical notion of canonicity so
that it refers to languages of arbitrary size. The definition of canonical frames and
2.4 Canonicity: the algebraic perspective
53
models can easily be parametrized by a set of proposition letters: the maximal consistent sets are supposed to be maximal within the induced set of formulas. We now
simply define a logic to be canonical if it is valid on each of its canonical frames.
With this definition we can indeed establish equivalence between the logical and
algebraic notions of canonicity. (An alternative, and mathematically quite interesting alternative, would be to introduce, both logically and algebraically, a hierarchy
of canonicity notions, parametrized by cardinal numbers. Such an approach has
indeed been studied in the literature, but this option will not be pursued here.)
Regardless of our approach towards this issue, the algebraic notion of canonicity
can do a lot of work for us. The important point is that it offers a genuinely new
perspective on what canonicity is, a perspective that will allow us to use algebraic
arguments. This will be demonstrated in the two papers forming part of these
Lecture Notes.
Finally, in the course we will concentrate on canonicity results of a syntactic
nature, proving that terms of various syntactic shape have various properties. But
there is an interesting result, and an intriguing open problem, concerning a ore
structural side of canonicity as well; both of these have to do with the relation
between canonical varieties and frame classes that are closed under the formation
of ultraproducts (such as elementary frame classes).
First we need the following definition.
Definition 2.48 Let be modal similarity type, and K be a class of -frames. The
variety generated by K (notation: VK ) is the class HSPCmK.
a
Theorem 2.49 Let be modal similarity type, and K be a class of -frames which
is closed under ultraproducts. Then the variety VK is canonical.
Example 2.50 Consider the modal similarity type fÆ; ; 1’g of arrow logic, where
The standard interpretation of this
language is given in terms of the squares (cf. Example 1.10). Recall that the square
SU = (W; C; R; I ) is defined as follows:
Æ is binary, is unary and 1’ is a constant.
W = U U;
C ((u; v); (w; x); (y; z )) iff u = w and v = z and x = y;
R((u; v); (w; x)) iff u = x and v = w;
I (u; v) iff u = v:
It may be shown that the class SQ of (isomorphic copies of) squares is first-order
definable in the frame language with predicates C , R and I . Therefore, Theorem 2.49 implies that the variety generated by SQ is canonical. This variety is well
known in the literature on algebraic logic as the variety RRA of Representable Relation Algebras.
a
54
2 Algebraizing Modal Logic
Rephrased in terminology from modal logic, Theorem 2.49 boils down to the following result.
Corollary 2.51 Let be a modal similarity type, and K be a class of -frames
which is closed under ultraproducts. Then the modal theory of K is a canonical
logic.
We conclude the section with the foremost open problem in this area: does the
converse of Theorem 2.49 holds as well?
Open Problem 2 Let be modal similarity type, and V a canonical variety of
boolean algebras with -operators. Is there a class K of -frames, closed under
taking ultraproducts, such that V is generated by K?
Notes
The main aim of algebraic logic is to gain a better understanding of logic by treating it in universal algebraic terms – in fact, the theory of universal algebra was
developed in tandem with that of algebraic logic. Given a logic, algebraic logicians try to find a class of algebras that algebraizes it in a natural way. When a
logic is algebraizable, natural properties of a logic will correspond to natural properties of the associated class of algebras, and the apparatus of universal algebra can
be applied to solve logical problems. For instance, we have seen that representation theorems are the algebraic counterpart of completeness theorems in modal
logic. The algebraic approach has had a profound influence on the development of
logic, especially non-classical logic; readers interested in the general methodology
of algebraic logic should consult Blok and Pigozzi’s [7] or Andréka et al. [1].
The field has a long and strong tradition dating back to the nineteenth century.
In fact, nineteenth century mathematical logic was algebraic logic: to use the terminology of Section 2.1, propositions were represented as algebraic terms, not
logical formulas. Boole is generally taken as the founding father of both propositional logic and modern algebra – the latter because, in his work, terms for the first
time refer to objects other than numbers, and operations very different from the
arithmetical ones are considered. The work of Boole was taken up by de Morgan,
Peirce, Schröder and others; their contributions to the theory of binary relations
formed the basis of Tarski’s development of relational algebra. In the Historical
Overview in Chapter 1 we mentioned MacColl, the first logician in this tradition to
treat modal logic. A discussion of the nineteenth century roots of algebraic logic is
given by Anellis and Houser in [4].
However, when the quantificational approach to logic became firmly established
in the early twentieth century, interest in algebraic logic waned, and it was only
the influence of a relatively small number of researchers such as Birkhoff, Stone,
Notes to Chapter 2
55
Tarski, and Rasiowa and Sikorski [43, 44], that ensured that the tradition was
passed on to the present day. The method of basing an algebra on a collection
of formulas (or equivalence classes of formulas), due to Lindenbaum and Tarski,
proved to be an essential research tool. This period also saw the distinction between
logical languages and their semantics being sharpened; an algebraic semantics for
non-classical logics was provided by Tarski’s matrix algebras. But the great success story of algebraic logic was its treatment of classical propositional logic in
the framework of boolean algebras, which we sketched in Section 2.1. Here, the
work of Stone [48] was a milestone: not only did he prove the representation theorem for boolean algebras (our Theorem 2.16), he also recognized the importance
of topological notions for the area (something we did not discuss in the text). This
enabled him to prove a duality theorem permitting boolean algebras to be viewed
as essentially the same objects as certain topologies (now called Stone spaces).
Stone’s work has influenced many fields of mathematics, as is witnessed by Johnstone [25].
McKinsey and Tarski [40] drew on Stone’s work in order to prove a representation theorem for so-called closure algebras (that is, S4-algebras); this result significantly extended McKinsey’s [39] which dealt with finite closure algebras. However, when it comes to the algebraization of modal logics, the reader is now in a
position to appreciate the full significance of the work of Jónsson and Tarski [29].
Although modal logic is not mentioned in their paper, the authors simultaneously
invented relational semantics, and showed (via their representation theorem) how
this new relational world related to the algebraic one. Both Theorem 2.45 from
the present chapter and important results on canonicity are proved here. It is also
obvious from their terminology (for instance, the use of the words ‘closed’ and
‘open’ for certain elements of the canonical embedding algebra) that hidden beneath the surface of the paper lies a duality theory that extends Stone’s result to
cover operators on boolean algebras.
For many years after the publication of the Jónsson-Tarski paper, research in
modal logic and in BAO theory pretty much took place in parallel universes. In
algebraic circles, the work of Jónsson and Tarski was certainly not neglected. The
paper fitted well with a line of work on relation algebras. These were introduced
by Tarski [49] to be to binary relations what boolean algebras are to unary ones;
the concrete, so-called representable relation algebras have, besides the boolean
repertoire, operations for taking the converse of a relation and the composition
of two relations, and as a distinguished element, the identity relation. (From the
perspective of our book, the class RRA of representable relation algebras is nothing
but the variety generated by the complex algebras of the two-dimensional arrow
frames; see Example 2.50. In fact, one of the motivations behind the introduction of
arrow logic was to give a modal account of the theory of relation algebras.) Much
attention was devoted to finding an analog of Stone’s result for boolean algebras:
56
2 Algebraizing Modal Logic
that is, a nice equational characterization of the representable relation algebras.
But this nut turned out to be hard to crack. Lyndon [34] showed that the axioms
that Tarski had proposed did not suffice, and Monk [41] proved that the variety
does not even have a finite first-order axiomatization. Later work showed that
equational axiomatizations will be very complex (Andréka [2]) and not in Sahlqvist
form (Hodkinson [23] and Venema [53]), although Tarski proved that RRA is a
canonical variety. As a positive result, a nice game-theoretical characterization
was given by Hirsch and Hodkinson [22]. But these Notes can only provide a lopsided account of one aspect of the theory of relation algebras; for more, the reader
is referred to Jónsson [26, 27], Maddux [35] or Hirsch and Hodkinson [24]. One
last remark on relation algebras: it is a very powerful theory. In fact, as Tarski and
Givant show in [50], one can formalize all of set theory in it.
Tarski and his students developed other branches of algebraic logic as well: for
example, the theory of cylindric algebras. The standard reference here is Henkin,
Monk and Tarski [21]. Cylindric algebras (and also the polyadic algebras of Halmos [20]), are boolean algebras with operators that were studied as algebraic counterparts of first-order logic. For an introductory survey of these and other algebras
of relations we refer the reader to Németi [42]; modal logic versions of these algebras are discussed in Marx and Venema [38].
But in modal circles, the status of algebraic methods was very different. Indeed,
with the advent of relational semantics for modal logic in the 1960s, it seemed that
algebraic methods were to be swept away: model theoretic tools seemed to be the
route to a brave new modal world. (Bull’s work was probably the most important exception to this trend. For example, his theorem that all normal extensions
of S4.3 are characterized by classes of finite models was proved using algebraic
arguments.) Indicative of the spirit of the times is the following remark made by
Lemmon in his two part paper on algebraic semantics for modal logic. After thanking Dana Scott for ideas and stimulus, he remarks:
. . . I alone am responsible for the ugly algebraic form into which I have cast
some of his elegant semantics. [33, page 191]
Such attitudes only seemed reasonable because Jónsson and Tarski’s work had been
overlooked by the leading modal logicians, and neither Jónsson nor Tarski had
drawn attention to its modal significance. Only when the frame incompleteness
results (which began to appear around 1972) showed that not all normal modal
logics could be characterized in terms of frames were modal logicians forced to
reappraise the utility of algebraic methods.
The work of Thomason and Goldblatt forms the next major milestone in the
story: Thomason [52] not only contains the first incompleteness results and uses
BAO s, it also introduced general frames (though similar, language dependent, structures had been used in earlier work by Makinson [36] and Fine [11]). Thomason
Notes to Chapter 2
57
showed that general frames can be regarded as simple set theoretic representations
of BAOs, and notes the connection between general frames and Henkin models for
second-order logic. In [51], Thomason developed a duality between the categories
of frames with bounded morphisms and that of complete and atomic modal algebras with homomorphisms that preserve infinite meets. It was Goldblatt, however,
who did the most influential work: in [17] the full duality between the categories of
modal algebras with homomorphisms and descriptive general frames with bounded
morphisms is proved, a result extending Stone’s. Independently, Esakia [10] came
up with such a duality for closure algebras. Goldblatt generalized his duality to arbitrary similarity types in [18]; a more explicitly topological version can be found
in Sambin and Vaccaro [46]. Ever since their introduction in the seventies, general frames, a nice compromise between algebraic and the relational semantics,
have occupied a central place in the theory of modal logic. Kracht [31, 32] developed an interesting calculus of internal descriptions which connects the algebraic
and first-order side of general frames. Zakharyaschev gave an extensive analysis
of transitive general frames in his work on canonical formulas – see for instance
Chagrov and Zakharyaschev [8]
The first proof of the canonicity of Sahlqvist formulas, for the basic modal similarity type, was given by Sahlqvist [45], although many particular examples and
less general classes of Sahlqvist axioms were known to be canonical. In particular, Jónsson and Tarski proved canonicity, not only for certain equations, but also
for various boolean combinations of suitable equations. This result overlaps with
Sahlqvist’s in the sense that canonicity for simple Sahlqvist formulas follows from
it, but on the other hand, Sahlqvist formulas allowing properly boxed atoms in
the antecedent do not seem to fall under the scope of the results in Jónsson and
Tarski [29]. Incidentally, Sahlqvist’s original proof is well worth consulting: it is
non-algebraic, and very different from the one given in the text. Our proof of the
Sahlqvist Completeness Theorem is partly based on the one given in Sambin and
Vaccaro [47].
Recent years have seen a revived interest in the notion of canonicity. De Rijke
and Venema [9] defined the notion of a Sahlqvist equation and generalized the
theory to arbitrary similarity types. Jónsson became active in the field again; in [28]
he gave a proof of the canonicity of Sahlqvist equations by purely algebraic means,
building on the techniques of his original paper with Tarski. Subsequent work of
Gehrke, Jónsson and Harding (see for instance [14, 13]) generalized the notion
even further by weakening the boolean base of the algebra to that of a (distributive)
lattice. Ghilardi and Meloni [15] prove canonicity of a wide class of formulas
using an entirely different representation of the canonical extension of algebras
with operators. Whereas the latter lines of research tend to separate the canonicity
of formulas from correspondence theoretic issues, a reverse trend is visible in the
work of Kracht (already mentioned) and Venema [54]. The latter work shows that
58
2 Algebraizing Modal Logic
for some non-Sahlqvist formulas there is still an algorithm generating a first-order
formula which is now not equivalent to the modal one, but does define a property
for which the modal formula is canonical.
The applications of universal algebraic techniques in modal logic go much further than we could indicate in this chapter. Various properties of modal logics have
been successfully studied from an algebraic perspective; of the many examples we
only mention the work of Maksimova [37] connecting interpolation with amalgamation properties, and the work by Goldblatt and Thomason [19] relating modal
definability to Birkhoff’s Theorem [6] identifying varieties with equational classes.
Also, most work by Blok, Kracht, Wolter and Zakharyaschev on mapping the lattice of modal logics makes essential use of algebraic concepts such as splitting
algebras; a good starting point for information on this line of research would be
Kracht [32].
Finally, the proof of Theorem 2.49 is a generalization and algebraization by
Goldblatt [18] of results due to Fine [12] and van Benthem [5]. The Open Problems 1 and 2 seem to have been formulated first in Fine [12]; readers who would
like to try and solve the second one should definitely consult Goldblatt [16].
Appendix A
An Algebraic Toolkit
In this appendix we review some basic (universal) algebraic notions. The first part
deals with algebras and operations on (classes of) algebras, the second part is about
algebraic model theory, and in the third part we discuss equational logic. Birkhoff’s
fundamental theorems are stated without proof.
Universal Algebra
An algebra is a set together with a collection of functions over the set; these functions are usually called operations. Algebras come in various similarity types,
determined by the number and arity of the operations.
Definition A.1 (Similarity Type) An algebraic similarity type is an ordered pair
F = (F; ) where F is a non-empty set and is a function F ! N . Elements
of F are called function symbols; the function assigns to each operator f 2 F a
finite arity or rank, indicating the number of arguments that f can be applied to.
Function symbols of rank zero are called constants. We will usually be sloppy in
our notation and terminology and write f 2 F instead of f 2 F .
a
Given a similarity type, it is obvious what an algebra of this type should be.
Definition A.2 (Algebras) Let A be some set, and n a natural number; an n-ary
operation on A is a function from An to A.
Let F be an algebraic similarity type. An algebra of type F is a pair A =
(A; I ) where A is a non-empty set called the carrier of the algebra, and I is an
interpretation, a function assigning, for every n, an n-ary operation fA on A to
each function symbol f of rank n. We often use the notation A = (A; fA )f 2F for
such an algebra. When no confusion is likely to arise, we omit the subscripts on
the operations.
a
We now define the standard constructions for forming new algebras from old. First
we define the natural notion of structure preserving maps between algebras.
59
60
A An Algebraic Toolkit
Definition A.3 (Homomorphisms) Let A = (A; fA )f 2F and B = (A; fB )f 2F
be two algebras of the same similarity type. A map : A ! B is a homomorphism
if for all f 2 F , and all a1 , . . . , an 2 A (where n is the rank of f ):
(fA (a1 ; : : : ; an )) = fB (a1 ; : : : ; an ):
(A.1)
(Here ai is shorthand (ai ).) The special case for constants c is
(cA ) = cB :
The kernel of a homomorphism f : A ! B is the relation ker f = f(a; a0 ) 2 A2 j
f (a) = f (a0 )g. We say that B is a homomorphic image of A (notation: A B),
if there is a surjective homomorphism from A onto A0 . Given a class C of algebras,
HC is the class of homomorphic images of algebras in C.
a
Definition A.4 (Isomorphisms) A bijective homomorphism is called an isomorphism. We say that two algebras are isomorphic if there is an isomorphism between
them. Usually we do not distinguish isomorphic algebras, but if we do, we write
IC for the class of isomorphic copies of algebras in C.
a
The second way of making new algebras from old is to find a small algebra inside
a larger one.
Definition A.5 (Subalgebras) Let A be an algebra, and B a subset of the carrier
A. If B is closed under every operation fA , then we call B = (B; fA B )f 2F a
subalgebra of A. We say that C is embeddable in A (notation: C A), if C is
isomorphic to a subalgebra of A; the isomorphism is called an embedding. Given
a class C of algebras, SC denotes the class of isomorphic copies of subalgebras of
algebras in C.
a
A third way of forming new algebras is to make a big algebra out of a collection of
small ones.
Definition A.6 (Products) Let (Aj )j 2J be a family of algebras. We define the
Q
product j 2J Aj of this family as the algebra A = (A; fA )f 2F where A is the
Q
cartesian product j 2J Aj of the carriers Aj , and the operation fA is defined coQ
ordinatewise; that is, for elements a1 ; : : : ; an 2 j 2J Aj , fA (a1 ; : : : ; an ) is the
Q
element of j 2J Aj given by:
fA (a1 ; : : : ; an )(j ) = fAj (a1 (j ); : : : ; an (j )):
Q
When all the algebras Aj are the same, say A, then we call j 2J A a power of
Q
A, and write AJ rather than j 2J A. Given a class C of algebras, PC denotes the
class of isomorphic copies of products of algebras in C.
a
An Algebraic Toolkit
61
Suppose you are working with a class of algebras from which you cannot obtain
new algebras by the three operations defined above. Intuitively, such a class is
‘complete’, for it is closed under the natural algebra-forming operations. Such
classes play an important role in universal algebra. They are called varieties:
Definition A.7 (Varieties) A class of algebras is called a variety if it is closed
under taking subalgebras, homomorphic images, and products. Given a class C
of algebras, VC denotes the variety generated by C; that is, the smallest variety
containing C.
a
A well-known result in universal algebra states that VC = HSPC. That is, in order
to obtain the variety generated by C, you can start by taking products of algebras
in C, then go on to take subalgebras, and finish off by forming homomorphic images. You do not need to do anything else: subsequent applications of any of these
operations will not produce anything new.
Homomorphisms and Congruences
Homomorphisms on an algebra
tions on the carrier of A.
A are closely related to special equivalence rela-
Definition A.8 (Congruences) Let A be an algebra for the similarity type F . An
equivalence relation on A (that is, a reflexive, symmetric and transitive relation)
is a congruence if it satisfies, for all f 2 F
if a1
b & : : : & an bn; then fA (a ; : : : ; an) fA(b ; : : : ; bn);
1
1
1
(A.2)
a
where n is the rank of f .
The standard examples of congruences are the ‘modulo’ relations on the integers.
Consider the algebra Z = (Z; +; ; 0; 1) of the integers under addition and multiplication, and, for a positive integer n, let the relation n be defined by z n z 0
if n divides z z 0 . We leave it to the reader to verify that these relations are all
congruences.
The importance of congruences is that they are precisely the kind of equivalence
relations that allow a natural algebraic structure to be defined on the collection of
equivalence classes.
Definition A.9 (Quotient Algebras) Let A be an F -algebra, and a congruence
on A. The quotient algebra of A by is the algebra A= whose carrier is the set
A= = f[a] j a 2 Ag
62
A An Algebraic Toolkit
of equivalence classes of A under , and whose operations are defined by
fA= ([a1 ]; : : : ; [an ]) = [fA (a1 ; : : : ; an )]:
(This is well-defined by (A.2).) The function taking an element a 2 A to its
equivalence class [a] is called the natural map associated with the congruence. a
As an example, taking the quotient of Z under the relation n makes the algebra
Zn of arithmetic modulo n.
The close connection between homomorphisms and congruences is given by the
following proposition (the proof of which we leave to the reader).
Proposition A.10 (Homomorphisms and Congruences) Let A be an F -algebra.
Then
(i) If f : A ! B is a homomorphism, its kernel is a congruence on A.
(ii) Conversely, if is a congruence on A, its associated natural map is a
surjective homomorphism from A onto A=.
Algebraic Model Theory
Universal algebra can be seen as a branch of model theory in which one is only
interested in structures where all relations are functions. The standard language
for talking about such structures is equational, where an equation is a statement
asserting that two terms denote the same element.
Definition A.11 (Terms and Equations) Given an algebraic similarity type F and
a set X of elements called variables, we define the set Ter F (X ) of F -terms over
X inductively: it is the smallest set T containing all constants and all variables in
X such that f (t1 ; : : : ; tn ) is in T whenever t1 , . . . , tn are in T and f is a function
symbol of rank n.
An equation is a pair of terms (s; t); the notation s t is usually used.
a
Having defined the algebraic language, we now consider the way it is interpreted
in algebras. Obviously terms refer to elements of algebras, but in order to calculate
the meaning of a term we need to know what the variables in the term stand for.
This information is provided by an assignment.
Definition A.12 (Algebraic Semantics) Let F be an algebraic similarity type, X
a set of variables, and A an F -algebra. An assignment on A is a function : X !
A associating an element of A with each variable in X . Given such an assignment
, we can calculate the meaning ~(t) of a term t in Ter F (X ) as follows:
An Algebraic Toolkit
~(x)
~(c)
~(f (t1 ; : : : ; tn ))
63
= (x);
= cA ;
= fA(~(t ); : : : ; ~(tn)):
a
1
The last equality bears an obvious resemblance to the condition (A.1) defining
homomorphisms. In fact, we can turn the meaning function into a genuine homomorphism by imposing a natural algebraic structure on the set Ter F (X ) of terms:
Definition A.13 (Term Algebras) Let F be an algebraic similarity type, and X
a set of variables. The term algebra of F over X is the algebra TerF (X ) =
(Ter F (X ); I ) where every function symbol f is interpreted as the operation I (f )
on Ter F (X ) given by
I (f )(t1 ; : : : ; tn ) = f (t1 ; : : : ; tn ):
(A.3)
a
In other words, the carrier of the term algebra over F is the set of F -terms over the
set of variables X , and the operation I (f ) or fTer F (X ) maps an n-tuple t1 ; : : : ; tn
of terms to the term f (t1 ; : : : ; tn ). Note the double role of f in (A.3): on the righthand side, f denotes a ‘static’ part of the syntactic term f (t1 ; : : : ; tn ), while on
the left-hand side I (f ) denotes a ‘dynamic’ interpretation of f as an operation on
terms.
The perspective on F -terms as constituting an F -algebra is extremely useful.
For example, we can view the operation of substituting terms for variables in terms
as an endomorphism on the term algebra, that is, a homomorphism from an algebra
to itself.
Definition A.14 Let F be a similarity type, and X a set of variables. A substitution
is a map : X ! Ter F (X ) mapping variables to terms. Such a substitution can
be extended to a map ~ : Ter F (X ) ! Ter F (X ) by the following inductive
definition:
~ (x)
~ (f (t1 ; : : : ; tn ))
:= (x);
:= f (~(t ); : : : ; ~ (tn )):
1
We sometimes use the word ‘substitution’ for a function mapping terms to terms
that satisfies the second of the conditions above (that is, we sometimes call ~a
substitution).
a
Proposition A.15 Let
: Ter F (X ) ! Ter F (X ) be a substitution.
TerF (X ) ! TerF (X ) is a homomorphism.
Then
:
64
A An Algebraic Toolkit
Moreover, the meaning function associated with an assignment
morphism:
is now a homo-
Proposition A.16 Given any assignment of variables X to elements of an algebra A, the corresponding meaning function ~ is a homomorphism from TerF (X )
to A.
The standard way of making statements about algebras is to compare the meaning
of two terms under the same valuation – that is, to use equations.
Definition A.17 (Truth and Validity) An equation s t is true or holds in an
algebra A (notation: A j= s t), if for all assignments , ~(s) = ~(t).
A set E of equations holds in an algebra A (notation: A j= E ), if each equation
in E holds in A. If A j= s t or A j= E we will also say that A is a model for
s t, or for E , respectively.
An equation s t is a semantic consequence of a set E of equations (notation:
E j= s t), if every model for E is a model for s t.
a
Algebraists are often interested in specific classes of algebras such as groups and
boolean algebras. Such classes are usually defined by sets of equations.
Definition A.18 (Equational Class) A class C of algebras is equationally definable, or an equational class, if there is a set E of equations such that C contains
precisely the models for E .
a
The following theorem, due to Birkhoff, is one of the most fundamental results of
universal algebra:
Theorem A.19 (Birkhoff) A class of algebras is equationally definable if and only
if it is a variety.
Unfortunately, we do not have the space to prove this theorem here. The reader
is advised to try proving the easy direction (that is, to show that any equationally
definable class is closed under taking homomorphic images, subalgebras and direct
products) for him- or herself.
Equational Logic
Equational logic arises when we formalize the rules that enable us to deduce new
equations from old. Although we do not make direct use of equational logic in the
text, it will be helpful if the reader is acquainted with it. Here is a fairly standard
system.
An Algebraic Toolkit
65
Definition A.20 (Equational Logic) Let F be an algebraic similarity type, and E
a set of equations. The set of equations that are derivable from E is inductively
defined by the following schema:
The equations in E are derivable from E ; they are called axioms.
Every equation t t is derivable from E .
If t1 t2 is derivable from E , then so is t2 t1 .
If the equations t1 t2 and t2 t3 are derivable from E , then so is
t1 t3 .
(congruence) Suppose that all equations t1 u1 , . . . , tn un are derivable from
E , and that f is a function symbol of rank n. Then the equation
f (t1 ; : : : ; tn ) f (u1 ; : : : ; un ) is derivable from E as well. This
schema is sometimes called replacement.
(substitution) If t1 t2 is derivable from E , then so is the equation t1 t2 ,
for every substitution .
(axioms)
(reflexivity)
(symmetry)
(transitivity)
The notation E ` t1 t2 means that the equation t1 t2 is derivable from E .
A derivation is a list of equations such that every element is either an axiom, or
has the form t t, or can be obtained from earlier elements of the list using the
symmetry, transitivity, congruence/replacement, or substitution rules.
a
A fundamental completeness result, also due to Birkhoff, links this deductive apparatus to the semantic consequence relation defined earlier.
Theorem A.21 Let E be a set of equations for the algebraic similarity type
Then for all equations s t, E j= s t iff E ` s t.
F.
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On the Canonicity of Sahlqvist Idendities
Bjarni Jónsson
Monotone Bounded Distributive Lattice
Expansions
Mai Gehrke and Bjarni Jónsson
Monotone Bounded Distributive Lattice
Expansions
Mai Gehrke and Bjarni Jonsson
Abstract
Monotone bounded distributive lattice expansions (DLMs) are bounded
distributive lattices augmented by nitary operations that are isotone or
antitone in each coordinate. Such algebras encompass most algebras with
bounded distributive lattice reducts that arise from logic, and generalize bounded distributive lattices with operators. We dene the canonical
extension for DLMs and show that this extension is functorial.
A class of compatible DLMs is said to be canonical provided it is
closed under canonical extensions. The main theorems are two criteria
for canonicity: (1) If K is a canonical class of compatible DLMs closed
under ultraproducts, then the variety generated by K is also canonical; in
particular every nitely generated variety of DLMs is canonical. (2) Any
variety of DLMs, for which the canonical extension of each basic operation
of each DLM in the class is both continuous and dually continuous, is
canonical. Both of these preservation theorems encompass varieties not
encompassed by previous results in the literature even in the setting of
Boolean algebras with additional operations.
1
Introduction
Birkho's Representation Theorem for nite distributive lattices states that
any nite distributive lattice is isomorphic to the ring of order ideals of a nite
partially ordered set. This makes nite distributive lattices particularly simple
to work with as this enables one to work with the underlying ordered set and
the set theoretic operations of intersection and union. The Stone and Priestley dualities [?, ?] generalize this representation theorem to arbitrary bounded
distributive lattices. Underlying these dualities, as well as Birkho's representation theorem is the embedding D ! P (P ) of a bounded distributive lattice
in the power set of its prime spectrum. In the nite case the image of this map
is completely determined by the order on P , namely it is the set O(P ) of order
ideals of P . In the innite case the image is not all of O(P ), and more structure
is necessary. The dualities show that one way of characterizing this embedding
is by generating a topology on P using the image of the map. However, in
working with the dualities one has to leave the realm of algebra and ordered
sets as the lattice is identied topologically. That is, algebraic questions must
1
be translated to this topological setting in order to be studied there. In [?], we
abstractly characterize the embedding D ! O(P ) within the realm of complete
lattices. This allows one to work with the useful set representation while keeping
an algebraic and order theoretic point of view. Moreover, the resulting canonical extension, is a complete lattice that is completely distributive and generated
by its completely join irreducible elements. The (complete lattice) theory of
such lattices is essentially the same as the theory of nite distributive lattices,
thus allowing the embedding of an arbitrary bounded distributive lattice in a
nite-acting one.
A setting in which the canonical extension is likely to oer an advantageous
point of view is in the presence of additional operations on the bounded distributive lattice. In order to make use of the extensions, the question is rst of
all how to extend the additional operations, and then in turn whether the extended algebras stay within the variety from which the original algebras come.
The canonical extension was rst dened and put to use in exactly this way
by B. Jonsson and A. Tarski's [?] for Boolean Algebras with Operators. The
real strength of their result is that they showed that any positive equational
property, that is, one not involving the negation, is satised in the extension if
(and only if) it is satised in the original algebra. Thus this theorem supplies
representation/extension theorems for all varieties of Boolean algebras with operators given by a set of positive equations. Due to the fact that the algebraic
counterparts of many logics are varieties (or quasi-varieties) of DLMs, these
types of algebras are very actively investigated. In particular, the tight relationship between Kripke semantics for such logics and either the topological
duals or the canonical extensions of the corresponding algebras, gives the representation/extension results a central role. For varieties of DLMs that are
BAOs, where the canonical extension and preservation of positive identities results have been available, these results have been used by some researchers, e.g.
[?, ?, ?, ?, ?]. Alternatively, one can use the topological dualities augmented
by additional relational structure. This approach has been used to some extent
for BAOs but especially for more general types of DLMs where the results and
methods of [?] do not apply, for example [?, ?, ?, ?, ?, ?, ?]. Of strong relation
to our work are the recent eorts to work out the right Kripke semantics for
non-classical logics, e.g. [?, ?, ?]. There are some general papers investigating
the relationship between varieties or quasivarieties of DLMs and their representations/extensions in topological or algebraic terms [?, ?, ?, ?, ?, ?, ?, ?]. Our
current work, including the results herein, is a further eort in this direction.
This paper contains results completed by the Spring of 1998. Since then
several conceptual advancements, including the introduction of two topologies
on canonical extensions as well as a new denition of extensions for maps have
allowed us to simplify proofs, and generalize and add to our results. In addition,
we are working on a comprehensive exposition of the consequences of these
results. Due to the enormous work involved in this endeavor, we oer these
rst results in published form to make at least some part of the ideas involved
available at this time.
In section two we briey state the results on canonical extensions of bounded
2
distributive lattices from [?] and give the denition of the most general type of
algebraic system we will consider, that is what we call monotone bounded distributive lattice expansions (DLMs). These are bounded distributive lattices
with additional operations whose restrictions to any one coordinate are either
isotone or antitone. In section three we make a detailed study of the interaction
between canonical extension, dualization, and composition for isotone maps culminating in Theorem 17 which states that the canonical extension is functorial
when the maps are taken to be algebra homomorphisms for the appropriate type
of algebras. Section four contains the main theorems on the preservation of identities. Theorem 21, which is an algebraic counterpart to Goldblatt's Theorem
3.6.7 in [?], states that if a class of compatible DLMs is closed under ultraproducts, then the variety generated by the class is also closed under canonical
extensions. As a consequence, any nitely generated variety of DLMs is closed
under canonical extensions. Theorem 23 states that any DLM for which the
canonical extension of each basic operation is both continuous and dually continuous has the same equational theory as its canonical extension. Neither of
these theorems implies or is implied by the preservation theorems in [?] and [?].
However, they allow applications to structures involving negation, implication,
and other such basic operations that these other theorems do not address.
2
Canonical Extensions of Bounded Distributive Lattices
All the lattices considered in this paper will be bounded and distributive, with
0 and 1 regarded as distinguished elements. For brevity, a bounded distributive
lattice will be referred to as a DL. The following results, with proofs can be
found in [?]. They are listed here for easy reference.
Theorem 2.1 Let A be a bounded distributive lattice (DL), then there is a
unique (up to isomorphism) DL A satisfying:
A is a DL-extension of A ;
2. A is complete, completely distributive, and join generated by the set
J (A ) of all completely join irreducible elements of A ;
3. J (A ) K (A ), where K (A ) is the complete meet closure of A in A ;
W
V
4. for all subsets K; L A, if W
K LVin A , then there are nite subsets
K 0 K , and L0 L so that K 0 L0 .
Theorem 2.2 For a complete DL L, the following conditions are equivalent:
1. L is completely distributive and join generated by J (L);
2. L is algebraic and generated by J (L);
3. L =O(J (L)) (the order ideals of J (L));
1.
3
4.
5.
L
=O(P; ) for some partially ordered set (P; );
L is doubly algebraic.
Notice that since the last condition clearly is self-dual, the dual of each of
the other conditions also are equivalent to these. In particular, it follows that
A also is meet generated by M (A ), the set of completely meet irreducible
elements in A . We call the elements of K (A ), of O(A ), and of A the closed
elements, the open elements, and the clopen elements of A , respectively. Given
a lattice A , the lattice obtained from A by reversing the order is called the dual
lattice of A and we denote it by A .
Corollary 2.3 Given a DL A , we have that (A ) = (A ) .
Proof It is clear that (A ) satises conditions 1,2, and 4 of Theorem 1
for A . To see that condition 3 is also satised let q; q 0 2 M (A ) with q q 0 .
0
Then there
V is p 2 J (A ) with p q but p q . Now since J (A ) K (A )
we have fa 2 A : p ag = p q . Since q is completely meet irreducible,
there is a 2 A with p a q , but p q 0 implies a q 0 . It follows that every
completely meet irreducible is the join of the elements of A below it, that is,
M (A ) O(A ).
Corollary 2.4 Given DLs A and B , we have that A B = (A B ) , K (A ) K (B ) = K ((A B ) ), and O(A ) O(B ) = O ((A B ) ).
Proof It is straight forward to verify that A B
satises the conditions
of Theorem 1 for A B .
We are interested in proving an extension theorem for DLs with additional
operations that ensures the preservation of identities as far as this is possible.
We want to make our class of DL expansions broad enough that it encompasses
those obtained by considering weakened negations, implications and such. This
means that we need to allow our maps to be antitone, or order reversing, in
some coordinates. We work with the following class of objects as our broadest
scope:
Denition 2.5 A DLM (monotone DL expansion) is an algebra A = (A 0 ; (fi )i2I )
so that
1.
A
0
is a DL;
i
i
2. fi : A "1 : : : A "ni ! A is isotone for the appropriate choices of "ij =
1
for each j , 1 j ni ; where ni is the arity of fi .
A class of DLMs is said to be a class of compatible algebras
provided all
the algebras in the class are similar and the indexed family "ij i2I is the same
for all algebras in the class. A compatibility type is a similarity type along
4
with the "ij 's. Notice that the DLMs of a particular compatibility type form a
subvariety of the class of all algebras of the corresponding similarity type.
If all the basic operations of a DLM are operators, that is, they preserve
nite joins in each coordinate (when viewed without the turning upside down
provided by the "ij 's), then it is called a DLO. A DLM (DLO) augmented by a
Boolean complementation of the underlying lattice is denoted by BLE (BAO).
3
Canonical Extensions of Isotone Maps
Let A and B be DL's, f : A ! B an isotone (or order preserving) map. We
want to dene f : A ! B that extends f . Recall that A is join generated
by J (A ), and that J (A ) lies in K (A ), the complete meet closure of A . For
x 2 K (A ), we let
f (x) =
^
ff (a) : x a 2 Ag.
Notice that :
The mapping f K (A ) is isotone and extends f ;
The mapping f K (A ) maps into K (B );
It is the largest possible isotone extension of f to K (A ).
Now we extend this map to all of A by dening, for u 2 A ,
f (u) =
_
ff (x) : u x 2 K (A )g.
Here again we have:
The mapping f is isotone and extends f ;
It is the least possible isotone extension of f K (A ) to all of A .
Since we will be dealing with functions of the form fi : A "1 : : : A "n ! A ,
and since we will need to compose these, it will be useful to use fi sometimes
instead of fi . So the question arises whether for isotone f : A ! B it holds that
(f ) = (f ) . Of course (f ) is the same function as f , just considered
with the order on its domain and codomain turned upside down. The function
(f ) however, is obtained by using the above denition but on the dual lattice.
The extension thus dened we call f . We spell out the details:
For y 2 O(A ), we let
i
f (y) =
_
ff (a) : y a 2 Ag.
Notice that :
The mapping f O(A ) is isotone and extends f ;
5
i
The mapping f O(A ) maps into O(B );
It is the least possible isotone extension of f to O(A ).
Now we extend this map to all of A by dening, for u 2 A ,
f (u) =
^
ff (y) : u y 2 O(A )g.
Here again we have:
The mapping f is isotone and extends f ;
It is the largest possible isotone extension of f O(A ) to all of A .
These two functions do not agree in general. We have the following relationship:
Theorem 3.1 Let A and B be DL's, f : A ! B an isotone map, then f f with equality holding for open as well as for closed elements (consequently,
we will at times denote both extensions by f when acting on closed or open
elements).
Proof Let x 2 K (A ), then f (x) f (x) since f is the
V largest extension
of f to K (A ). We need to show that f (x) f (x) = ff (y ) : x y 2
O(A )g. So let x y 2 O(A ); then since x is closed and y is open, there is
a 2 A with x a y. Thus we have f (x) f (a) = f (a) = f (a) f (y),
and we have shown that f and f agree on closed elements. By duality they
must agree on the open elements as well. Finally, it now follows that f f holds on A since f is the least isotone extension of f K (A ) to all of A .
Remark 3.2 Notice that since f maps closed elements to closed elements, it
follows from this theorem that the same is true for f . Similarly we conclude
that f maps open elements to open elements.
Denition 3.3 An isotone map f : A
provided f = f .
!B
between DL's is said to be smooth
Denition 3.4 A map f : K
! L between complete lattices is said to be (Scott)
continuous provided it preserves (up-)directed joins. The dual notion, that is, a
map that preserves down-directed meets is said to be dually continuous.
Theorem 3.5 If f is continuous, or f is dually continuous, then f is smooth.
W
Proof For each u 2 A , by denition f (u) = ff (x) : u x 2 K (A )g.
Since the set D = fx : u W
x 2 K (WA )g is directed, it follows that if f is
continuous, then f (u) = f ( W
D) = ff (x) : u x 2 K (A )g. But f = f on closed elements, so f (u) = ff (x) : u x 2 K (A )g = f (u).
6
Theorem 3.6 If f is join preserving, then f is completely join preserving,
and thus continuous. Dually, if f is meet preserving, then f is completely
meet preserving, and therefore dually continuous. In either case f is smooth.
W
W
Proof Let U be any subset of A , with U = u0 , then u2U f (u) f (u0 )
as f is isotone. Now using the denition of f and complete distributivity of
A
we get
0
_
u2U
_ B
B
@
f (u) =
u2U
1
^
y2O(A )
uy
^
=
C
f (y )C
A
_
: U!O(A ) u2U
(u)u
f ( (u))
!
we need to show that for each W: U ! O(A ) with
(u) u we have
( (u)) f (u0 ). We have
( (u)) = Wff (a) : a 2 A and
f
f
u2U
u2U
a (u) for some u 2 Ug by the denition of f on open elements. We now
claim
joins implies that this last
W that the fact that f preserves
W
W join is in fact equal
to ff (a) : a 2 A and a W u2U (u)g. It is clear Wthat ff (a) : a 2 A and
a (u) for some u 2 Ug Wff (a) : a 2 AWand a u2U (u)g. For the other
inequality, take a0 2 A, a0 u2U (u) = fa 2 A : a (u) for some u 2 Ug.
Since a0 is clopen, there are a1 ; a2 ; a3; : : : ; an 2 A and u1; u2 ; u3 ; : : : ; un 2 U
with ai (ui ) and a0 a1 _ a2 _ a3 _ : : : _ an . Now f being join preserving
implies
So
W
f (a0 )
f ( a _ a _ a _ : : : _ an )
= f (a ) _ f (a ) _ f (a ) _ : : : _ f (an )
_
ff (a) : a 2 A and a (u) for some u 2 Ug
1
2
1
3
2
3
thus proving the other inequality. So
_
u2U
f ( (u)) =
_
ff (a) : a 2 A and a _
= f (
u2U
(u))
_
u2U
(u)g
f (u )
0
W
and we can conclude that u2U f (u) f (u0 ), and consequently equality
holds, thus proving that f is completely join-preserving.
As we proved in [?], the weaker condition of f being an operator implies
that f is completely join preserving in each coordinate, and thus, as it has
only nitely many coordinates, f is continuous. However, f being an operator
is not enough to guarantee smoothness.
7
Example 3.7 This is an example of a binary operation on a DL which is both
an operator and a dual operator but f 6= f . Let A = f n1 ; n1 : n 2 Ng as
ordered in Q . Dene f : A A ! A by
f (a; b) =
8
>
>
<
>
>
:
1 if b > 0
a if a > b > 0
b if 0 < a b
1 if a; b < 0
In order to check that f is meet and join preserving in both coordinates, we just
need to check that f is isotone in both coordinates since A is a chain. This can
be done by checking in all the possible cases for each coordinate. The canonical
extension A of A is the chain obtained when adding two elements x and y to
A ; both between the positive and the negative pieces of A and with y < x. It can
be checked that f (x; y ) = y < x = f (x; y ).
We now turn to the question of composability of and . This will play an
important role in determining whether or not equational properties are preserved
by taking canonical extensions.
Theorem 3.8 Let g : A
(fg )
! B, f : B ! C
f g
f g
f g
be isotone maps on DL's. Then
f g (fg)
with equality holding on closed elements and on open elements of A .
Proof If we just prove that (fg)
f g
holds, then the rest follows by
Theorem 6 and duality. Furthermore, in order to prove that (fg ) f g holds
on all of A it is enough to show that it holds on K (A ) since (fg ) is the least
isotone extension of (fg ) from K (A ) to A . For x 2 K (A )
(fg ) (x) =
^
ffg(a) : x a 2 Ag
f g (x) =
ff (b) : g (x) b 2 B g
V
Let b 2 B with g (x) = fg (a) : x a 2 Ag b. Then, since b is clopen, there
are a ; : : : ; an 2 A, all greater than or equal to x, with g (a ) ^ : : : ^ g (an ) b. Now we let a = a ^ : : : ^ an , and we have a 2 A, x a, and g (a) =
g (a ^ : : : ^ an ) g (a ) ^ : : : ^ g (an ) b, and we conclude that (fg) (x) ^
1
1
1
1
f g (x).
1
Example 3.9 We give an example to show that we don't always have equality in
the above theorem. Let B be a Boolean algebra, let f : B ! B be the function that
sends everything to zero, except 1 which gets sent to 1. Also, let g : B B ! B
denote the join operation in B . Then it is straight forward to check that (fg ) <
f g , for example, (fg) (x; x0 ) = 0 < 1 = f g (x; x0 ) as soon as x is not in B .
Notice that f is meet preserving, and that g is both join preserving and a dual
operator.
8
Proposition 3.10 Let g : A ! B , f : B ! C be isotone maps on DL's. If
f is continuous, then (fg) = f g . Dually, if f is dually continuous, then
(fg ) = f g .
Proof Recall that
(fg ) (u) =
_
f(fg) (x) : u x 2 K (A )g
_
=
ff g (x) : u x 2 K (A )g
and notice that D = fg (x) : u x 2 K (A )g is a directed set in B W
D = g (u). Using the fact that f is continuous, we get:
_
(fg ) (u) =
ff (d) : d 2 Dg
_ = f
D
with
= f (g (u))
As mentioned above, if f is an operator, then f is continuous and then
is compositional. This is the essence of the proof of the canonical extension
theorem for DLO's in [?]. However, a new result is that we also can guarantee
compositionality if the rst (or inner) map is 'nice' { it needs to be very nice
though:
Theorem 3.11 Let g : A ! B , f : B ! C be isotone maps on DL's. If g
is meet preserving, then (fg ) = f g . Dually, if g is join preserving, then
(fg ) = f g .
Proof We know that (fg) f g with equality on closed elements and on
open elements holds in general. For u 2 A (fg ) (u) =
_
ff g (x) : u x 2 K (A )g
_
f g (u) =
ff (s) : g (u) s 2 K (B )g
We show that for each s 2 K (B )\ # g (u), there is x 2 K (A )\ # u with
s g (x). Let s 2 K (B )\ # g (u), then
^
s g (u) = g (u) = fg (y) : u y 2 O(A )g
W
That is, for each y 2 O(A )\ " u, s g (y ) = fg (a) : y a 2 Ag and since
s is closed, there are a ; : : : ; an 2 A\ # y with s g (a ) _ : : : _ g (an ). We let
ay = a _ : : : _ aVn 2 A\ # y, then s g (a ) _ : : : _ g (V
an ) g (a _ : : : _ an ) =
g (ay ), so s fg(ay ) : y 2 O(A )\ " ug = g ( fay : y 2 O(A )\ " ug)
1
1
1
1
1
since g is V
meet preserving, making g completely
meet preserving. So we nally
V
let x = fay : y 2 O(A )\ " ug fy : y 2 O(A )\ " ug = u then
x 2 K (A )\ # u and s g (x).
For the proof of the next theorem we need a denition:
9
Denition 3.12 Let fi : A i ! B i for i = 1; : : : ; n. Then we dene the superposition of f1 ; : : : ; fn to be the function hf1 ; : : : ; fn i : A 1 : : : A n ! B 1 : : : B n
dened by hf1 ; : : : ; fn i (a1 ; : : : ; an ) = (f1 (a1 ); : : : ; fn (an )).
Remark 3.13 It is straightforward to check that hf1 ; : : : ; fn i = hf1 ; : : : ; fn i
whenever the fi 's are isotone.
Theorem 3.14 Let be a compatibility type for DLMs, and let A be the category of all DLMs of compatibility type with algebra homomorphisms of the
corresponding type, then : A ! A is a functor. Moreover, preserves and
reects injections and surjections.
Proof On objects : Ob(A) ! Ob(A) is dened by (A 0 ; (fi )i2I ) =
(A 0 ; (fi )i2I ), and on maps : Hom(A ; B ) ! Hom(A ; B ) is dened by
(h) = h . Since the homomorphisms are meet preserving, is compositional.
The only thing to check is whether h is a DLM homomorphism. Let f A :
A "01 : : :A "0n ! A 0 and f B : B "01 : : :B "0n ! B 0 be corresponding operations of
A
B
"1
"n
A and B , respectively.
Suppose
h : A 0 !
B 0 with h Æ f = f Æ hh ; : : : ; h i.
A
A
A
Then h Æ f = h Æ f
= hÆf
since
h is join preserving, so that
A
A
h is continuous. Thus h Æ f = h Æ f
= f B Æ hh"1 ; : : : ; h"n i . Now
hh"1 ; : : : ; h"n i : A "01 : : : A "0n ! B "01 : : : B "0n is a homomorphism since h is
a homomorphism and since the dual of a homomorphism is again a homomorphism. So hh"1 ; : : : ;h"n i is meet
preserving and thus, by
the previous theorem,
f B Æ hh"1 ; : : : ; h"n i = f B Æ (hh"1 ; : : : ; h"n i) = f B Æh(h"1 ) ; : : : ; (h"n ) i.
Finally, since h"i is a homomorphism, it is smooth,
and (h"i ) = (h )"i for each
i 2 f1; : : : ; ng. We have shown that h Æ f A = f B Æ h(h )"1 ; : : : ; (h )"n i, that
is, h is a DLM homomorphism.
We now need to show that preserves injections and surjections. This
is independent of any consideration involving the additional operations and
we consider h : A ! B , a DL homomorphism. First suppose h is injective.
Let s; x 2 K (A ) and suppose
that h (x) = h (s). Let a 2 A\ " x, then
V
h(a) h (x) = h (s) = fh(b) : s b 2 Ag. Since h(a) is clopen there
are b1 ; : : : ; bn 2 A\ " s with h(a) h(b1 ) ^ : : : ^ h(bn ) h(b1 ^ : : : ^
V bn ).
So b = b1 ^ : : : ^ bn s and a b since h is injective. That is, s fa :
x a 2 Ag = x. By symmetry, s = x and h is injective on the closed
elements. Now let u; v 2 A with h (u)V= h (v ). Let s 2 K (A )\ # u,
then h (s) h (u) = h (v ) = h (v ) = fh (y ) : v W
y 2 O(A )g. Thus,
for each y 2 O(A )\ " v we have h (s) h (y ) = fh(a) : y a 2 Ag.
Since s is closed, so is h (s), and thus there are a1 ; : : : ; an 2 A\ # y with
h (s) h(a1 ) _ : : : _ h(an ) h(a1 _ : : : _ an ). For each y 2 O(A )V\ " v we
let ay = a1 _ : : : _ an , then
h (s) fh(ay ) :
V ay y and h (s) h(ay ). So
y 2 O(A )\ " vg = h ( fay V
: y 2 O(A )\ " v g) since h is completely Vmeet
preserving. Finally we let x = fay : y 2 O(A )\ " v g 2 K (A ), and x fy :
y 2 O(A )\ " vg = v while h (s) h (x). Now, since
W h is injective on closed
elements, it follows that s x v . That is, u = fs : u s 2 K (A )g v
and by symmetry u = v . We have shown that h is injective.
10
We
V show that if h : A ! B is surjective, then so is h . Let s 2 K (B ), then
s = fbV: s b 2 B g. For each b 2 B \ " s, there is ab 2 A with h(ab ) = b.
Let x = fab : s b 2 B g. Since
h is meet preserving,V h is completely meetV
preserving and thus h (x) = fh(ab ) : s b 2WB g = fb : s b 2 B g = s, so
h is onto K (B ). Now let v 2 B , then v = fs : v s 2 K (B )g. Since h
is onto K (B ), we haveWthat for each s 2 K (B )\ # v there is xs 2 K (A ) with
h (xs ) = s. Let u =W fxs : s 2 K (B )\ # vg, then
join
W as h is completely
preserving, h (u) = fh(xs ) : s 2 K (B )\ # v g = fs : s 2 K (B )\ # v g = v ,
and h is surjective.
The fact that reects injections is clear; the fact that reects surjections
is also quite easy to show as there being a u 2 A with h (u) = b 2 B easily
is seen to imply that there also must be a 2 A with h(a) = b by compactness
arguments similar to the ones given above.
Remark 3.15 Notice that the image actually falls in the category
DL
of all
doubly algebraic distributive lattices (or equivalently, complete, completely distributive lattices, join generated by their completely join irreducible elements)
with complete homomorphisms. It is however not clear exactly what the image
objects are. In particular, the underlying distributive lattices are exactly those
lattices O(P ) for which P is a spectral set, that is, a poset that is order isomorphic to the set of prime ideals or lters of a DL ordered by inclusion. No order
theoretic description of these posets is known.
+
Our goal is to determine when properties, such as identities, are preserved
when going to the canonical extension. The above theorem gives an immediate
corollary in this direction. We need the following terminology:
Denition 3.16 A class K of (compatible) DLMs is canonical provided it is
closed under . A single DLM A is canonical provided V (A ), the variety generated by A , is canonical. A set of identities , or more generally a property, is
canonical provided its class of models is canonical.
Corollary 3.17 If K is a class of compatible DLMs that is closed under products and that is canonical, then V (K ) is canonical.
In the following section we explore the preservation of identities further.
4
Preservation of identities
We rst sharpen the above corollary by using the fact that any product can be
viewed
as a Boolean product of ultraproducts. We rst sketch this fact: Let
B = Qi2I B i , and let X be the Stone space of the Boolean
algebra P (I ). Then
Q
each 2 X is an ultralter of P (I ). We let B = i2I B i =, the ultraproduct
of fB i gi2I Q
with respect to the ultralter . Then it is straight forward to check
that B ,! 2X B where x 7! (x=)2X is a Boolean product.
We also need the following theorem which appeared in its topological form
in [?].
11
Theorem 4.1 Q
Let fA x gx2X be a family of compatible DLMs, X Q
a Boolean
space, and A x2X A x a Boolean product of fA x gx2X . Then A = x2X A x .
Q
x2X A x satises the conditions for
being
the
canonical
extension
of
the
DL
reduct
of
A
. We use the same notation
Q
for x2X A x , A and
their
DL
reducts
in
order
to
simplify
the notation.
Q
Q
It is clear that x2X A x is a DL extension of A . It is also
Q clear that x2X A x
is complete, completely distributive, and generated by J ( x2X A x ) (which consists of those X -tuples that have a completely join irreducible in one coordinate,
and 0's in all theQothers).
Let p; q 2 J ( x2X A x ) with p q . Then there are x; y 2 X with pz = 0
for each z 6= x, and qz = 0 for each z 6= y . If x 6= y then there is a clopen set
V X with x 2= V and y 2 V . Since 0 = (0x )x2X and 1 = (1x )x2X belong
to A , a = 0 V { [ 1 V also belongs to A by the patching property. It is
now clear that a satises p a while q a. On the other hand, if x = y then
px; qx 2 J (A x ) and px qx , so there is ax 2 A x with px ax and qx ax .
Since A is a subdirect product of fA x gx2X , there is a 2 A with ax = ax , and
we have p a but q a.
Finally we need to check the generalized compactness property. This is
the part thatWrequiresVthat the
product be Boolean. Let K; L A
Q subdirect
. Then for each x 2 X , there are nite
and suppose K L in x2X A W
x
V
subsets Kx0 KW, L0x LVso that ( VKx0 )x ( L0x )x . For each x 2 X , let
Vx = fy 2 X : ( Kx0 )y ^ ( L0x )y = ( L0x )y g then the Vx 's are open and they
cover X . Since X is compact, it follows that there are x1 ; : : : ; xn 2 X so that
0
0
0
VWx1 ; : : : ; VxnVcover X . Let K 0 = Kx0 1 [ : : :W[ Kx0 n and
V 0L = Lx1 [ : : : [ Lxn , then
0
0
0
( K )y ( L )y for all y 2 X , that is, K L as desired.
In addition, for an n-ary order preserving operation f on A we must show
that for all u 2 A n , f (u) = (fx (ux ))x2X . We have
Proof We show that the DL reduct of
f (u) =
whereas for x 2 X
fx (ux ) =
_ ^
f ff (a) : s a 2 An g : u s 2 K (A )n g
_ ^
f ff (b) : t b 2 Anx g : ux t 2 K (A x )n g.
We show that for x 2 X and u 2 A ,
ft 2 K (A x ) : ux tg = fsx : u s 2 K (A )g
and for x 2 X and s 2 K (A ),
fb 2 Ax : sx bg = fax : s a 2 Ag.
Once this has been showed it follows that f = (fx )x2X . Let x 2 X and
u 2 A . Clearly if u s 2 K (A ), then uQx sx 2 K (A x ). For the reverse
inclusion, let ux t 2 K (A x ). Let s 2 y2X A y be given by sx = t and
sy = 0 for y 6= x. Then clearly u s, and we show that s 2 K (A ). This
12
V
S
follows since s = fa V 0 V { : a 2 A with ax t and V is a clopen
neighborhood of x in X g. Finally let x 2 X and s 2 K (A ). If s a 2 A
then certainly sx ax 2 Ax . For the reverse inclusion, let sx b 2 Ax , then
V
fax : s a 2 Ag = sx b. So by compactness in A x , there is a 2 A with
s a and ax b. Also there is c 2 A with cx = b. Now take d = a _ c. Then
d 2 A and s d and dx = b.
Theorem 4.2 If K is a class of compatible DLMs that is closed under ultraproducts and canonical, then V (K ) is also canonical.
Corollary 4.3 Any nitely generated variety of compatible DLMs is canonical.
This applies to quite a few varieties, such as for example
Corollary 4.4 All the proper subvarieties of the variety of pseudo-complemented
DL's are canonical.
Corollary 4.5 Varieties of de Morgan algebras are all canonical.
The second main result we present here is based on knowledge of the smoothness, continuity and dual continuity of the operations in the DL expansions.
For a single DLM, A , preservation of identities when going to the canonical
extension can be phrased in terms of the relationship between the two kernels:
T(X )
Clon (A )
%
j
j
#
&
Clon (A )
where X = fx1 ; : : : ; xn g. The identities that hold in A , Id(A ), is exactly the
kernel of the surjection T(X ) ! Clon (A ), and the identities that hold in A ,
Id(A ), is exactly the kernel of the surjection T(X ) ! Clon (A ). So saying
that Id(A ) Id(A ) is equivalent to showing that there is a homomorphism
: Clon (A ) ! Clon (A )
making the diagram commute (on generators). In order to show this for a given
map , we need:
1. For each j = 1; : : : ; n, the map commutes on the j th projection: jA
jA ;
2. For each i 2 I , and for h1 ; : : : ; hni
fiA Æ [h1 ; : : : ; hni ]
=
2 Clon (A ) we have fiA Æ [h ; : : : ; hni ]
13
1
=
Here [h1 ; : : : ; hni ] : A n
! A ni
is dened by
[h1 ; : : : ; hni ] (a) = (h1 (a) ; : : : ; hni (a)) :
On the isotone part of the clone we have a map ; namely our extension (or
if appropriate).
It is easy to check that the rst condition above holds. We
also have fiA = fiA for all the basic operations: it is true for the additional
operations in the DL expansion by denition, and it is straight forward to check
for meet and join. So, as long as the whole clone is isotone, we just need to
know that is compositional, i.e., fiA Æ [h1 ; : : : ; hni ] = fiA Æ [h1 ; : : : ; hni ]
for each i 2 I and for h1 ; : : : ; hni 2 Clon (A ).
Here we see how the proof works for DLO's: the whole clone
is isotone, the
compositionality works because fiA is an operator, that is, fiA is continuous
for each i 2 I . We can also use our knowledge about and on isotone maps
to get a theorem that encompasses some DLMs with maps that are antitone in
some coordinates:
A is a DLM for which the extension of each basic operation
is both continuous and dually continuous, then A is canonical.
Theorem 4.6 If
To simplify the discussion we give this hypothesis on maps a name:
Denition 4.7 An isotone map f : A ! B is said to be -doubly continuous
provided f is both continuous and dually continuous.
Notice that if f is -doubly continuous then f is smooth. Also, notice that
for an isotone map to be -doubly continuous it is suÆcient that one of the
following conditions hold:
1. f is an operator and is meet preserving;
2. f is a dual operator and is join preserving.
This is of course not at all necessary. It does however show that the meet and
join operations of a DL always are -doubly continuous, so that the hypothesis
of the theorem does not restrict the class of DL reducts allowed. In addition,
this clearly holds for operations that are either lattice homomorphisms from a
product of copies of the underlying lattice and its dual to the underlying lattice.
This holds, for example, for all Ockham algebras. Thus
Corollary 4.8 All varieties of Ockham algebras are canonical.
In addition, it can be shown that -double continuity of the basic operations
is preserved by taking subalgebras, homomorphic images, and Boolean products.
Thus, the basic operations of any DLM that lies in the variety generated by a
nite DLM (as in Corollary 21) are -doubly continuous.
For the proof of the theorem we need two lemmas, the rst of which follows
from our results in the previous section.
14
Lemma 4.9 Let g : A
! B and f : B ! C be isotone maps that are -doubly
continuous then the same holds true for the composition fg . Also, superposition
preserves -double continuity.
Proof This follows since continuity of f implies (fg) = f g , dual continuity of f = f implies(fg ) = f g , and continuity and dual continuity
is preserved by composition. The statement for superposition is clear since
everything is done coordinate-wise.
The additional complication in this setting is that not all of Clon (A ) may
be considered as isotone even with the use of dualization of coordinates. For
example, if is the pseudo-complementation in a pseudo-complemented DL,
x 7! x _ x is in general neither isotone nor antitone.
However,
every function
in Clon (A ) can be obtained in the form f A Æ iA1 ; : : : ; iAk where f is obtained
from the basic operations using superposition and composition, and iAj is the
ij th projection of An onto A;where ij 2 f1; : : : ; ng for each j 2 f1; : : : ; kg (given
a term t, obtaining the term f corresponds to replacing repeated variables by
distinct, previously unused, variables).
The commutativity of the diagram corresponding to
(A ) Id
(A ) tells us
Id
A
A
A
that we must extend to Clon (A ) by letting f Æ i1 ; : : : ; ik
= fA Æ
A
A
A
i1 ; : : : ; ik . But since f is the composition of -doubly continuous operations
that
be viewed as isotone
with
domains
and ranges,
can all
matching
f A = f A , so f A Æ iA1 ; : : : ; iAk = f A Æ iA1 ; : : : ; iAk . What remains
to be seen is that this last step yields a well-dened extension of to Clon (A ).
For this purpose we need the following lemma:
1
Lemma 4.10 Let f : A "1
: : : A "1k ! A and g : A "21 : : : A "2l ! A
isotone maps, and suppose that for each a 2 An
f iA1 (a) ; : : : ; iAk (a) g jA1 (a) ; : : : ; jAl (a) ,
where 1 i ; : : : ; ik ; j ; : : : ; jl n
then for each u 2 (A )n
1
f iA1 (u) ; : : : ; iAk (u)
be
1
g jA1 (u) ; : : : ; jAl (u)
.
Before we prove this lemma, we see how this proves that as extended to
Clon (A ) above is well-dened. Let t be an
n-ary term, and suppose
that for
all a 2 An tA (a) = f A iA1 (a) ; : : : ; iAk (a) = g A jA1 (a) ; : : : ; jAl (a) where f
and g are terms without repeated variables. Then f A and g A can be considered
as isotone and by lemma 26 we have for all u 2 (A )n
A
A
(f A ) iA1 (u) ; : : : ; iAk (u)
(g
A
)
j1 (u) ; : : : ; jl (u)
(gA ) jA1 (u) ; : : : ; jAl (u)
15
and (f A ) iA1 (u) ; : : : ; iAk (u) .
Now since f A and g A are -doubly continuous, it follows that
f (A
)
iA1 (u) ; : : : ; iAk (u) = g(A
)
jA1 (u) ; : : : ; jAl (u)
for all u 2 (A )n and the last step of the extension of is well-dened. Now for
the proof of the lemma:
Proof Suppose the conclusion not to hold. That is, there are u 2 (A )n and
p 2 J (A ) with
p f iA1 (u) ; : : : ; iAk (u)
but
p g jA1 (u) ; : : : ; jAl (u)
Suppose (WLOG) that "11 ; : : : ; "1k1 = 1, and "1k1 +1 ; : : : ; "1k = , that is, f is
isotone in the rst k1 variables, and antitone in the last k2 = k k1 variables.
Similarly for g with l1 + l2 = l. Given a k -tuple ( l-tuple) we write it as
v = (v 1 ; v2 ) where vi is the ki -tuple (li -tuple) consisting of the rst, or last ki
(li ) entries of v for i = 1; 2. There should be no confusion as to whether we are
talking about k -tuples or l-tuples as this will depend on whether we are talking
about f or g .
For v 2 (A )k we have
f (v) = f (v 1 ; v2 )
_
=
ff (x; y) : v1 x 2 K (A )k1 ; v 2 y 2 O(A )k2 g
f iA1 (u) ; : : : ; iAk (u) means that there exist x2 K (A )k1 and
y 2 O(A )k2 with p f (x; y) and x (iA1 (u) ; : : : ; iAk1 (u)), and y (iAk1 +1 (u) ; : : : ; iAk (u)). Similarly for v 2 (A )l we have
Now p
g (v) = g (v1 ; v2 )
^
=
fg (y; x) : v 1 y 2 O(A )l1 ; v2 x 2 K (A )l2 g
A Now p g jA1 (u) ; : : : ; jAl (u) means that there exist y0 2 O(A )l1
and x0 2 K (A )l2 with p g (y0 ; x0 ) and y 0 (jA1 (u) ; : : : ; jAl1 (u)), and
x0 (jAl1 +1 (u) ; : : : ; jAl (u)). For each m between 1 and n we let
xbm =
_
fx : i = m; 1 k g
_
_ fx0 : jl1 = m; 1 l g
1
+
and
ybm =
2
^
fy : ik1 = m; 1 k g
^
^ fy0 : j = m; 1 l g
+
2
1
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Then since x iA (u) and x0 jAl1 + (u) we get xbm um . Similarly
um ybm . Let x0 2 K (A )k1 where (x0 ) = xbm if and only if i = m for
1 k1 . Then
x x0 (iA1 (u) ; : : : ; iAk1 (u))
(1)
Similarly, we let y 0
1 k2 . Then
2 O(A )k2
where (y 0 ) = ybm if and only if i +k1 = m for
y y0 (iAk1 +1 (u) ; : : : ; iAk (u)).
(2)
We go through the same constructions for g , y 0 , and x0 thus getting y 00 2 O(A )l1
where (y 00 ) = ybm if and only if j = m for 1 l1 so that
y0 y00 (jA1 (u) ; : : : ; jAl1 (u))
(1')
and x00 2 K (A )l2 where (x00 ) = xbm if and only if j +l1 = m for 1 l2 so
that
x0 x00 (jAl1 +1 (u) ; : : : ; jAl (u)).
(2')
>From (1) and (2) we get f (x; y ) f (x0 ; y 0 ) so that p f (x0 ; y0 ). From
(1') and (2') we get g (y 0 ; x0 ) g (y 00 ; x00 ) so that p g (y 00 ; x00 ). Now for each
m between 1 and n, since xbm um ybm , there is am 2 A with xbm am ybm .
We let a = (a1 ; : : : ; an ), then by the denitions of x0 , y 0 , y00 , x00 , and a we get
x0
y0
y00
x00
(iA1 (a) ; : : : ; iAk1 (a))
(iAk1 +1 (a) ; : : : ; iAk (a))
(jA1 (a) ; : : : ; jAl1 (a))
(jA11 +1 (a) ; : : : ; jAl (a))
so that
p
f (x ; y ) f (iA1 (a ) ; : : : ; iAk (a ))
p
0
0
0
0
and
g (y00 ; x00 ) g(jA1 (a0 ) ; : : : ; jAl (a0 ))
thus contradicting our hypothesis that for each a 2 An we have
f iA1 (a) ; : : : ; iAk (a)
5
g jA1 (a) ; : : : ; jAl (a)
.
Further work
This paper gives an idea of the types of general canonicity results one can
obtain for distributive lattices with additional operations that are either order
17
reversing or preserving in each coordinate. In ongoing work we have generalized
the denition of canonical extensions of maps to encompass all maps on DLs,
and through the introduction of two topologies on canonical extensions of DLs,
we are able to greatly simplify the proofs of the results given here as well as
generalize them. For example, one can show that all nitely generated varieties
of algebras with a DL reduct are canonical. A comprehensive paper detailing
these new concepts and generalized results and giving applications to many
specic situations of interest, such as residuation in one and two variables,
quasi-identities etcetera is currently in preparation.
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Mai Gehrke,
New Mexico State University,
Department of Mathematical Sciences,
Las Cruces, NM88003,
USA
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Work phone 505-646-4218
Work fax 505-646-1064
e-mail: [email protected]
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