Stability of The Blok Theorem
Tadeusz Litak
Abstract. Blok [2] proved that varieties of boolean algebras with a single unary operator uniquely determined by their class of perfect algebras (i.e., duals of Kripke frames)
are exactly those which are intersections of conjugate varieties of splitting algebras. The
remaining ones share their class of perfect algebras with uncountably many other varieties.
This theorem is known as The Blok Dichotomy or The Blok Alternative. We show that The
Blok Dichotomy holds when perfect algebras in the formulation are replaced by ω-complete
algebras, atomic algebras with completely additive operators or algebras admitting residuals. We also generalize The Blok Dichotomy for lattices of varieties of boolean algebras
with finitely many unary operators. In addition, we answer a question posed by Dziobiak
[10] and show that classes of lattice-complete algebras or duals of Scott-Montague frames
in a given variety are not determined by their subdirectly irreducible members.
1. Introduction
One of the most fascinating results obtained by Wim Blok was a complete characterization of degrees of Kripke incompleteness of normal modal logics [2]. The
theorem, known as The Blok Alternative or The Blok Dichotomy, can be expressed
in the algebraic language as follows: a variety V of boolean algebras with a single
normal unary operator is either uniquely determined by its class of duals of Kripke
frames (perfect algebras) or there are uncountably many other varieties which cannot be distinguished from V this way. The main goal of the present research is to
examine how relevant the restriction to perfect algebras is, i.e., whether the theorem holds when duals of Kripke frames in the formulation are replaced by other
classes of algebras. Below, we sketch some background, but the reader may feel
the need for a more detailed introduction: Chagrov and Zakharyaschev [6] (see also
[36] and [35]) provide all the necessary mathematical information on modal logic,
Venema [33] explains the connection between modal logic and algebra, Goldblatt
[13] describes the development of the field, while Rautenberg et al. [23] gives an
account of the motivation behind The Blok Theorem itself, the troubled history of
its publication and its influence on the modal logic community.
1.1. Motivation and historical background. We are going to work with normal modal logics based on classical propositional calculus. Algebraically, these
systems correspond to boolean algebras with normal operators (baos); thorough
1991 Mathematics Subject Classification: 06E25, 03G05, 03B45.
Key words and phrases: BAOs, ω-complete algebras, atomic algebras, completely additive
operators, residuals, duality, modal logic.
1
2
TADEUSZ LITAK
the paper, we are assuming that all operators are unary and there are only finitely
many of them. The concept of an operator on a boolean algebra was introduced
by Jónsson and Tarski’s papers [16, 17] (see also Jónsson [15]): it is an operation
which distributes over finite joins on every coordinate. An operator is normal if it
preserves bottom; thorough the paper, we restrict attention to normal operators.
If, in addition, the operator distributes over all existing (not only finite) joins, it
is called completely additive. One of the main results of Jónsson and Tarski was
an extension of Stone’s representation theorem for boolean algebras. They proved
that every bao can be embedded into a perfect algebra, i.e., a lattice-complete and
atomic bao where all operators are completely additive. The class of perfect algebras will be denoted as CAV. A variety V is called (CAV-) complete if it is of the
form V = HSP (V ∩ CAV), i.e., if it is generated as a variety from a set of CAV’s.
Jónsson and Tarski [16, 17] proved that CAV’s are exactly those baos which
arise as dual algebras of relational structures. As relational structures in modal
logic came to be known as Kripke frames, perfect algebras are also sometimes called
Kripke algebras. This duality between CAV’s and frames can be, in fact, extended
to the level of (complete) morphisms to obtain a full-blown category-theoretical
duality—see Remark 2.19 below. This explains the use of the notion completeness
above: a variety is CAV-complete iff the corresponding logic is Kripke complete (cf.
any of references mentioned at the beginning of the paper for definitions of logical
notions appearing henceforth). Thomason [28] showed that there is a nontrivial
variety of boolean algebras with two conjugated unary operators which does not
contain any perfect algebra, i.e., there are no Kripke frames adequate for the corresponding logic. It follows from Makinson’s Theorem (cf. Theorem 3.1 below) that
there is no such variety of boolean algebras with one unary operator, i.e., modal
algebras. Nevertheless, Fine [11] found an example of an incomplete variety of closure algebras (or S4-algebras), that is, modal algebras satisfying all the equations
which hold for the closure operator Int on the powersets of arbitrary topological
spaces.
The final section of Fine’s paper posed a problem, which in algebraic terms may
be formulated as follows. If a variety V of boolean algebras with n operators—
strictly speaking, Fine was interested in the case of modal algebras—is not complete,
there is another variety V 0 6= V satisfying the following ’equation’:
V 0 ∩ CAV = V ∩ CAV
(cav-inc)
For example, the largest complete subvariety V 0 = HSP (V ∩ CAV) of V satisfies
cav-inc. More generally, we can say V 0 is CAV-equivalent to V if cav-inc holds. If
one feels ontologically uncomfortable about forming equivalence classes of varieties,
i.e., classes of proper classes, it is possible to represent varieties by their equational
theories. Denote by δCAV the function which assigns to every V the cardinality
of its CAV-equivalence class. Fine called the value of that function the degree of
(Kripke) incompleteness. Thomason and Fine’s results showed that in general δCAV
can have a value different than one. It is clear that the variety of all modal algebras
has the degree of incompleteness equal to one. Fine called such varieties strictly
STABILITY OF THE BLOK THEOREM
3
complete and asked if there are other strictly complete varieties. More generally,
he asked how δCAV behaves and what possible values it can take. A solution would
make clear how rare the incomplete varieties are and in what regions of the lattice
of varieties of baos they can appear.
Wim Blok [2] (cf. also [3]) provided a totally unexpected answer. Just like Fine,
he studied only modal algebras, but as we shall see, his analysis extends to the case
of finitely many
unary operators. He showed that the only strictly complete varieties
V
are the Π -prime ones, i.e., varieties corresponding to join-splitting logics; the
exact definition and explanation of their special status is given in Section 2.2 below.
All the other varieties—and these include most modal varieties of independent
interest, such as T, B, K4, the variety of closure algebras, diagonalizable algebras,
monadic algebras or Grzegorczyk algebras—have
V the degree of incompleteness equal
to continuum. That is, if a variety V is not Π -prime, then there are uncountably
many other varieties V 0 satisfying cav-inc.
How one is to understand such an amazing result? It would be tempting to
say that Blok’s result implies that most varieties of modal algebras are incomplete,
but—as pointed out by the referee of the present paper—that would be doubly
misleading. First, varieties of modal algebras introduced ’in a natural way’ are
complete, even if not strictly complete. Second, there are uncountably many joinsplitting logics, so in terms of cardinality there are exactly as many strictly complete
varieties as incomplete ones. An interesting analogy is suggested by Blackburn et al.
[1]. An incomplete variety can be compared to a continuous nowhere-differentiable
function. There are no such functions among those encountered at first stages of
the study of calculus: polynomials or other functions appearing in basic physics.
Yet more advanced students of calculus soon learn that being continuous does not
guarantee being differentiable at any point whatsoever—and still more advanced
students learn that the set of functions differentiable at at least one point is actually of measure zero among all continuous ones. One may loosely say that every
differentiable function is surrounded by a cloud of nowhere-differentiable continuous ones—and in this sense, being differentiable is a pathology among continuous
functions. The same can be said about the relationship between complete and incomplete varieties. Except for the case of varieties corresponding to join-splitting
logics, no variety is uniquely characterized by its class of perfect algebras. Complete
varieties are, in a sense, surrounded by an uncountable cloud of incomplete ones:
one has to look at non-perfect algebras to find out the difference.
This immediately raises a question: what happens if we do not restrict our attention to perfect algebras and consider larger classes of structures? Can we obtain a
class of algebras which allows for finer distinction between varieties than CAV’s and
yet preserves some of their nice properties? Is the conjunction of atomicity, latticecompleteness and complete additivity necessary to prove The Blok Dichotomy?
All three properties hold trivially for finite algebras; as was said, they also hold
for duals of relational structures. Atomicity means that every element is determined
by the class of atoms below it; we may think of atoms as individuals or points.
Combination of atomicity and complete additivity means that not only elements
of algebra can be thought of as collections of atoms, but also the behaviour of all
4
TADEUSZ LITAK
operators is determined by their behaviour on atoms. On the other hand, for latticecomplete algebras, an operator is completely additive iff it is conjugated (see Section
2 for a definition) iff it is residuated. For non-complete algebras, conjugation (or
residuation) implies complete additivity, but the converse does not hold even under
the assumption of atomicity. Algebraic importance of all these properties is clear,
but they are also of independent interest to logicians studying extended modal
formalisms like hybrid logic or tense logic; cf. Remark 2.7 for more information. If
we focus on algebras which are atomic, completely additive or residuated but not
necessary lattice-complete, can we still prove The Blok Dichotomy? Or perhaps
this results depends more on the fact that the duals of relational structures are
closed under joins of arbitrary families of elements?
Some partial results in this direction have been obtained before. The most interesting observation is contained in Zakharyaschev et al. [36] The authors note that
The Blok Dichotomy for varieties of modal algebras holds when perfect algebras
are replaced by algebras admitting residuals. Also, there is a set-theoretical semantics for modal logic which generalizes the Kripke semantics of relational structures:
this is the Scott-Montague semantics of neighborhood frames. These structures are
dual representations of atomic and lattice-complete (but not necessarily completely
additive!) algebras, cf. Došen [9] for details. Dziobiak [10] attempted to generalize The Blok Dichotomy for neighborhood semantics, obtaining interesting partial
results. Nevertheless, Blok’s proof used the fact that every perfect algebra can be
subdirectly embedded into a product of its complete-homomorphic images—which
are also CAV’s and hence (isomorphic to) perfect algebras. Dziobiak [10] claimed
it is not possible to complete the proof along the lines of Blok [2] without an analogous result for duals of Scott-Montague frames. It was left as an open question
whether such a result can be obtained. Later, Chagrova [7] generalized The Blok
Blok Dichotomy for degrees of neighborhood incompleteness using a different technique developed by Chagrov and Zakharyaschev [6] and thus avoiding the problem
which plagued Dziobiak. His question, however, was left unanswered.
Another question is whether the restriction to varieties of boolean algebras with
just one unary operator is difficult to overcome. Zakharyaschev et al. [36] claimed,
without proof, that it can be removed. But this claim requires some justification.
All the existing proofs of The Blok Dichotomy, including the one provided in [36],
used The Makinson Theorem describing the atoms of the lattice of varieties of
normal modal algebras. As was mentioned above, the theorem does not hold in the
setting with at least two operators, but Zakharyaschev et al. [36] do not indicate
how to deal with this problem.
1.2. The present paper. The paper paper extends The Blok Theorem both
in width and in depth using the model-theoretic methods of Chagrov and Zakharyaschev [6], [36], [35]. Concerning the number of operators, we show that
the theorem holds for varieties of modal algebras with finitely many normal unary
operators. The failure of Makinson Theorem is overcome by the use of Minimal
Variety Theorem obtained in Section 3, which seems a new observation.
STABILITY OF THE BLOK THEOREM
5
As for completeness with respect to broader classes of algebras than CAV, we
strengthen the result of Chagrova [7] to degrees of incompleteness with respect to
ω-complete baos. This is probably the most important new result in the paper.
It shows that as long as we are considering completeness with respect to algebras
closed under joins of relatively small (i.e., countable) families of elements, The
Blok Dichotomy cannot be avoided. Existence of sufficiently many atoms (’points’,
’individuals’) or lack thereof is of no importance in such a situation; ditto for
distributivity of operators over infinite joins.
Nevertheless, it does not mean that dropping lattice-completeness while keeping both atomicity and complete additivity is a safe strategy. We also show The
Blok Dichotomy holds for discrete baos: atomic algebras with completely additive operators. The proof establishing this fact is similar to the one establishing
that the theorem holds for baos admitting residuals, which for n = 1 has been
already observed by Zakharyaschev et al. [36]. Observe, however, that both admissibility of residuals and discreteness can be considered to be weak notions of
lattice-completeness: see Remark 2.5 for further discussion.
We also answer the question posed by Dziobiak [10]. Section 6 gives an example
of atomic and complete algebra generating a variety which is not determined by any
class of subdirectly-irreducible ω-complete algebras. The result means that Blok’s
original method, which relies on techniques applicable in arbitrary congruencedistributive varieties and focuses on subdirectly irreducible algebras, is not useful
when investigating completeness with respect to classes of algebras which enjoy
some form of lattice-completeness without complete additivity. Blok himself [3]
hinted that using general frames one might probably provide an alternative proof
of his result. It turns out that it was necessary for Chagrova [7] and the present
paper to use such methods in order to generalize The Blok Dichotomy. As long as
one follows Blok’s strategy, results similar to those obtained by Dziobiak [10] are
best possible.
The paper is based on results obtained by the author in his PhD Thesis [21].
The author wishes to thank Balder ten Cate for his comments, Wieslaw Dziobiak
for discussion of results in Section 6, Tomasz Kowalski for pointing out an error in
an early version and suggesting a short proof of Lemma 3.5, the anonymous referee
for a number of comments greatly improving the presentation and Ian Hodkinson
for his criticism and patience during the editing process.
2. Preliminaries
2.1. Classes of n-baos, notions of completeness.
Definition 2.1. A n-bao is a boolean algebra with n normal unary operators
♦1 , . . . , ♦n : i.e., boolean algebra extended with n unary operations which preserve
the bottom element and distribute over finite joins.
Thorough the paper, we use Greek letters to denote terms of n-baos. The
choice is motivated by strict correspondence between algebraic terms and formulas
of modal logic. In fact, even though the paper is written in the language of algebra,
6
TADEUSZ LITAK
we attempt to make the connection with logic as explicit as possible and the notation
was chosen with this goal in mind. We define i ϕ := ¬♦i ¬ϕ. If n = 1, we drop
the subscript as a rule. The modal degree md(ϕ) of a term is defined in a standard
way, analogously as, e.g., quantifier depth in the first-order case. Assume n is fixed,
∆ is a set of terms, ϕ a term and let
[]0 ∆ := ∆,
[]∆ := {♦i δ | δ ∈ ∆, i ≤ n},
[]k+1 ∆ := [][]k ∆
[]0 ∆ := ∆,
[]∆ := {i δ | δ ∈ ∆, i ≤ n},
[]k+1 ∆ := [][]k ∆
k ϕ :=
_
[]i {ϕ},
⊕k ϕ :=
^
[]i {ϕ},
k ϕ :=
^
[]k {ϕ}
i≤k
k ϕ :=
^
[]k {ϕ}
i≤k
Observe that even though is the dual of , is in general not a dual of
⊕. Intuitive reading of k ϕ is that φ can be accessed at most k steps by some
compound modality, ⊕k ϕ—φ can be accessed by every compound modality whose
length is exactly k, k ϕ—φ holds at every point accessible by a compound modality
of length k and k ϕ — φ holds at every point accessible by a compound modality
of length no greater than k.
For an equation σ (a set of equations Σ), we denote the class of all n-baos where
σ (Σ) holds as Kn (σ) (Kn (Σ))—in particular, the class of n-baos is denoted by
Kn . By Birkhoff’s Theorem, a class of n-baos is of the form Kn (Σ) iff it is closed
under homomorphisms, subalgebras and products (HSP -closed).
Definition 2.2 (Classes of n-baos).
• A is ω-complete W
(A ∈ ωC) if for every countable family of elements {ai }i∈ω
has supremum
ai , i.e., the smallest element which is above all ai ’s.
i∈ω
• A is lattice-complete (A ∈ C) if every family of elements (including uncountable ones) has a supremum.
• A is atomic (A ∈ A) if every element is above an atom, minimal element
distinct from the bottom.
• A is completely additive (A ∈ V) if all its operators are. W
That is, for every
i ≤ n and every family of elements {aj }j∈J ⊆ A s.t.
aj exists in A,
j∈J
W
W
W
♦i aj also exists in A and, in addition,
♦i aj = ♦i
aj .
j∈J
j∈J
j∈J
• A is discrete (A ∈ AV) if it is both atomic and completely additive. In
a similar vein, we define CA (CAV) to be the class of algebras which are
lattice-complete and atomic (and completely additive).
• A admits residuals (A ∈ T ) if for every i ≤ n there is a function hi : A 7→ A
s.t. for every a, b ∈ A, a ≤ hi b iff ♦i a ≤ b.
STABILITY OF THE BLOK THEOREM
7
Remark 2.3. Our definition of an ω-complete bao coincides with the one in Sikorski [27] but not with the one in Koppelberg [18]. In the definition of algebras
admitting residuals, we don’t require that residuals are term-definable. This is the
difference between T -baos and Jipsen’s [14] residuated baos.
Admissibility of residuals can be reformulated as follows. We say that n-bao A
admits conjugates if for every i ≤ n there is a function pi : A 7→ A s.t. for every
a, b ∈ A, a ∧ ♦i b = ⊥ iff pi a ∧ b = ⊥. These two definitions are equivalent, as
hi a = ¬pi ¬a. The following is a basic fact about residuated operations in general,
but for the particular case of boolean algebras it can be already found in Jónsson
and Tarski [16, Theorem 1.14].
Fact 2.4. Every algebra admitting residuals is completely additive, i.e., T ⊆ V.
For lattice-complete algebras, the converse also holds: a complete n-bao admits
residuals iff it is completely additive, i.e., C ∩ T = C ∩ V.
Remark 2.5. Both admissibility of residuals and discreteness can be, in a sense,
considered to be a combination of complete additivity with a very weak form of
lattice-completeness: that is, the existence of suprema of certain special families of
elements. TheVfact that operators are residuated means that for every a ∈ A and
every i ≤ n, {x | a ≤ i x} exists (this is exactly pi a). The class of discrete
algebras is defined
in the class of atomic ones by the fact that for every a ∈ A and
W
every i ≤ n, {♦i x | x ∈ AtA, x ≤ a} exists and is equal to ♦i a (AtA, as usual,
denotes the family of atoms of A).
Definition 2.6. A variety V is countably complete, discretely complete, residually
complete (ωC−, AV−, T −complete, respectively) if it satisfies V = HSP (V ∩ ωC),
V = HSP (V ∩ AV), V = HSP (V ∩ T ), respectively. Definitions of C−, V−, A−,
CA− and CAV-completeness are analogous.
The notion of CAV-completeness is the standard notion of Kripke completeness
in modal logic.
Remark 2.7. The notions of AV-completeness, T -completeness and ωC-completeness—
although much weaker than standard Kripke completeness—can be of independent
interest to modal logicians. A variety corresponding to a normal modal logic L
is AV-complete iff its minimal extension either with nominals or with difference
modality is conservative. These extensions are axiomatized with the addition of
Gabbay-style or non-orthodox rules: cf. Gargov [12] or Blackburn et al. [1, Chapter 7.3] for axiomatization of logics with nominals and Venema [34, Section 6] for
axiomatization of logics with difference modality. In a similar vein, T -completeness
corresponds to conservativity of minimal tense (or conjugate) extensions—see Zakharyaschev et al. [36, Section 2.3]. Finally, ωC-completeness can be characterized
as conservativity of minimal extensions with infinitary conjunctions and disjunctions over countable sets of formulas.
Remark 2.8. There are examples of AV- and T -complete varieties which are not
ωC-complete, CT -complete varieties which are not A-complete, CA-complete varieties which are neither AV nor T -complete, ωC-complete varieties which are not
8
TADEUSZ LITAK
C-complete and AV-complete varieties which are not T -complete (see [21], [22]).
Thus, all the completeness notions introduced above are different and all of them
properly generalize the notion of CAV-completeness.
Definition 2.9 (Degrees of Incompleteness). Let X be one of the classes of algebras introduced above. For any variety V of n-baos, we define its degree of
X -incompleteness δX (V ) as the number of varieties V 0 satisfying the ’equation’
V 0 ∩ X = V ∩ X,
(x-inc)
i.e., as the number of varieties X -equivalent to V : those ones which cannot be
distinguished from V by means of algebras from X . If δX (V ) = 1, V is called strictly
X -complete. δωC , δAV , δT , δCA and δCAV will be called the degree of countable,
discrete, residual, neighborhood and Kripke incompleteness, respectively.
We can also introduce a similar notion for subvarieties of Kn .
Definition 2.10 (Relative Degrees of Incompleteness). For W a variety (more
generally, any class of algebras), V a subvariety of W and X being one of classes
W
of algebras introduced above, the W -degree of X -incompleteness of V δX
(V ) is the
0
number of subvarieties V ⊆ W satisfying x-inc.
General lattice-theoretical arguments ensure that certain logics are natural candidates for examples of strictly X -complete algebras—for any class X including
finite baos. This is the subject of the next subsection.
2.2. The role of splittings. For any variety V and its subvariety V 0 , the class of
all subvarieties of V X -equivalent to V 0 has the smallest element: it is the unique X complete member of this class (in a pathological case, it can be the trivial variety).
If this member also happens to be the largest variety in the same equivalence class,
then of course it must be its only member. When such cases are possible? Recall the
notion of a splitting subdirectly irreducible algebra A in the lattice of subvarieties
of a given variety V : an algebra for which there exists largest subvariety W ⊆ V
which does not contain A. This variety is usually called by algebraists the conjugate
variety of VA (cf., e.g., Jónsson [15]) and denoted by V /A. Here,
V we will use the
name the -prime counterpart of A, as such varietyVmust be a -prime element of
the lattice of subvarieties of V . A variety W is
V Π -prime among
V subvarieties of
V if it is an intersection of varieties which are -prime: if those -prime varieties
are of the form V /Ai for some i ∈ I, then W is denoted as V /{Ai | i ∈ I}.
Remark 2.11. It is a consequence of Jónsson’s Lemma that when a variety is generated by a HSPw -closed class of algebras K (Pw denotes closure under ultrapowers), then every subdirectly irreducible algebra splitting the lattice of its subvarieties
belongs to K. If K is a class of finite algebras, it is of course enough to stipulate
that K is HS-closed.
V
V
What does being -prime and Π -prime mean for X -incompleteness? If a
variety is the largest one which does not contain a family of finite algebras, it must
be the largest element of its X -equivalence class, for arbitrary class X containing
the class of finite algebras. If we can prove that the variety also generated by its
STABILITY OF THE BLOK THEOREM
9
finite members (hence X -complete), it must be strictly
X -complete. And this is
V
exactly the positive part of Blok’s Theorem: all Π -prime varieties of baos have
the finite model property.
Call a finite n-bao A a dual acyclic n-graph (see Remark 2.21 for an explanation
of the name) if for every a ∈ A, a 6= ⊥ implies a 6≤ 1 a. Observe that dual acyclic
n-graphs must be subdirectly irreducible.
Lemma 2.12. For every n ∈ ω, Kn is generated by the class of all dual acyclic ngraphs. Moreover, the class of dual acyclic n-graphs is HS-closed and hence every
V
-prime variety of n-baos is of the form Kn /A for some dual acyclic n-graph A.
For n = 1 this is, e.g., Lemma 3.28 in Chagrov and Zakharyaschev [6]
Theorem
2.13 (Blok [2]). For every dual acyclic n-graph A, {B | B 6∈ HSP (A)}
V
is a -prime variety. Thus, the “moreover”Vpart of Lemma 2.12 can be strengthened
to an equivalence: a variety of n-baos is -prime iff it is of the form Kn /A for
some dual acyclic n-graph A.
Strictly speaking, Blok has proven this result for the unimodal case. The first
author to note that the restriction to K1 is not relevant was Kracht [20].
V
Theorem 2.14 (Positive Part of Blok Dichotomy [2]). Every Π -prime subvariety
of Kn has the finite embeddability property. Consequently, its degree of Kripke
(countable, discrete, residual . . . ) incompleteness is 1.
Again, Blok formulated this result in a restricted case: as finite generation of
V
Π -subvarieties of ExtK1 . Close inspection of the proof ensures, however, that in
fact finite embeddability property is proven and it is possible to conduct the proof
for any finite n in an analogous manner.
As stated above, our main concern in this paper is the other half of Blok’s result.
Theorem 2.15 (Negative
Part of Blok Dichotomy [2]). For every subvariety V of
V
K1 which is not Π -prime, δCAV (V ) = 2ℵ0 .
This theorem is generalized by Theorem 5.1 below.
2.3. Duality theory.
Definition 2.16. A relational n-structure or a Kripke n-frame is a pair F :=
hW, {R1 , . . . Rn }i where Ri ⊆ W 2 for every i ≤ n. With every frame, we can
associate its complex algebra F∗ whose universe is the family of all subsets of W .
The boolean connectives are interpreted as set-theoretical operations and for every
i ≤ n, the corresponding operator is defined as
♦i X := {y ∈ W | ∃x ∈ X.yRi x}.
Definition 2.17. For any n-bao A, the atom structure of A is A∗ := hAtA, R♦1 , . . . , R♦n i,
where aR♦i b if a ≤ ♦i b. With every A, we can also associate the atomic embedding
fA : A 7→ (A∗ )∗ defined as fA (a) = {x ∈ AtA | x ≤ a}.
Theorem 2.18.
10
TADEUSZ LITAK
• For every relational structure F, F∗ ∈ CAV and (F∗ )∗ ∼
= F.
• For every n-bao A, fA is an isomorphic embedding iff A ∈ AV and fA is
an isomorphism onto iff A ∈ CAV.
Proof: The first claim and the second equivalence of the second claim by Jónsson
and Tarski. The first equivalence of the second claim by Venema [31].
a
Relational structures allow us to introduce many notions from the graph theory
and the model theory, such as path, cycle, point-generated subframe and so on. The
definitions are standard and can be found in any of references mentioned at the
beginning of this paper.
Thomason [28] introduced general frames and from now on, the word frame will
be used in this sense. A (general) n-frame is a structure F := hW, {R0 , . . . Rn−1 }, Ai
where A is any family of subsets of W closed under boolean connectives and ♦i for
every i < n. A is called the family of admissible sets of F. hW, {R0 , . . . Rn−1 }, Ai+
is the boolean algebra whose universe is A with induced operations. It is known
that every algebra can be represented this way via the the atom structure of its
canonical extension A+ [15, 33]. Frames which are of the form A+ for some A are
called descriptive.
We can introduce a notion of morphism between frames which is dual to the
notions of morphism between baos. A bounded morphism between Kripke frames
hW, {R1 , . . . Rn }i and hV, {S1 , . . . Sn }i is a function f satisfying for every i ≤ n
f (x)Si f (y) iff there is v ∈ V s.t. xRi v and f (v) = f (y).
A mapping from hW, {R1 , . . . Rn }, Ai to hV, {S1 , . . . Sn }, Bi is a frame morphism
if it is a bounded morphism of underlying frames and for every B ∈ B, f −1 (B) ∈ A.
Remark 2.19 (Discrete Duality). Thomason [29] observes that the category of
CAV’s with complete homomorphisms and the category of Kripke frames with bounded
morphisms are dually equivalent in the sense of Clark and Davey [8]; Thomason
[29] uses the name equivalent by contravariant functors. He also introduces a notion of morphism between Kripke frames which yields a duality with the category
of CAV’s with arbitrary morphisms. Both of Thomason’s dualities can be extended
to the categories of discrete algebras and discrete frames, i.e., frames where every
singleton subset is admissible. For objects, the duality has been already established
as the second claim of Theorem 2.18.
Remark 2.20 (Canonical Duality). Except for the duality between the categories of
CAV’s with complete homomorphism and Kripke frames with bounded morphisms
(we can call it ∗-duality or the discrete duality), there is also a duality between the
class of all n-baos with arbitrary homomorphisms and descriptive n-frames with
frame morphisms ( +-duality or the canonical duality). This one is in fact more
important from the algebraic perspective, as it allows for a relational representation
of arbitrary algebra. See Venema [33, Section 5] for a discussion of both dualities.
Remark 2.21. From the present perspective we can justify the name dual acyclic
graph introduced above. A is a dual acyclic graph iff A ∼
= F+ for a finite F = hW, Ri
STABILITY OF THE BLOK THEOREM
11
s.t. S
for no element x ∈ F there exists a R1 ∪ . . . ∪ Rn -path from x to x, i.e.,
hW,
Ri i is an acyclic graph in a standard sense.
i≤n
3. The Minimal Variety theorem
All known proofs of the Blok Dichotomy for the unimodal case used in an essential way the Makinson Theorem, which describes the minimal subvarieties of K1 .
There are, up to isomorphism, only two 1-baos, whose boolean reduct is the twoelement algebra: one satisfies ♦> = >, the other ♦> = ⊥. The variety generated
by the former algebra is denoted by Triv, the one generated by the latter algebra
— by Ver.
Theorem 3.1 (Makinson). Triv and Ver are the only minimal subvarieties of K1 .
The problem with the Makinson Theorem is that it does not generalize even to
the bimodal case.
Lemma 3.2. K2 has uncountably many minimal subvarieties.
It is hard to say who was the first author to note this. As claimed by the referee,
the result has been a folklore as early as in the 1970’s. Similar results have been
obtained for Post-complete extensions of non-normal modal logics, see, e.g., [25],
[26]. From our point point of view, this is a negative result: it shows that Blok’s
method cannot be used without any modification when investigating degrees of
incompleteness of subvarieties of Kn . It turns out, however, that one does not need
the full strength of The Makinson Theorem.
Definition 3.3. A n-bao is strictly simple if is simple and zero-generated, i.e., it
has neither proper nontrivial homomorphic images nor proper subalgebras.
Lemma 3.4. A subvariety V of Kn is minimal iff V is HSP -generated by a strictly
simple n-bao.
Proof: For “if” direction see, for example, Kowalski [19]; this direction is of no
relevance for us here. For the converse direction, take any non-trivial algebra A ∈ V
and let B be its zero-generated subalgebra. As V is minimal, it is generated by B
as well. Assume that B has a non-trivial proper homomorphic image C. C is also
zero-generated. It means that for some variable-free terms ϕ1 and ϕ2 , B 2 ϕ1 = ϕ2
and C ϕ1 = ϕ2 . Thus, the variety generated by B is not minimal, a contradiction.
a
The following is a variant of a well-known theorem of Rautenberg, see, e.g.,
Venema [33] for a more standard formulation of this result.
Lemma 3.5. An algebra A is simple iff for every a 6= ⊥ there is k ∈ ω s.t.
k a = >.
Proof: An algebra is simple iff its only congruences are the identity relation and the
total equivalence relation. For n-baos, we can identify congruences with filters—
equivalence classes of >. Hence, n-bao is simple iff for any a 6= >, the congruence
12
TADEUSZ LITAK
filter generated by a is not proper, i.e., contains ⊥. Recall that the congruence
filter generated by a is of the form {x | k a ≤ x}. Thus, by properties of complementation in boolean algebras, simplicity is equivalent to the fact that for every
a 6= ⊥ there is k ∈ ω s.t. k ¬a = ⊥.
a
Theorem 3.6 (Minimal Variety). For every minimal variety of n-baos V , either
there exists k ∈ ω and i < n s.t. V k i ⊥ = > or V is generated by the two
element algebra satisfying ♦i > = > for every i < n.
Proof: By Lemma 3.4, V = HSP (A) for some zero-generated and simple A. Assume first i ⊥ =
6 ⊥ for some i < n. Then by Lemma 3.5 there exists k ∈ ω s.t.
k i ⊥ = >.
Assume now ♦i > = > for every i < n. But then, as A is zero-generated, its
universe must be {>, ⊥}.
a
Remark 3.7. The Makinson Theorem implies also that both minimal subvarieties
of K1 are determined by a single finite algebra—hence, they are CAV-complete.
Nothing like this holds in general. In fact, there are plenty of examples of varieties
of n-baos with do not contain any perfect algebras—the first example obtained by
Thomason [28]. Cf. Venema [32] or Litak [22] for examples of varieties without any
discrete, atomic or ω-complete algebras. Thus, one cannot hope for generalizations
of Makinson’s result stronger than the Minimal Variety Theorem.
4. A variety with continual K4-degree of ωC-incompleteness
We begin with a construction of a variety with a continual degree of countable
incompleteness. Actually, we will obtain a stronger result. Recall the equation
p ≤ p
(4)
axiomatizing the subvariety K4 of K1 . This is one of best studied varieties of modal
algebras, containing the variety of closure algebras S4 and the variety of monadic
algebras S5. It is also a paradigm example of a modal variety with EDPC. On the
duality side, it corresponds to the class of transitive frames. Yet the question of
K4-degrees of incompleteness (cf. Definition 2.10) remains a challenging problem
in the metatheory of unimodal logics (cf. [6, Problem 10.5] or open problems posed
in [23]).
K4
(Λ) = 2ℵ0 . That such subvarieties of K4
We show there is a variety Λ s.t. δωC
exist seems a new observation. There were examples of continua of incomplete subvarieties of K4—cf., e.g., Rybakov [24] or Chagrov and Zakharyaschev [6, Chapter
6.4]—but no two distinct varieties V , V 0 from these continua satisfied cav-inc. Nevertheless, our example is based on the construction used in the proof of Theorem
6.22 in [6] and this construction in turn is a modification of the original construction
of Fine [11], so nothing revolutionary happens here. This construction, however,
will prove useful in what follows.
Remark 4.1. We would like to observe, however, that the existence of varieties of
closure algebras with a continual degree of S4-incompleteness or similar varieties
STABILITY OF THE BLOK THEOREM
a4
ppp •
6
a3
-•
6
a2
-•
6
a1
-•
6
13
a0
-•
6
◦ b4
6
◦ b3
6
◦ b2
6
◦ b1
6
◦ b0
6
p p p ◦
c4
◦
c3
•
c2
•
c1
◦
c0
Figure 1. Frame FineI for I = {0, 3, 4, . . .}
of Heyting algebras seems a more challenging problem and the answer may be in
the negative here.
For arbitrary I ⊆ ω, let FineI := hW, RI , Fi be the general frame defined as
• W := A ∪ B ∪ C, where A := {an }n∈ω , B := {bn }n∈ω , C := {cn }n∈ω ,
• RI := R ∪ {hci , ci i | i ∈ I}, where R := {hai , aj i | i > j} ∪ {hbi , aj i | i ≥
j} ∪ {hbi , bi i | i ∈ ω} ∪ {hci , bj i | i ≤ j} ∪ {hci , cj i | i < j} ∪ C × A,
• F is the set of finite and cofinite subsets of W .
See Figure 1. It is straightforward to observe that F is closed under ♦ RI for any
I and thus every FineI is a general frame. Moreover, it is a K4 frame, as it is based
on a transitive structure.
Theorem 4.2. For every I ⊆ ω, there exists a K4-variety ΛI , s.t. for every J ⊆ ω,
Fine+
J ∈ ΛI iff I = J and
ΛI ∩ ωC = ΛJ ∩ ωC
Proof: Define the following three sequences of terms in one variable:
α0 (p)
αk+1 (p)
:= p,
:= ♦αk (p) ∧ 2 ¬αk (p) ∧ ♦α0 (p),
βk (p) := ♦2 αk (p) ∧ ¬αk+1 (p) ∧ ♦α0 (p),
γ0 (p) := ♦β0 (p) ∧ ♦β1 (p) ∧ ♦α0 (p),
γi (p)
:= ¬1 γi−1 (p) ∧ ♦βi (p) ∧ ♦i βi+1 (p) ∧ ♦α0 (p)
(k ∈ ω, i > 0).
Clearly, for subvarieties of K4 the last conjunct ♦α0 (p) is redundant; we are
going to need it in the proof of Theorem 5.1, however. Define also
p ≤ 1 (p ∧ (p ∨ ¬p)).
(wgrz)
Claim 1: For arbitrary I ⊆ ω, k ∈ ω and arbitrary valuation V in Fine+
I ,
V(αk (⊥)) = {ak }, V(βk (⊥)) = {bk }, V(γk (⊥)) = {ck }.
Proof of claim: A straightforward verification, analogous to the one found in
Chagrov and Zakharyaschev [6], section 6.3.
a
14
TADEUSZ LITAK
This justifies the following convention:
ak := αk (⊥),
bk := βk (⊥),
ck := γk (⊥).
Claim 2: For arbitrary I ⊆ ω and arbitrary k ∈ ω, Fine+
I ck ≤ ♦ck+1 .
+
+
FineI ck ≤ ♦ck iff k ∈ I; otherwise, FineI ck ≤ ¬ck .
Proof of claim: A consequence of Claim 1.
a
+
Claim 3: FineI wgrz.
Proof of claim: Assume V(p) 6= ∅. For x ∈ V(p)
• either x ∈ A ∪ B and hence there must be a R-maximal point y ∈ V(p) s.t.
either xRy or x = y. y ∈ V(p ∧ (p ∨ ¬p))), hence x ∈ V(1 (p ∧ (p ∨
¬p)))
• or x ∈ C. If x has a successor in (A ∪ B) ∩ V(p), then the above argument
applies. But if there is no such point, then there must exist a maximal
successor of x in C ∩ V(p), otherwise V(p) would be neither finite nor
cofinite.
a
Let LF := {4} ∪ {ci ≤ ♦ci+1 | i ∈ ω}.
Claim 4: For any A ∈ ωC ∩ Kn (LF ) and arbitrary k ∈ ω, A 2 ck = ⊥ implies
A 2 wgrz.
W
Proof of claim: Define V(p) :=
c2i . For every i ≥ k, c2i ≤ ♦c2i+1 and
i≥k
c2i+1 ≤ ♦c2i+2 . Thus, V(p) ≤ V(♦(¬p ∧ ♦p)).
a
Now let ΛI := Kn (LF ∪ {wgrz} ∪ {ck ≤ ♦ck | k ∈ I} ∪ {ck ≤ ¬ck | k 6∈ I}).
Claim 5: Fine+
I ∈ ΛJ iff I = J.
Proof of claim: A consequence of Claims 2 and 3.
The theorem now follows by Claim 4.
a
a
5. Degrees of incompleteness
This section proves the main result of the paper.
V
Theorem 5.1. If Λ is not a Π -prime subvariety of Kn , then
δωC (Λ) = δAV (Λ) = δT (Λ) = 2ℵ0 .
Proof: First, let us deal with countable incompleteness—as was explained in the
Introduction, this is in fact the major
V new contribution of the paper. Assume Λ 6=
Λ0 , where Λ0 is an intersection of all -prime varieties contained in Λ. By Theorem
2.14, Λ0 is generated by a family of finite subdirectly irreducible algebras which are
not dual acyclic graphs. Take one of those A ∈ Λ0 and an equation ϕ = > which
holds in Λ s.t. A 2 ϕ = > but every proper homomorphic image of A belongs to Λ.
As A is finite, we can assume A = F∗ for a finite Kripke frame F := hV, S1 , . . . Sn i.
As A is not acyclic, we can assume F contains a minimal cycle, i.e., a sequence of
STABILITY OF THE BLOK THEOREM
15
elements {xi }i<k s.t. for every i < k there is m ≤ n s.t. xi Sm xi+1modk and no
proper subsequence of {xi }i<k is a cycle. Let F0 := hV 0 , S10 , . . . Sn0 i be a variant of
F with the cycle {xi }i<k multiplied l := md(ϕ) + 1 times. In other words, {xi }i<k
is replaced by {xji }i<k,j<l . xi is identified with x0i . The relations on the extended
frame are defined in the natural way using
xi : x = xji for some i < k, j < l,
f (x) :=
x : otherwise.
0
Then for every m ≤ n and u, v ∈ V 0 , uSm
v if either v is not in the cycle and
j
f (u)Sm f (v) or for some j < l, u = xi and v = xji+1modk . The very definition forces
that f is a bounded morphism; equivalently, that A is (isomorphic to) a subalgebra
∗
of A0 := F0 . This, in turn, implies that
(*) for any valuation V, any x ∈ W and term ψ, x ∈ V(ψ) iff x ∈ V0 (ψ),
where V0 (p) is a valuation in A0 defined as V0 := f −1 [V0 (p)] for every variable p.
Moreover, (*) holds for those ψ’s which are subterms of ϕ also if F0 is replaced
0
by any general frame G := hX, T1 , . . . , Tn , Ai s.t. V 0 ⊆ X, Sm
⊆ Tm (m ≤ n),
V0
0
0
2 ⊆ A and for every x ∈ V , there is y ∈ X − V s.t. xTm y for some m ≤ n only
if x = x0 := xlk . This follows from the fact that the cycle was chosen to be minimal
and hence nontransitive; strictly speaking, even antitransitive.
At this point, the proofs of The Blok Dichotomy for the unimodal case such as
the one in Chagrov and Zakharyaschev [6] use The Makinson Theorem. We have to
use Lemma 3.4 and Theorem 3.6 instead. Thus, either there exists B ∈ HSP (A),
k ∈ ω and o ≤ n s.t. B k o ⊥ = > or B is the two-element algebra satisfying
♦o > = > for every o ≤ n. Let us consider the former case first. We can choose k to
be minimal, i.e., s.t. B 2 k−1 o ⊥ = >. Let τ := k o ⊥∧k−1 ♦o >, w := |F0 |+1,
v := max{k, w} and let G := hX, T1 , . . . Tn , Bi be any frame s.t. B = G+ . Also, let
r be the root of F0 .
I
I
Let Fine0I := hW 0 , R0 1 , . . . R0 n , Fi be a general frame s.t. W 0 := W ∪ V 0 ∪ X ∪
I
0
I
∪ Tm ∪ {ha0 , dv i} ∪ {hdi+1 , di i |
∪ Sm
{d0 , . . . , dv }, for every m ≤ n, R0 m := Rm
0
i ≤ v − 1} ∪ {hx , d0 i, hd0 , ri, hd1 , xi} where x is an arbitrarily chosen element of the
subset of X corresponding to τ and F is the family of all sets whose intersection
with, respectively, W and X belongs to A and B; see Figure 2. Define the following
constant terms:
d0
d1
dm+1
:= ⊕τ,
:= ¬d0 ∧ ⊕d0 ,
:= ¬dm ∧ ⊕dm ∧
^
¬di
(m ∈ {1, . . . , v − 1}),
i<m
a0
:= ⊕1 (dv ∧ ⊕(dv−1 ∧ . . . ∧ ⊕d0 ) . . .) ∧ 2 ¬(dv ∧ ⊕(dv−1 ∧ . . . ∧ ⊕d0 ) . . .) ∧ v ⊕ >.
+
Claim 1: In Fine0 I , a0 = {a0 }.
Proof of claim: It follows from the choice of v and τ .
a
16
TADEUSZ LITAK
F0
transitive
a4
ppp •
6
a3
-•
6
a2
-•
6
a1
-•
6
a0
-•
6
◦ b3
6
◦ b2
6
◦ b1
6
◦ b0
6
p p p ◦
c4
◦
c3
•
c2
•
c1
+
◦
c0
d0 +
-• A
A
AU
◦ b4
6
dv
-•
r
x0
ppp
G
Figure 2. Frame Fine0I for I = {0, 3, 4, . . .}
Constant a0 is defined in a different way than in the Section 4. Nevertheless, we
can use sequences of terms {αm , βm , γm }m∈ω as defined in that section replacing ♦
with ⊕, with and setting
ai := αi (a0 ),
bi := βi (a0 ),
ci := γi (a0 )
(i ∈ ω).
+
Just like in the proof of Theorem 4, it is easily provable now that in Fine0 I ,
am = {am }, bm = {bm } and cm = {cm }. For arbitrary I ⊆ ω, let ΛI be the
+
+
smallest variety containing both Fine0 I and Λ, i.e., the HSP -closure of Λ ∪ Fine0 I .
Claim 2: ΛI ci ∧ ⊕ci ∧ ⊕v+3 ¬ϕ = ⊥ iff i 6∈ I.
Claim 3:
ΛI ∩ ωC = Λ ∩ ωC.
Proof of claim: Λ ⊆ ΛI , so it is enough to prove ΛI ∩ ωC ⊆ Λ ∩ ωC. Assume
that for some D ∈ ΛI ∩ ωC there is a term ψ s.t. Λ ψ = > but D 2 ψ = >.
+
By the definition of ΛI , it must be the case that Fine0 I 2 ψ = >. Moreover,
the point x where ψ is refuted under some valuation cannot belong to G (as
G+ ∈ Λ) and if it belongs to F0 , then some point from W 0 must be accessible
from x in finitely many steps by some compound modality—otherwise the dual
algebra of the rooted subframe generated by x would belong to Λ. Thus, it can
+
be proved there exists l s.t. for arbitrary term σ s.t. G σ = >, Fine0 I 2 σ = >
l
iff x ∈ V( ¬σ) for some valuation V. l can be taken to be w + v + 4 for the
sake of definiteness. The following equations hold then in Fine+
I :
STABILITY OF THE BLOK THEOREM
(σ)
(χi )
(θi,j )
(wgrzψ )
¬ψ
ci ∧ ψ0
ci ∧ ψ0
ψ0 ∧ p
≤
≤
≤
≤
17
l (c0 ∧ ψ0 ),
⊕(ci+1 ∧ ψ0 ) (i ∈ ω),
¬cj (i 6= j),
¬1 (ψ0 ∧ p → ⊕(ψ0 ∧ ¬p ∧ ⊕(ψ0 ∧ p))),
where ψ0 := ⊕a0 ∧ l ¬ψ. We assume that p does not appear in ψ; otherwise
we replace it in wgrzψ by the first variable which does not appear in ψ. That
+
Fine0 I wgrzψ = > can be established by an argument analogous to the one
in the proof of Theorem 4.2. The only additional fact one needs is that the
universe of FineI is exactly the subset corresponding to the constant ⊕a0 . In
addition, it was assumed that Λ ψ = >, thus all of σ, χi , θi,j , wgrzψ must
hold also in Λ. We can conclude they belong to the equationalWtheory of ΛI .
By σ, V(c0 ∧ l ¬ψ) 6= ⊥ for some V in D. Define V0 (p) := i∈ω V(c2i ∧ ψ0 ).
{θi,j }i6=j and {χi }i∈ω guarantee we can show that V0 refutes wgrzψ in the same
way as in the proof of Theorem 4.2.
a
What is left is the situation when the only minimal subvariety of Λ is the one
determined by the two-element algebra with ♦i > = > for every i ≤ n. Let G be
the single point frame where all Ri ’s are reflexive. This time, we set v := |F| + 1.
The definition of Fine0 I for arbitrary I ⊆ ω is analogous to the preceding case, with
a suitable replacement of G. As before, for arbitrary I ⊆ ω let ΛI := HSP (Λ ∪
+
{Fine0 I }).
Now we cannot use variable-free terms, but this obstacle can be overcome by
using of a trick analogous to the one used in Chagrov and Zakharyaschev [6] or
Chagrova [7]. Let δ := 2∗v+4 ¬s ∧ 2∗v+4 s, where s is an arbitrary variable which
does not occur in ϕ and define the following one-variable terms:
δ0 (s)
:= (⊕2∗v+4 ¬s ∧ ¬2∗v+4 ¬s) ∨ (⊕≤2∗v+4 s ∧ ¬2∗v+4 s),
δ1 (s)
:= ¬δ0 ∧ ⊕1 δ0 ,
δm+1 (s)
:= ¬δk ∧ ⊕δm ∧
^
¬δi
(m ∈ {1, . . . , v − 1}),
i<m
α00 (s)
:= ⊕(δv ∧ ⊕(δv−1 ∧ . . . ∧ ⊕δ0 ) . . .) ∧ 2 ¬(δv ∧ ⊕(δv−1 ∧ . . . ∧ ⊕δ0 ) . . .),
αi0 (s) := αi (α00 (s)),
βi0 (s) := βi (α00 (s)),
γi0 (s) := γi (α00 (s)) (i ∈ ω),
where αi , βi and γi are as in the Section 4.
+
Claim 4: In Fine0 I , if V(δ) 6= ⊥ for some V, then V(α00 ) = {a0 }.
+
Claim 5: Assume V(δ) 6= ⊥ for some V in Fine0 I . Then V(αi0 ) = {ai },
V(βi0 ) = {bi }, V(γi0 ) = {ci } for arbitrary i ∈ ω. Hence, ΛI γi0 ∧⊕γi0 ∧⊕v+3 ¬ϕ =
⊥ iff i 6∈ I.
18
TADEUSZ LITAK
Claim 6:
ΛI ∩ ωC = Λ ∩ ωC.
Proof of claim: Again, it is enough to prove ΛI ∩ ωC ⊆ Λ ∩ ωC. Assume that
for some D ∈ ΛI ∩ ωC there is a term ψ s.t. Λ ψ = > but D 2 ψ = >.
+
By the definition of ΛI , it must be the case that Fine0 I 2 ψ = >. As in the
proof of Claim 3, the point x where ψ is refuted under some valuation cannot
belong to G (as G+ ∈ Λ) and if it belongs to F0 , then some point from G must
be accessible from x in less than by v steps by some compound modality. The
+
following equations hold then in Fine0 I :
(ρ0 )
(σ 0 )
(χ0i )
0
(θi,j
)
0
(wgrzψ )
¬ψ
¬ψ ∧ δ 0
γi0 ∧ ψ00
γi0 ∧ ψ00
ψ00 ∧ p
≤ 2∗v+4 ψ,
≤ 2∗v+4 (γ00 ∧ ψ00 ),
0
≤ ⊕(γi+1
∧ ψ00 ),
≤ ¬γj0 (i, j ∈ ω, i 6= j),
≤ ¬1 (ψ00 ∧ p → ⊕1 (ψ00 ∧ ¬p ∧ ⊕(ψ00 ∧ p))),
where ψ00 := ⊕α00 ∧ 2∗v+4 ¬ψ ∧ δ 0 . We assume that neither p nor s appear
in ψ. Thus, if V(ψ) 6= > for some V, then by ρ0 , V0 (s) = V(¬ψ) implies
V0 (¬ψ) ≤ V0 (δ 0 ). Therefore, σ 0 gives that V0 (ψ00 ∧ γ00 ) 6= ⊥ for any such V0 and
we can repeat the reasoning from the proof of Claim 3.
a
Now, let us deal with δT (Λ) and δAV (Λ). In fact, there is no need to provide all
details of proof. It was observed by Zakharyaschev et al. [36] that the construction
of van Benthem [30] can be used to prove the Blok Dichotomy for degrees of residual
incompleteness of modal algebras, i.e., 1-baos. The above proof shows how to
remove the restriction of n = 1. Finally, Litak [22] observed that the variety
generated by the van Benthem algebra is also discretely incomplete. Thus, the
proof of The Blok Dichotomy for degrees of discrete incompleteness can be carried
out in exactly the same way as the one for residual incompleteness.
To make the paper more self-contained, let us sketch the first part of the proof:
a construction of a variety with continual degree of discrete incompleteness and
residual incompleteness. As in Section 4, let us work with just one modality. Fix
arbitrary I ⊆ ω − {0, 1}. For arbitrary i > 1, define
{ai }
: i 6∈ I,
a•i :=
{ai , a0i } : i ∈ I.
S •
Now let vB I := hWI , RI , FI i, where WI := {b} ∪ {c} ∪
ak , RI = {hc, bi} ∪
k∈ω
S •
S
({b} ×
ak ) ∪
(a•k × a•l ) ∪ {ha1 , a0 i} ∪ {ha1 , ci}, FI consists of finite sets which
k∈ω
k>l>1
do not contain b and their complements (cf. Figure 3) and let VBI = HSP (vB +
I ).
The sequences of terms {αk (p)}k∈ω and {ak }k∈ω are defined in the same way
as in Section 4. In addition, define γ(p) := ♦2 α1 (p) ∧ ¬♦α1 (p) and c := γ(>).
Finally, for arbitrary ϕ let α1 ϕ := (1 a1 → ϕ).
STABILITY OF THE BLOK THEOREM
19
transitive
b
a6
ppp •
•
6
a05
a04
•
•
@ @
a5 @@
a4 @@
a3
R
R
-•
-•
-•
a02
•
@
a2 @@
a1
R
-•
-•
a0
-•
•
c
Figure 3. Frame vB I for I = {2, 4, 5, . . .}
Claim 7: For arbitrary I ⊆ ω − {0, 1}, k ∈ ω and arbitrary valuation V in
•
2
vB +
I , V(ak ) = ak and V(c) = {c}. Hence, for arbitrary i ∈ ω, VBI (ai →
2
p) ∨ (ai → ¬p) = > iff i 6∈ I and thus VBI = VBJ iff I = J.
Define
()
a0 ∨ 2 a1 = >,
(ζ)
a1
≤ ♦c,
(η)
c
≤ ¬(p ∧ α1 (¬p ∨ ♦α1 p)).
The conjunction of these three equations is denoted as λ.
Claim 8: For arbitrary I ⊆ ω − {0, 1}, VBI λ.
Claim 9: For arbitrary A, if A is discrete or admits residuals and A λ, then
A a0 = >.
Proof of claim: Assume A λ, A ∈ AV ∪ T and yet A 2 a0 = >. To obtain
a contradiction, it is enough to find b s.t.
⊥ < c ∧ ♦(α1 (♦α1 b ∨ ¬b) ∧ b),
as it would imply that A 2 η. Observe that by , a1 6= ⊥ and hence by ζ, c 6= ⊥;
by definition, c ≤ ♦2 a1 If A is discrete, then let b be any atom below ♦a1 s.t
c ∧ ♦b 6= ⊥; its existence is guaranteed by discreteness of A. It is easy to see that
b, being an atom, must be below α1 (♦α1 b ∨ ¬b). If A admits residuals, then it
is either the case that pc ≤ (♦α1 pc ∨ ¬pc) or pc ∧ ♦(α1 ¬pc ∧ pc) 6= ⊥. In
a
the first case, let b := pc. In the second case, b := pc ∧ p2 c.
V
In order to prove the main theorem now, given a non-Π -prime Λ, a suitable
F, G and ϕ s.t. Λ ϕ = > and F+ 2 ϕ = >, we can construct vB 0I out of vB I
in the same way we constructed Fine0 I out of FineI in the proof for degrees of
ωC-incompleteness. Details can be left as an exercise now; cf. also Chagrov and
Zakharyaschev [6], Zakharyaschev et al. [36] or Chagrova [7].
a
20
TADEUSZ LITAK
6. Subdirectly irreducible algebras
Blok’s original proof was based on a different strategy than the technique of
Chagrov and Zakharyaschev used above. One of basic observations used was that
every CAV can be represented as a subdirect product of its complete-homomorphic
images, which are CAV’s again—recall that complete homomorphisms are the ones
preserving arbitrary existing joins and meets. In other words, Kripke completeness is equivalent to completeness with respect to subdirectly irreducible perfect
algebras—duals of rooted frames. But then, as varieties of baos are congruencedistributive, we can use powerful algebraic tools like Jónsson’s Lemma. In particular, in order to find which equations hold in all CAV’s from the variety generated
by a given algebra A, it is enough to investigate those CAV’s which can be obtained
from ultrapowers of A by taking homomorphic images and subalgebras.
The basic observation on which this strategy relies can be generalized as follows.
Lemma 6.1. Every AV is a subdirect product of its complete-homomorphic images,
which are themselves AV’s.
Proof: It is immediate that the class of AV is closed under complete-homomorphic
images. A homomorphism onto preserves existing joins and meets iff the corresponding congruence filter is closed under arbitrary existing meets—we call such
filters complete. For arbitrary atom a ∈ A ∈ AV, let F (a) := {b | ∀n ∈ ω.a ≤ n b}.
F (a) is closed under , hence it is a congruence filter. Moreover, complete additivity implies that if a subset of A has an infimum, it is again an element of F (a).
Thus A/F (a) is a complete-homomorphic image of A. Finally, A is a subdirect
product of {A/F (a) | a ∈ AtA}.
a
In the above proof, it is enough to assume distribute over arbitrary meets—it
is not necessary that all operators are completely distributive. Also, the lemma
holds for any subclass of AV’s closed under complete-homomorphic images. But
can it be generalized further? And is it true that a variety generated by a subclass
of X only if it is generated by a family of subdirectly irreducible elements of X , for
“natural” X ’s?
We are going to show that the answer is in the negative, thus solving the problem
posed by Dziobiak [10]. Actually, we are going to prove a much stronger result.
Recall that an algebra is finitely subdirectly irreducible if the intersection of any
pair of nontrivial congruence filters is again nontrivial, i.e., distinct from {>}.
Theorem 6.2. There exists a modal algebra in CA which generates a K4-variety
not generated by any class of finitely subdirectly irreducible ωC’s.
Proof: Let Sic be the powerset algebra of
W := {an , bn | n ∈ ω} ∪ {cij | j ∈ ω − {0}, 1 ≤ i ≤ j} ∪ {d}.
Let Ξ is the set of all cofinite subsets of the set of natural numbers without 0.
W
Define an auxiliary function Ng : W −→ 22 as
STABILITY OF THE BLOK THEOREM
21
• a0
6
I
@
@
@ a1
•
6
I
@
@
@ a2
◦
◦ b1
•
c11
6
I
@
@
-◦ b2 @• a3
◦ -◦
c12 c22
6
I
@
@
@ a4
◦ ◦
◦ b3
•
◦
c13 c23 c33
6
I
@
@
- -◦ b4 @• a5
◦
◦
◦ -◦
c14
c24 c34 c44
6 p
pp
◦
◦
◦
◦
◦
◦ b5
c15
c25
c35
c45
c55
◦ b0
Figure 4. The atom structure of Sic with d removed.
N ∈ Ng(x)
iff















∪{cik
N ⊇ {ai | 0 ≤ i ≤ n − 1}
N ⊇ {bn } ∪ {ai | i ≤ n}
N ⊇ {ckj | i ≤ k ≤ j}∪
∪{bk | i ≤ k ≤ j} ∪ {ak | k ≤ j}
∃K ∈ Ξ N ⊇ {d}∪
| k ∈ K, 1 ≤ i ≤ k} ∪ {an , bn | n ∈ ω}
: x = an ,
: x = bn ,
: x = cij ,
: x = d.
Now, Ng X := {x | X ∈ Ng(x)}. See Figure 4.
Claim 1: Sic is not finitely subdirectly irreducible.
Proof of claim: Note that, for example, A := W − {d} and B := W − {c11 }
are both coatoms of 2W such that A = ♦Ng A and B = ♦Ng B. Thus, we have
two non-trivial congruence filters whose intersection is trivial.
a
In the remaining part of the proof, we drop the subscript Ng. The sequences of
terms {an }n∈ω , {bn }n∈ω are defined as in Section 4. {cn }n∈ω are replaced by the
sequence cn+1 := ¬♦bn ∧ ♦bn+1 ∧ ♦bn+2 , n ∈ ω. Define also d := ♦b0 ∧ ♦b1 .
(ζn+1 )
d ≤ ♦cn+1 ∧ 1 (cn+1 → ♦cn+2 )
(n ∈ ω).
Claim 2: Sic ∈ K4. Also, in Sic, an = {an }, b0 = {b0 }, bm = {bm , cim |
1 ≤ i ≤ m}, cm = {cmj | j > m}, ♦cn = {d} ∪ {cij | i ≤ m, j > m} and
d = ♦d = {d}, for any n ∈ ω and any m ≥ 1.
Claim 3: For any n ≥ 1, Sic ζn .
22
TADEUSZ LITAK
Proof of claim: We have shown that d = {d} ⊆ ♦cn and ¬{cnn+1 } = ¬cn ∨
♦cn+1 . The set W −{cnn+1 } belongs to Ng(d) and hence {d} ⊆ 1 (cn → ♦cn+1 ).
a
Claim 4: Sic wgrz.
Proof of claim: For any x ∈ W − {d} and any valuation V, it is clear that
x ∈ V(wgrz). Assume now that for some V, d 6∈ V(wgrz). This means that
d ∈ V(p). Let M := V(p → ♦(¬p∧♦p)). Then M ∈ Ng(d) and M ∩V(¬p∧♦p) is
a non-empty set which does not contain d. Pick arbitrary u ∈ M ∩ V(¬p ∧ ♦p).
It follows that there is v ∈ M − {d} such that u ∈ ♦v and v ∈ V(q). But
M ∈ Ng(v) and hence v 6∈ V(wgrz), a contradiction.
a
Claim 5: If A ∈ ωC ∩ K4 is finitely subdirectly irreducible, A d = ♦d,
A ζn for any n ≥ 1 and A 2 d 6= ⊥, then A cn ≤ ♦cn+1 , for arbitrary n ≥ 1.
Proof of claim: Assume for some n c := ¬cn ∨ ♦cn+1 6= >. ζn guarantees
that the congruence filter generated by c—that is, the principal filter generated
by 1 c—is contained in the principal filter generated by d. However, another
consequence of the validity of ζn is that ¬c ≤ ¬d and by assumption the principal
filter generated by ¬d is a congruence filter. Thus, we have two non-trivial
congruence filters whose intersection is {>}, a contradiction with finite subdirect
irreducibility of A.
a
The proof is completed in the same way as the one in Section 4.
Claim 6: If A is as assumed in Claim 5, then A 2 wgrz.
Proof of claim: See the proof of Theorem 4.2.
a
a
7. Conclusion and open problems
The results of Section 5 show that The Blok Dichotomy is stable and cannot be
avoided even when the notion of Kripke completeness is considerably generalized. It
is natural to ask if there is any non-trivial completeness notion for which The Blok
Dichotomy does not hold. The notion of discrete completeness can be generalized
in two different ways: to A-completeness, i.e., completeness with respect to atomic
baos and V-completeness, i.e., completeness with respect to completely additive
baos; the latter notion also generalizes residual completeness. In neither case it is
known whether The Blok Dichotomy can survive such a vast generalization.
Problem 1. Is every variety of n-baos V-complete?
As for atomic algebras, examples of incomplete varieties from existing literature
are cited above. But even the existence of incompleteness results does not necessarily mean that The Blok Dichotomy holds for such a weak notion. To see the
difference with the Kripke and even the discrete case, consider the following result
of Buszkowski [4], [5]:
STABILITY OF THE BLOK THEOREM
23
Theorem 7.1 (Buszkowski). Every variety of n-baos axiomatized by a set of formulas where every occurrence of a variable lies within the scope of some modal
operator is complete with respect to atomic baos.
This is a strong result. It shows that generalization of proofs of The Blok Dichotomy sketched in this paper can be problematic for atomic algebras. If the term
ψ in the proof of Theorem 5.1 satisfy the assumption of The Buszkowski Theorem,
then equations used in that proof axiomatize A-complete variety.
Problem 2. Investigate the degrees of A-incompleteness.
Another intriguing challenge is to investigate K4-degrees of incompleteness. As
was mentioned in Section 4, the present paper is the first which shows that there
are extensions of K4 whose K4-degree of Kripke incompleteness is 2ℵ0 , but general
picture remains almost as unclear as it was in the 1970’s.
Problem 3. Investigate the K4-degrees of incompleteness.
In particular, the following question seems intriguing.
Problem 4. Are all varieties of K4-algebras AV-complete and T -complete?
Almost needless to say, the problem of existence of Blok-like theorems for varieties corresponding to logics different than modal logics with classical propositional
base —e.g., for superintuitionistic, relevant or substructural logics—is a terra incognita. To the best of our knowledge, there were no attempts whatsoever to investigate questions of this kind. Even in the modal case, our results can still be
improved upon: the proofs do not work for logics with infinitely many operators or
with operators whose arity is greater than one.
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School of Information Science, JAIST, Asahidai 1-1, Nomi-shi, Ishikawa 923-1292 JAPAN
E-mail address: [email protected]
URL: http://www.jaist.ac.jp/ litak