Arch. Math. Logic 45, 149–177 (2006)
Digital Object Identifier (DOI):
10.1007/s00153-005-0303-1
Mathematical Logic
Radim Bělohlávek · Vilém Vychodil
Fuzzy Horn logic II
implicationally defined classes
Received: 20 January 2004 / Revised version: 22 November 2004
Published online: 27 September 2005 – © Springer-Verlag 2005
Abstract. The paper studies closure properties of classes of fuzzy structures defined by
fuzzy implicational theories, i.e. theories whose formulas are implications between fuzzy
identities. We present generalizations of results from the bivalent case. Namely, we characterize model classes of general implicational theories, finitary implicational theories, and
Horn theories by means of closedness under suitable algebraic constructions.
1. Introduction and preliminaries
The present paper is a follow up to [10] where we introduced fuzzy Horn logic and
presented its general completeness theorem plus some completeness theorems for
some important special cases. The main aim of this paper is to study implicationally
defined model classes. In particular, we characterize the model classes by means
of closedness under suitable algebraic constructions. Our results generalize analogous results for the ordinary case which are well-known in universal algebra. Note
that a characterization of equationally defined classes generalizing the well-known
Birkhoff variety theorem was presented in [8]. As equationally defined classes are
special cases of implicationally defined classes (equations correspond to implications with empty premises), the present paper shifts the results of [8] to a more
general setting.
In the rest of this section we briefly recall the necessary notions. For further
details, we refer mainly to [10] (implications between identities), [9, 24] (constructions related to algebras with fuzzy equalities), [7] (general first-order fuzzy
structures), [15, 17, 20] (fuzzy logic).
We use complete residuated lattices as the structures of truth degrees. A
(complete) residuated lattice is an algebra L = L, ∧, ∨, ⊗, →, 0, 1 of type
2, 2, 2, 2, 0, 0 such that (i) L, ∧, ∨, 0, 1 is a (complete) lattice with the least
element 0 and the greatest element 1, (ii) L, ⊗, 1 is a commutative monoid,
R. Bělohlávek: Department Computer Science, Palacký University, Tomkova 40, 779 00,
Olomouc, Czech Republic. e-mail: [email protected]
V. Vychodil: Department Computer Science, Palacký University, Tomkova 40, 779 00,
Olomouc, Czech Republic. e-mail: [email protected]
Key words or phrases: Fuzzy logic – Equational Logic – Horn logic – Implication – Degree
of provability
150
R. Bělohlávek, V. Vychodil
(iii) ⊗, → is an adjoint pair, i.e. a ⊗ b ≤ c iff a ≤ b → c is valid for each
a, b, c ∈ L (so called adjointness property).
An L-set A (or fuzzy set with truth degrees in L) in a universe set U is a
mapping A : U → L, A(u) ∈ L being interpreted as the truth value of “element u belongs to A”. Let LU denote the collection of all L-sets in universe
U . A mapping ∅U : U → L with ∅U (u) = 0 (u ∈ U ) is called an empty Lset in U . For every L-set A : U → L, we define an ordinary set Supp(A) by
Supp(A) = {u ∈ U | A(u) > 0}. Supp(A) is called the support set of A. L-set
A is called finite if Supp(A) is finite. For every L-set A : U → L and a ∈ L,
we define an ordinary set aA by aA = {u ∈ U | A(u) ≥ a}. aA is called an a-cut
of A. L-sets can be represented by L-indexed
systems of a-cuts
[7]. For L-sets A
and B, S(A, B) defined by S(A, B) = u∈U A(u) → B(u) is called a degree
of subsethood of A in B. We write A ⊆ B (A is a subset of B) iff S(A, B) = 1,
i.e. iff for each u ∈ U , A(u) ≤ B(u). Basic operations with L-sets are defined
componentwise using operations of L. A binary L-relation R (or binary fuzzy relation with truth degrees in L) on U is an L-set in the universe set U × U , i.e. it
is a mapping R : U × U → L. An L-equivalence (or similarity) E on U is a
binary L-relation on U satisfying E(u, u) = 1 (reflexivity), E(u, v) = E(v, u)
(symmetry), E(u, v) ⊗ E(v, w) ≤ E(u, w) (transitivity) for all u, v, w ∈ U . An
L-equivalence on U for which E(u, v) = 1 implies u = v is called an L-equality. Function f : U n → U is said to be compatible
with binary L-relation
R on
U if R(u1 , v1 ) ⊗ · · · ⊗ R(un , vm ) ≤ R f (u1 , . . . , un ), f (v1 , . . . , vn ) for all
u1 , v1 . . . un , vn ∈ U . An L-set A is called crisp if A(u) ∈ {0, 1} for each u ∈ U .
As usual, we sometimes identify crisp L-sets with the corresponding ordinary sets.
A type is a collection F of function symbols, each with its arity. Given a complete
residuated lattice L, the language of L-Horn logic consists of (at least denumerable) set X of variables, a type F , a binary predicate symbol ≈ standing for (fuzzy)
equality, a set {a; a ∈ L} of symbols of truth values (however, for the sake of convenience and since there is no danger of misunderstanding, we identify a V
with a),
and symbols of logical connectives i (implication), c (conjunction) and (generalized conjunction). The set T (X) of all terms over F and X is defined as usual.
Terms are denoted by p, q, . . . , t, possibly with indices. The set of all variables
occurring in t is denoted by var(t).
AnMalgebra
with L-equality
(shortly an L-algebra) of type F is a triplet M =
M, ≈ , F M , where M, F M is an (ordinary) algebra of type F and ≈M is an
L-equality on M such that each f M ∈ F M is compatible with ≈M .
A subuniverse of M is any
subset N ⊆ M which is closed under all operations of M. L-algebra N = N, ≈N , F N is called a subalgebra of M (denoted
N ∈ Sub(M)), if N, F N is a subalgebra of M, F M (as an ordinary algebra) and
≈N is a restriction of ≈M to N . The notion of an L-algebra
generated by a set of
elements is defined as in ordinary case. We denote N M the least subuniverse
of M containing N . If N M = ∅, the corresponding L-algebra N with universe
N M is called an L-algebra generated by N . If |N | < κ for an infinite cardinal
κ then N is called κ-generated. ω-generated L-algebra is called finitely generated.
Fuzzy Horn logic II
151
Let M, N be L-algebras of the same type. A mapping h : M → N satisfying
a ≈M b ≤ h(a) ≈N h(b) is called an ≈-morphism. An ≈-morphism h : M → N
is
called
a morphism
(of L-algebras) if h is a morphism between ordinary algebras
M, F M and N, F N . A morphism h : M → M is called an endomorphism. A morphism h : M → N satisfying a ≈M b = h(a) ≈N h(b) (a, b ∈ M) is called an
embedding. Surjective embedding is called an isomorphism. For an isomorphism
h : M → N, M and N are called isomorphic (M ∼
= N). If h : M → N is surjective
(epimorphism), then N is called an image of M. For morphisms h : M → M and
g : M → M we consider a composed morphism (h◦g) : M → M as a composed
mapping.
An L-relation θ on M such that (i) θ is an L-equivalence on M, (ii) ≈M ⊆ θ ,
(iii) all functions f M ∈ F M are compatible with θ, is called a congruence on M.
Congruences on an L-algebra M form a complete lattice [9] denoted by ConL (M).
θ (R) ∈ ConL (M) denotes the congruence generated by R : M ×M → L. For a congruence θ on an L-algebra M, an L-algebra M/θ = M/θ, ≈M/θ , F M/θ , where (i)
M/θ, F M/θ is an ordinary factor algebra of M, F M modulo {a, b | θ(a, b) =
1}, (ii) [a]θ ≈M/θ [b]θ = θ (a, b) for all a, b ∈ M, is called a factor L-algebra of
M modulo θ . An epimorphism hθ : M → M/θ, where hθ (a) = [a]θ (a ∈ M), is
called a natural morphism. For a morphism h : M → N let θh ∈ L be a congruence
defined by θh (a1 , a2 ) = h(a1 ) ≈N h(a2 ) (a, b ∈ M). θh is called a kernel of h.
The above-introduced concepts obey the usual theorems on morphisms (see [7] for
first-order fuzzy structures
for L-algebras
Q in particular).
Q
Q and [9]Q
Mi , F i∈I Mi of L-algebras M
A direct product i∈I Mi =
M
,
≈
i Q i∈I
i
i∈I
Q
i∈I Mi is a direct product of ordinary
(i ∈ I ) is an L-algebra such
that
M
,
F
i
i∈I
Q
Q
algebras MiQ
, F Mi and ≈ i∈I Mi is defined by a ≈ i∈I Mi b = i∈I a(i) ≈Mi b(i).
For I = ∅, i∈IQMi is a trivial (one-element) L-algebra. For every j ∈ I an epimorphism πjQ: i∈I Mi → Mj , where πj (a) = a(j ) (a ∈ M) is called a j -th
projection of i∈I Mi . A subdirect product of L-algebras Mi (i ∈ I ) is a sublagebra
of the direct product of Mi ’s such that each πj is a surjective mapping.
A partially ordered index set I, ≤ is called directed, if I = ∅ and for every
i, j ∈ I there is k ∈ I such that i, j ≤ k. A family {Mi | i ∈ I } of L-algebras, where
I, ≤ is a directed index set and Mi ∈ Sub(Mj ) for i ≤ jis called adirected family
i∈I Mi , F i∈I Mi of a diof L-algebras. A direct union i∈I Mi =
i∈I Mi , ≈
rected family {Mi | i ∈ I } of L-algebras is an L-algebra, where i∈I Mi , F i∈I Mi
is a direct union of ordinary algebras Mi , F Mi (that is, for a, . . . , b ∈ i∈I Mi ,
f i∈I Mi (a, . . . , b) is defined to be f Mi (a, . . . , b) for i suchthat a, . . . , b ∈ Mi )
and for a, b ∈ i∈I Mi with a ∈ Mi and b ∈ Mk we put a ≈ i∈I Mi b = a ≈Mk b
for k ≥ i, j . Every L-algebra is isomorphic to a direct union of its finitely generated
subalgebras [9].
A weak direct family of L-algebras of type F consists of: (i) a directed index set I, ≤; (ii) a family {Mi | i ∈ I
} of pairwise disjoint L-algebras of type F ;
(iii) a family hij : Mi → Mj | i ≤ j of morphisms, where hii = idMi (i ∈ I )
and hik = hij ◦ hj k (i ≤ j ≤ k). A weak direct family is called a direct
family if for every a ∈ Mi , b ∈ Mj there exists k ∈ I , i, j ≤ k such that
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R. Bělohlávek, V. Vychodil
for each l ∈ I , k ≤ l we have hik (a) ≈Mk hj k (b) = hil (a) ≈Ml hj l (b)
(this condition is automatically satisfied if L is a finite residuated lattice). For
every weakdirect family of L-algebras {Mi | i ∈ I } let θ∞ denote an L-equivalence on i∈I Mi defined by θ∞ (a, b) = k≥i,j hik (a) ≈Mk hj k (b) (a ∈
Mi /θ∞ , ≈lim Mi
Mi , b ∈ Mj ). Furthermore, an L-algebra lim Mi =
i∈I
,
F lim Mi , where (i) i∈I Mi /θ∞ is a factorization of i∈I Mi by 1 θ∞
, i.e.
/θ∞ = [a]θ∞ | a ∈ i∈I Mi , [a]θ∞ = a | θ∞ (a, a ) = 1 ; (ii)
i
i∈I M
f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ = f Mk (hi1 k (a1 ), . . . , hin k (an )) θ for every n∞
ary f ∈ F and arbitrary [a1 ]θ∞ , . . . , [an ]θ∞ ∈
i∈I Mi /θ∞ such that a1 ∈
, k ≥ i1, . . . , in ; (iii) [a]θ∞ ≈lim Mi [b]θ∞ =
Mi1 , . . . , an ∈ Min , and k ∈ I
θ∞ (a, b) for all [a]θ∞ , [b]θ∞ ∈
i∈I Mi /θ∞ ; is called a direct limit of a (weak)
direct family {Mi | i ∈ I }. A family {hi : Mi → lim Mi | i ∈ I } of morphisms
hi : Mi → lim Mi , where hi (a) = [a]θ∞ (i ∈ I , a ∈ Mi ) is called a limit
cone of {Mi | i ∈ I }.
L-algebras of the same type and let F be a
Let {Mi | i ∈ I } be a family of Q
filter over I . Then for every a, b ∈ i∈I M i and X ∈ F we define a truth degree
Mi b(i). We define a congruence
[[a ≈ b]]X ∈Q L by [[a ≈ b]]X = i∈X a(i) ≈
Q
θQF ∈ ConL ( i∈I Mi ) by θF (a, b) = X∈F [[a ≈ b]]X . i∈I Mi /θF denoted by
F Mi is called a reduced product of {Mi | i ∈ I } modulo F . Reduced products
of L-algebras correspond in a natural way to direct
limits [24]. A filter F over I
Q
is safe w.r.t. {Mi | i ∈ I } if for every a, b ∈ i∈I M i there is X ∈ F such that
θF (a, b) = [[a ≈ b]]X . If K is a class of L-algebras of the same type and F is safe
w.r.t. every family of L-algebras taken from K then F is said to be K-safe. If F is
K-safe for arbitrary class K of L-algebras
then F is called safe. If F is safe with
Q
{M
}
respect to a family
|
i
∈
I
then
M
is called a safe reduced product.
i
i
F
T(X) = T (X), ≈T(X) , F T(X) denotes the term L-algebra of type F (in X),
i.e. T (X), F T(X) is an ordinary term algebra and ≈T(X) is crisp L-equality. Every
≈-morphism h : X → M (M being a universe of M) has a uniquely determined
homomorphic extension h : T(X) → M. If M ∼
= T(X)/θ (R), where X and R
are finite, then M is called a finitely presented L-algebra. Every L-algebra is isomorphic to a direct limit of a direct family of finitely presented L-algebras [24].
For every morphism h : M → lim Mi from a finitely presented L-algebra M to a
direct limit lim Mi of a direct family of L-algebras there is k ∈ I and a morphism
g : M → Mk such that h = g ◦ hk . The previous claim does not apply for direct
limits of general weak direct families of L-algebras [24].
Classes of L-algebras are denoted by K, K , . . . A class K of L-algebras is
an abstract class, if it is closed under isomorphic images. In the sequel we work
mostly with abstract classes of L-algebras. Class operators corresponding to the
above presented constructions are denoted by H (homomorphic images), I (isomorphic images), S (subalgebras), P (direct products), U (direct unions), L (direct
limits of direct families), PR (safe reduced products). That is, H(K) denotes the
class of all homomorphic images of L-algebras from K etc.
The paper is organized as follows. Section 2 introduces implicationally defined
classes and shows basic closure properties. Section 3 characterizes classes defined
Fuzzy Horn logic II
153
by general implications and classes defined by finitary implications, Section 4 characterizes classes defined by Horn implications.
2. Implicationally defined classes and closure properties
The notion of an implication of identities in fuzzy setting was introduced in [10];
we recall the basic concepts.
For P ∈ LT (X)×T (X) , and endomorphism h : T(X) → T(X) we define an
endomorphic image h(P ) ∈ LT (X)×T (X) by
h(P )(t, t ) =
P (s, s ) | h(s) = t, h(s ) = t T (X)×T (X) . For P ∈ LT (X)×T (X) , put
for every t,
t ∈ T (X). Let ∅ = P ⊆ L var(P ) =
var(t) ∪ var(t ) | P (t, t ) > 0 , i.e. var(P ) is a set of variables occurring in identities which belong to P in some nonzero degree.
A family P ⊆ LT (X)×T (X) is called a proper family of premises of type F (in
variables X) if for every P ∈ P and every endomorphism h on T(X) we have
h(P ) ∈ P. Every P ∈ P is then called an L-set of premises. A P-implication is an
expression of the form
V
P (s,s )>0 s ≈ s , P (s, s ) i (t ≈ t ).
where P ∈ P and t, t ∈ T (X). For a P-implication ϕ and a truth value a ∈ L, the
couple ϕ, a is called a weighted P-implication.
P-implications
(weighted
P-impli
cations) will be denoted by P ⇒ (t ≈ t ) ( P ⇒ t ≈ t ), a or P i t ≈ t , a ).
L-sets of P-implications are denoted usually by , , . . . . Note that L-sets of Pimplications naturally correspond to (ordinary) sets of weighted P-implications and
vice versa.
Any finite L-relation P ∈ LT (X)×T (X) , where P (t, t ) > 0 implies P (t, t ) =
P (t, t ) for every terms t, t ∈ T (X), is called a finite restriction of P ∈ P. In the
sequel, a set of all finite restrictions of P ∈ P will be denoted by Fin(P ).
A general semantics of P-implications can be approached in a unified way using
an additional unary operation “∗ ” on L. A unary operation ∗ : L → L satisfying
1∗ = 1,
a ∗ ≤ a,
(2)
(a → b)∗ ≤ a ∗ → b∗ ,
(3)
(1)
for every a, b ∈ L, is called a truth stresser for L [17, 18]. Let L∗ denote L equipped
with a truth stresser ∗ . A truth stresser ∗ satisfying
a ∗∗ = a ∗ ,
a∗ ⊗ a∗ = a∗,
∗
∗
i∈I ai =
i∈I ai ,
(4)
(5)
(6)
for every a ∈ L, ai ∈ L for all i ∈ I , is called an implicational truth stresser.
Unless stated otherwise, we assume that ∗ is an implicational truth stresser.
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R. Bělohlávek, V. Vychodil
For examples of truth stressers in our context, see [10]. An important example
of an implicational truth stresser is a so-called globalization [23]:
1
if a = 1,
∗
a =
(7)
0
otherwise.
If L is a chain, the globalization coincides with Baaz’s operation
[2, 17]. For an
Lalgebra M and a valuation v : X → M, we define a truth degree P ⇒ (t ≈ t )M,v
of P ⇒ (t ≈ t ) in M under v with respect to ∗ by
P ⇒ (t ≈ t )
= P M,v → t ≈ t M,v ,
M,v
where
∗
P (s, s ) → s ≈ s M,v .
For technicalreasons, for every weighted P-implication
P ⇒ (t ≈ t ), a , we de
fine a degree P ⇒ (t ≈ t ), a M,v by a → P ⇒ (t ≈ t )M,v . The truth stresser
∗ plays a role of a thresholding function. For instance, for ∗ defined by (7) we have


if P (s, s ) ≤ s ≈ s M,v
 t ≈ t M,v
P ⇒ (t ≈ t )
=
for all s, s ∈ T (X),
M,v


1
otherwise.
P M,v =
s,s ∈T (X)
As usual, for an L-algebra M and a class K of L-algebras we define the truth
degrees of P ⇒ (t ≈ t ) in M and K by
P ⇒ (t ≈ t ) = v:X→M P ⇒ (t ≈ t ) M,v ,
M
P ⇒ (t ≈ t ) = P ⇒ (t ≈ t ) .
K
M∈K
M
Mod() denotes the class of all models of , i.e.
Mod() = M | (ϕ) ≤ ϕM for every P-implication ϕ .
Let P be a proper family of premises. Put PFin = {P ∈ P | P is finite}, where
P finite means that Supp(P ) is a finite set. We will use the term P-Horn clauses
instead of PFin -implications. Analogously, we can define a restriction Pω of P on
premises with finitely many variables:
Pω = {P ∈ P | var(P ) is finite} .
Pω is a proper family of premises, see [10]. In what follows, we will use the term
P-finitary implications instead of Pω -implications. More generally, for an infinite
cardinal κ we put
Pκ = {P ∈ P | | var(P )| < κ} .
Clearly, Pκ is proper since | var(h(P ))| ≤ | var(h(var(P )))| < κ for every P ∈ Pκ
(each var(h(x)) is finite).
Fuzzy Horn logic II
155
Definition 1. A class K of L-algebras is called an P -implicational class determined by ∗ if there is an L-set of P-implications such that K = Mod(). A
P-Horn class)
P-implicational class K is called a P -finitary implicational class (P
if can be taken so that it is an L-set of P-finitary implications (P-Horn clauses).
For a class K of L-algebras of a type F and Q ⊆ P we define an L-set ImplQ(K)
of P-implications by
P ⇒ (t ≈ t )
if P ∈ Q,
K
ImplQ(K) P ⇒ (t ≈ t ) =
0
otherwise.
We write Impl(K), Implκ (K), and Horn(K) instead of ImplP(K), ImplPκ (K), and
ImplPFin (K), respectively.
Remark 1. Since PFin ⊆ Pω ⊆ P, every P-Horn class is a P-finitary class and every
P-finitary class is a P-implicational class.
We are now going to show some closure properties of implicational classes.
Lemma 1. Let ⇒ P t ≈ t be a P-implication. Then,
(i) every N ∈ Sub(M),
P ⇒ (t ≈ t )M ≤ P ⇒(t ≈ t ) N for
(ii) P ⇒ (t ≈ t ){M | i∈I } ≤ P ⇒ (t ≈ t )Q M for all Mi ’s (i ∈ I ),
i
i∈I i
(iii) P ⇒ (t ≈ t )
≤ P ⇒ (t ≈ t )
{Mi | i∈I }
M
for M being a subdirect product of Mi ’s (i ∈ I ).
Proof. (i): Follows from the fact that ≈N is a restriction of ≈M on N . Note that (i)
holds true for general truth stressers (conditions (4)–(6) are not required).
by πi , we have
(ii): Denoting thei-th projection
r ≈ r r ≈ r Q
=
. Using (6),
i∈I
Mi ,v◦πi
i∈I Mi ,v
∗
i∈I P Mi ,v◦πi =
i∈I
s,s ∈T (X) P (s, s ) → s ≈ s Mi ,v◦πi
∗
=
i∈I
s,s ∈T (X) P (s, s ) → s ≈ s Mi ,v◦πi
∗
=
s,s ∈T (X)
i∈I P (s, s ) → s ≈ s Mi ,v◦πi
∗
=
s,s ∈T (X) P (s, s ) →
i∈I s ≈ s Mi ,v◦πi
∗
= s,s ∈T (X) P(s, s ) → s ≈ s Q M ,v =P Qi∈I Mi ,v .
i∈I
i
We further obtain
P ⇒ (t ≈ t )
i∈I
vi :X→Mi P ⇒ (t ≈ t ) Mi ,vi
{Mi | i∈I }
= v:X→Q Mi i∈I P ⇒ (t ≈ t )M ,v◦π
i
i∈I
i
= v:X→Q Mi i∈I P Mi ,v◦πi → t ≈ t M ,v◦π ≤
i
i
i∈I
≤ v:X→Q Mi i∈I P Mi ,v◦πi → i∈I t ≈ t M ,v◦π
i
i
i∈I
= v:X→Q Mi P Qi∈I Mi ,v → t ≈ t Q M ,v
i∈I
i∈I i
Q
P ⇒ (t ≈ t )Q
=
= P ⇒ (t ≈ t )Q
v:X→
i∈I Mi
(iii): Consequence of (i) and (ii).
i∈I Mi ,v
i∈I Mi
.
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R. Bělohlávek, V. Vychodil
Lemma 2. For an L-set of P-implications, Mod() is an abstract class of
L-algebras closed under the formations of subalgebras and direct products.
Proof. It is almost immediate that if M ∈ I(Mod()) then M ∈ Mod(), i.e.
Mod() is an abstract class of L-algebras.
The facts that S(Mod()) ⊆ Mod() and P(Mod()) ⊆ Mod() follow by
Lemma 1. For instance, if N ∈ S(Mod()),
i.e. N ∈ Sub(M)
for some L-algebra
M ∈Mod() then (P ⇒(t ≈ t )) ≤ P ⇒ (t ≈ t )M since M ∈ Mod()
and P ⇒ (t ≈ t )M ≤ P ⇒ (t ≈ t )N by Lemma 1, whence (P ⇒
(t ≈ t )) ≤ P ⇒ (t ≈ t )N proving N ∈ Mod().
Analogously as in the ordinary case, one may easily show that P-implicational
classes are not closed under homomorphic images (e.g. using implications expressing cancellation rule).
Lemma 3. Let i∈I Mi be a direct union of a directed family {Mi | i ∈ I } of Lalgebras. Then
P ⇒ (t ≈ t )
{M | i∈I } ≤ P ⇒ (t ≈ t )
M
i
i
i∈I
for every P-finitary implication P ⇒ (t ≈ t ).
Proof. Since P ⇒ (t ≈ t ) is a P-finitary implication,
Y = var(P ) ∪ var(t) ∪
, we can consider a
var(t ) is a finite set. Thus, for every valuation v of X in i∈I Mi finite set M = {v(x) | x ∈ Y }. Moreover, M is a finite subset of i∈I Mi , i.e. there
the restriction
is some index k ∈ I such that M ⊆ Mk . Hence
vY of
v on Y can
be thought of as a valuation in Mk ∈ Sub( i∈I Mi ). Thus, r ≈ r M ,v =
i Y
i∈I
r ≈ r for every r, r ∈ s, s | P (s, s ) > 0 ∪ t, t . Furthermore,
Mk ,vY
P ⇒ (t ≈ t )
{Mi | i∈I } ≤ P ⇒ (t ≈ t ) Mk ,vY
= P ⇒ (t ≈ t ) M ,v = P ⇒ (t ≈ t ) M ,v .
i
Y
i
i∈I
i∈I
As a consequence, P ⇒ (t ≈ t ){M | i∈I } ≤ P ⇒ (t ≈ t ) M .
i
i∈I
i
Note that Lemma 3 is true for general truth stressers.
Lemma 4. For an L-set of P-finitary implications, Mod() is an abstract class
of L-algebras closed under the formations of subalgebras, direct products, and
direct unions.
Proof. Closedness under direct unions follows from Lemma 3. The rest follows
from Lemma 2.
Lemma 5. Let lim Mi be a direct limit of a direct family {Mi | i ∈ I } of L-algebras.
Then
P ⇒ (t ≈ t )
{M | i∈I } ≤ P ⇒ (t ≈ t ) lim M
i
for every P-Horn clause P ⇒ (t ≈
i
t ).
Fuzzy Horn logic II
157
Proof. Let us have a valuation v : X→ i∈I Mi /θ∞ and
let P ⇒ (t ≈ t ) be a
| m = 1, . . . , n . Take Y = var(P ) ∪
P-Horn clause, where Supp(P ) = tm , tm
var(t) ∪ var(t ) and consider a restriction vY of v on Y and the homomorphic
extension vY : T(Y ) → lim Mi . Since Y is finite, T(Y ) is finitely presented. Due
to the image factorization (see [24]), there is an index k ∈ I and a morphism
g : T(Y ) → Mk such that vY = g ◦ hk . Let gY denote a restriction of g on Y . We
have
rlim Mi ,v = rlim Mi ,vY = vY (r) = hk (g(r)) = hk rMk ,gY
for any r ∈ T (Y ). Moreover for r, r ∈ T (Y ) it follows that
r ≈ r = rlim Mi ,v ≈lim Mi r lim M ,v
lim Mi ,v
i
lim M
i
= hk rMk ,gY ≈
hk r M ,g
k Y
= rMk ,gY θ ≈lim Mi r M ,g θ
∞
k Y ∞
= θ∞ rMk ,gY, r M ,g .
k
Y
For every r, r ∈ T (Y ) there is an index l ∈ I , k ≤ l such that
r ≈ r = θ∞ rMk ,gY, r M ,g
lim Mi ,v
k Y
= hkl rMk ,gY ≈Ml hkl r M ,g
k Y
= rMl ,gY ◦hkl ≈Ml r M ,g ◦h = r ≈ r M ,g
l
Y
kl
l
Y ◦hkl
.
Therefore there are indices j0 , j1 , . . . , jn ≥ k such that
tm ≈ t for each m = 1, . . . , n,
m lim Mi ,v = tm ≈ tm Mjm,gY ◦hkjm
t ≈ t = t ≈ t M ,g ◦h .
lim M ,v
i
j0
Y
kj0
Moreover, I is directed, i.e. there is an index j ∈ I with j0 , j1 , . . . , jn ≤ j . Using
properties of direct families it follows that
tm ≈ t for each m = 1, . . . , n,
m lim Mi ,v = tm ≈ tm Mj ,(gY ◦hkjm )◦hjm j
t ≈ t = t ≈ t M ,(g ◦h )◦h .
lim M ,v
i
j
Y
kj0
j0 j
Now observe that for each m = 0, . . . , n we have
(gY ◦ hkjm ) ◦ hjm j = gY ◦ (hkjm ◦ hjm j ) = gY ◦ hkj .
Denoting gY ◦ hkj by w, we get
P ⇒ (t ≈ t )
{M | i∈I } ≤ P ⇒ (t ≈ t ) M
j ,w
i
Hence, P ⇒ (t ≈ t ){M
i
| i∈I }
= P ⇒ (t ≈ t )lim M ,v .
i
≤ P ⇒ (t ≈ t )lim M .
i
As one can see from the proof, Lemma 5 holds true for general truth stressers.
158
R. Bělohlávek, V. Vychodil
Lemma 6. For an L-set of P-Horn clauses, Mod() is an abstract class of
L-algebras closed under the formations of subalgebras, direct products, direct
unions, and direct limits.
Proof. Closedness under direct limits follows from Lemma 5. The rest follows
from Lemma 4.
∗
∗
∗
In ordinary case, validity of an implication can be expressed using the notion of
injectivity. Namely, an algebra M is injective w.r.t. an implication P ⇒ (t ≈ t ) iff
P ⇒ (t ≈ t ) is valid in M. This criterion is known as the Banaschewski-Herrlich
criterion. Such a criterion can be used to prove that a Horn class is closed under L.
In what follows, we present an analogy to Banaschewski-Herrlich criterion.
Definition 2. Let L∗ be a complete residuated
lattice with a truth stresser ∗ . For a
weighted P-implication P ⇒ (t ≈ t ), a we define
Q(s, s ) =
P (t, t ) ∨ a
P (s, s )
if s = t, and s = t ,
otherwise.
Let hP Q : T(X)/θ (P ) → T(X)/θ (Q) be a morphism defined by
hP Q [t]θ(P ) = [t]θ(Q)
(8)
for all t ∈ T (X). An L-algebra M is said to be injective w.r.t. P ⇒ (t ≈ t ), a if for
every morphism h : T(X)/θ (P ) → M, there exists a morphism g : T(X)/θ (Q) →
M such that h = hP Q ◦ g.
Remark 2. One can check that hP Q defined by (8) is a well-defined morphism.
∗
∗
Theorem
1. Let L
be a complete residuated lattice with a truth stresser and let
P ⇒ (t ≈ t ), a be a weighted P-implication.
(i) If a ≤ P ⇒ (t ≈ t )M , then M is injective w.r.t. P ⇒ (t ≈ t ), a .
(ii) For ∗ defined
by (7) if Mis injective w.r.t. P ⇒ (t ≈ t ), a ,
then a ≤ P ⇒ (t ≈ t )M .
Proof. (i): By assumption, a ≤ P ⇒ (t ≈ t )M,v for every v : X → M.
Consider
h : T(X)/θ (P ) → M and a valuation v : X → M, where
a morphism
v of v we have
v(x) = h [x]θ(P ) , x ∈ X. Hence, for a homomorphic
extension
v = hθ(P ) ◦ h. Furthermore, it follows that s ≈ s M,v = v (s) ≈M v (s ) =
θv (s, s ) = θhθ (P ) ◦h (s, s ) for all terms s, s ∈ T (X). Therefore,
P (s, s ) ≤ θ (P )(s, s ) = [s]θ(P ) ≈T(X)/θ(P ) s θ(P )
≤ h [s]θ(P ) ≈M h s θ(P ) = θhθ (P ) ◦h (s, s ) = θv (s, s )
Fuzzy Horn logic II
159
for all s, s ∈ T (X). By (1),
a ≤ P ⇒ (t ≈ t )M,v
∗
=
→ t ≈ t M,v
s,s ∈T (X) P (s, s ) → s ≈ s M,v
∗
=
→ θv (t, t )
s,s ∈T (X) P (s, s ) → θv (s, s )
= 1∗ → θv (t, t ) = 1 → θv (t, t ) = θv (t, t ).
We thus have P (t, t ) ≤ θv (t, t ) and a ≤ θv (t, t ), i.e. Q(t, t ) = P (t, t ) ∨ a ≤
θv (t, t ). Since θ (Q) ∈ ConL (T(X)) is generated by Q, it readily follows that
θ (Q) ⊆ θv . Now, by standard argument, there is a morphism g : T(X)/θ (Q) → M
such that v = hθ(Q) ◦ g. Hence, hθ(P ) ◦ h = hθ(Q) ◦ g, that is hθ(P ) ◦ h =
of hθ(P ) implies h = hP Q ◦ g. Hence, M is injective
(hθ(P ) ◦ hP Q ) ◦ g. Surjectivity
w.r.t. P ⇒ (t ≈ t ), a .
(ii): Let ∗ be defined by (7)
and let M be injective w.r.t. P ⇒ (t ≈ t ), a .
We have to show that a ≤ P ⇒ (t ≈ t )M,v for every valuation v of X in
M. Take a valuation
v : X → M. If there are terms s, s ∈ T (X)
such that
P (s, s ) s ≈ s M,v , then P M,v = 0 and thus a ≤ P ⇒ (t ≈ t )M,v = 1.
Hence, we focus on the nontrivial case. Let P (s, s ) ≤ s ≈ s M,v for all
s, s ∈ T (X).
From (7) it follows that P M,v = 1. Hence, we need to check
a ≤ t ≈ t M,v . For a homomorphic extension v of v we have P ⊆ θv , that is
g : T(X)/θ (P ) → M
θ (P ) ⊆ θv . Furthermore, it follows that there is a morphism
such that v = hθ(P ) ◦ g . Since M is injective w.r.t. P ⇒ (t ≈ t ), a , there is a
morphism g : T(X)/θ (Q) → M with g = hP Q ◦ g. Thus, v = hθ(P ) ◦ g =
hθ(P ) ◦ hP Q ◦ g = hθ(Q) ◦ g. As a consequence,
a ≤ θ (Q)(t, t ) = [t]θ(Q) ≈T(X)/θ(Q) t θ(Q) ≤ g [t]θ(Q) ≈M g [t]θ(Q)
= v (t) ≈M v (t ) = tM,v ≈M t M,v = t ≈ t M,v .
By (1), we obtain a ≤ t ≈ t M,v = 1∗ → t ≈ t M,v = P ⇒ (t ≈ t )M,v
showing a ≤ P ⇒ (t ≈ t )M .
Remark 3. (1) For ∗ being the globalization, Theorem 1 gives an “if and only if”
criterion for a P-implication to be true in M in degree at least a.
(2) Since identities can be thought of as P-implications for P = {∅}, the truth
stresser does not play any role and we can apply Theorem 1 without mentioning ∗ .
(3) Theorem 1 can be used to show that for globalization,
P ⇒ (t ≈ t )
≤ P ⇒ (t ≈ t )
{Mi | i∈I }
lim Mi
is true for every direct family
{Mi | i ∈ I }of L-algebras (this is already covered
by Theorem 5): Put a = P ⇒ (t ≈ t ){M | i∈I } , i.e. a ≤ P ⇒ (t ≈ t )M
i
i
for all i ∈ I . That is, every Mi isinjective w.r.t. P ⇒ (t ≈ t ), a . It remains to
show that lim Mi is injective w.r.t. P ⇒ (t ≈ t ), a as well. Consider a morphism
h : T(X)/θ (P ) → lim Mi . Since P ⇒ (t ≈ t ) is a P-Horn clause, T(X)/θ (P ),
where X = var(P )∪var(t)∪var(t ) is a finitely presented L-algebra. Due to image
160
R. Bělohlávek, V. Vychodil
factorization (see [24]), for some index k ∈ I the mapping h factorizes through
(P
some component of lim Mi , i.e. h = h ◦ hk , where
h : T(X)/θ
) → Mk is
a morphism. By assumption, Mk is injective w.r.t. P ⇒ (t ≈ t ), a , thus there is
a morphism g : T(X)/θ (Q) → Mk such that h = hP Q ◦ g. As a consequence,
h = h ◦ hk = hP Q ◦ (g ◦ hk ), i.e. g ◦ hk is
the desired morphism. Thus, lim Mi is
injective with respect to P ⇒ (t ≈ t ), a .
3. Sur-reflections and sur-reflective classes
In this section, we characterize P-implicational classes as abstract classes of
L-algebras closed under subalgebras and direct products. We start by sur-reflections.
Note that in ordinary implicational classes, sur-reflections play a role analogous to
that of free algebras in the theory of varieties.
Definition 3. Let K be an abstract class of L-algebras of type F , M be an L
-algebra of type F . A morphism r : M → R, where R ∈ K, is called a reflection of
M in K, if for every morphism h : M → N, N ∈ K, there exists a uniquely determined morphism h : R → N such that h = r ◦ h . Moreover, if r is an epimorphism
(surjective morphism), then r is called a sur-reflection of M in K.
An abstract class K of L-algebras of type F is called sur-reflective if every
L-algebra M of type F admits a sur-reflection r : M → R in K.
Example 1. Let K be a an abstract class of L-algebras. If FK (X) ∈ K [8, 9] then the
natural mapping hθK (X): T(X) → FK (X) is a sur-reflection of the term L-algebra
T(X) in K. Indeed, hθK (X) is an epimorphism. Moreover, for every morphism
h : T(X) → N, where N ∈ K, we can consider a mapping g : X → N defined
by g(x) = h(x) for all x ∈ X. Since N ∈ K, we have T(X)/θh ∈ IS(K), thus
θK (X) ⊆ θh . As a consequence,
x ≈FK (X) y = θK (X)(x, y) ≤ θh (x, y) = h(x) ≈N h(y) = g(x) ≈N g(y).
That is, g is an ≈-morphism. As a consequence, g has a uniquely determined
homomorphic extension g : FK (X) → N (see [9]). Hence, h(t) = g [t]θK (X) =
(hθK (X) ◦ g )(t) holds for all t ∈ T (X), i.e. h = hθK (X) ◦ g . To sum up, hθK (X) :
T(X) → FK (X) is a sur-reflection of T(X) in K.
Theorem 2. Let K be an abstract class of L-algebras and let r1 : M → R1 ,
r2 : M → R2 be sur-reflections of M in K. Then R1 ∼
= R2 .
Proof. Since R1 , R2 ∈ K by the definition of sur-reflections, there are uniquely
determined morphisms r1 : R2 → R1 , r2 : R1 → R2 such that r1 = r2 ◦ r1 , and
r2 = r1 ◦ r2 . Thus, r1 = (r1 ◦ r2 ) ◦ r1 , and r2 = (r2 ◦ r1 ) ◦ r2 . As a consequence,
r2 ◦ r1 = idR1 , r1 ◦ r2 = idR2 . Hence, R1 ∼
= R2 .
Remark 4. According to Theorem 2, a sur-reflection of M in K is determined up
to an isomorphism. This observation enables us to denote a sur-reflection of M in
K by rM : M → RK (M). Moreover, when considering sur-reflections, we sometimes omit the surjective mapping rM and use the term “sur-reflection” for RK (M)
instead. In such a case we assume that rM is the corresponding mapping.
Fuzzy Horn logic II
161
A sur-reflection of M in an abstract class K can be thought of as the greatest
image of M in K. The notion of a greatest image can be defined as follows. An
L-algebra M ∈ K is said to be the greatest image of M in K if there is an epimorphism h : M → M and for every epimorphism g : M → N, N ∈ K we have
θh ⊆ θg . Obviously, M ∼
= M/θh , and M/θh is a “greater factor L-algebra” than
M/θg ∼
= N. That is, the definition of the greatest image corresponds well to the intuition. The following theorem characterizes the relationship between sur-reflections
and greatest images in more detail.
Theorem 3. Suppose K is an abstract class of L-algebras, S(K) ⊆ K. Then an
epimorphism r : M → R is a sur-reflection of M in K iff R is the greatest image of
M in K.
Proof. “⇒”: Let r : M → R be a sur-reflection of M in K. That is, for every
epimorphism g : M → N there is a morphism g : R → N such that g = r ◦ g .
Thus, θr ⊆ θr◦g = θg , i.e. R is the greatest image of M in K.
“⇐”: Let R be the greatest image of M in K. Take arbitrary N ∈ K and a
morphism h : M → N. Note that h is not supposed to be surjective. On the other
hand, the first isomorphism theorem [9] yields h = h ◦ g, where h : M → M/θh
is an epimorphism, and g : M/θh → N is an embedding. Since K is closed under
subalgebras, we have M/θh ∈ IS(K) = K. Moreover, R is supposed to be the
greatest image, i.e. θr ⊆ θh for some epimorphism r : M → R. Since R ∼
= M/θr ,
it follows that there is a uniquely determined morphism g : R → M/θh such that
h = r ◦ g . Therefore, h = h ◦ g = (r ◦ g ) ◦ g = r ◦ (g ◦ g). Altogether,
r : M → R is a sur-reflection of M in K.
Theorem 4. An abstract class of L-algebras is sur-reflective iff it is closed under
the formations of subalgebras and direct products.
Proof. “⇒”: Let K be a sur-reflective class of L-algebras. We check closedness
under S and P. Take M ∈ K, and N ∈ Sub(M). We show N ∈ K by checking that
N is isomorphic to its sur-reflection RK (N) with rN : N → RK (N). Consider an
embedding h : N → M. Then h = rN ◦ h for some morphism h : RK (N) → M.
Since h is an embedding, we have
a ≈N b ≤ rN (a) ≈RK (N) rN (b)
≤ h (rN (a)) ≈M h (rN (b)) = h(a) ≈M h(b) = a ≈N b
for every a, b ∈ N. Thus, rN is an embedding. Since rN is a sur-reflection it is also
an epimorphism. Hence, N ∼
= RK (N) ∈ K proving N ∈QK, i.e. S(K) ⊆ K.
{M
|
i
∈
I } ⊆ K. Q
We will show Q
Take
a
family
i∈I Mi ∈ K by proving
Q i
Q
∼
M
(
M
),
where
r
:
M
→
R
(
R
=
K
K
i∈I iQ
i∈I i
i∈I Qi
i∈I Mi ) is a sur-reflection of i∈I Mi in K. Every projection πj : i∈I Mi → M
Qj is an epimorphism.
Hence, for every j ∈ I there exists a morphism pj : RK ( i∈I Mi ) Q
→ Mj such
that
π
=
r
◦
p
.
By
standard
argument,
there
is
a
morphism
h
:
R
(
j
j
K
i∈I Mi ) →
Q
M
such
that
h
◦
π
=
p
for
every
j
∈
I
.
Thus,
r
◦
h
◦
π
=
r
◦
pj = πj
j
j
j
i∈I i
and so r ◦ h = idQi∈I Mi . Now, we have
162
R. Bělohlávek, V. Vychodil
Q
Q
b ≤ r(a) ≈RK ( i∈I Mi ) r(b)
Q
Q
≤ h(r(a)) ≈ i∈I Mi h(r(b)) = a ≈ i∈I Mi b
Q
Q
Q
for all
Q a, b ∈ i∈I M i . Hence, r : i∈I Mi → RK ( i∈I Mi ) is an isomorphism,
i.e. i∈I Mi ∈ K.
“⇐”: Suppose K is an abstract class of L-algebras of type F and let S(K) ⊆ K
and P(K) ⊆ K. We show that every L-algebra M of type F has a sur-reflection in
K. Let
a≈
i∈I Mi
HK (M) = {θ ∈ ConL (M) | M/θ ∈ K} .
HK (M) is nonempty since K is closed under P. Putting,
Q
PK (M) = θ∈HK (M) M/θ,
P(K) ⊆ K implies PK (M) ∈ K. A family {hθ : M → M/θ | θ ∈ HK (M)} of natural morphisms induces a uniquely determined morphism p : M → PK (M) with
p ◦ πθ = hθ . Finally, we have p = r ◦ s for an epimorphism r : M → R and an
embedding s : R → PK (M). That is, R ∈ IS(K) ⊆ K.
We claim that r : M → R is a sur-reflection of M in K. Take a morphism
h : M → N, N ∈ K. Using the first isomorphism theorem [9] we have h = hθh ◦ g,
where hθh : M → M/θh is a natural morphism, and g : M/θh → N is an embedding, see Fig. 1. Thus, M/θh ∈ K, i.e. θh ∈ HK (M). We have,
h = hθh ◦ g = p ◦ πθh ◦ g = r ◦ s ◦ πθh ◦ g.
Hence, for h : R → N being s ◦ πθh ◦ g we have h = r ◦ h . Since r is surjective,
the uniqueness of h is immediate. Altogether, r : M → R is a sur-reflection of M
in K, i.e. K is sur-reflective.
Corollary 1. Suppose L∗ is a complete residuated lattice equipped with an implicational truth stresser ∗ . Then every P-implicational class of L-algebras is surreflective.
Proof. Consequence of Lemma 2 and Theorem 4.
For an L-algebra M = M, ≈M , F M we can consider a set of variables X
with |X| = |M|. For the sake of convenience, we can assume X = M. Then
M
r
R
s
p
PK (M)
h θh
πθ h
M/θh
h
g
K
N
Fig. 1. Construction of a sur-reflection of M in K
Fuzzy Horn logic II
163
T(M) is a term L-algebra of type F . The terms of type F over M are denoted by a, b, f (a1 , . . . , an ), and so on while the elements of M are denoted by
a, b, f M (a1 , . . . , an ).
Evidently, for the identical mapping idM : M → M and the corresponding
homomorphic extension idM : T(M) → M we have
idM (f (a1 , . . . , an )) = f M (a1 , . . . , an ).
We will need the following L-sets of P-implications.
Definition 4. Suppose K is a sur-reflective class of L-algebras of type F . For every
L-algebra M of type F let PM = LT (M)×T (M) . Moreover, we define an L-set
PM ∈ PM of premises by
PM (s, s ) = idM (s) ≈M idM (s )
K of P -implications
for all s, s ∈ T (M). A PM -theory of M over K is an L-set M
M
defined by
rM idM (t) ≈RK (M) rM idM (t ) if P = PM ,
K
M P ⇒ (t ≈ t ) =
0
otherwise,
where rM : M → RK (M) is a sur-reflection of M in K.
Lemma 7. Suppose L∗ is a complete residuated lattice with a truth stresser ∗
defined
K be a sur-reflective class of L-algebras of type F . Then
by (7). Let
K ) where M ranges over all L-algebras of type F .
K = M Mod(M
K
K
Proof. “K ⊆
M Mod(M )”: It suffices to show K ⊆ Mod(
M ) for every
K
L-algebra M of type F , i.e. to check that M (P ⇒ (t ≈ t )) ≤ P ⇒ (t ≈ t )N
holds for every PM -implication P ⇒ (t ≈ t ) and every N ∈ K.
So, let us have N ∈ K, and let v : M → N be a valuation of M in N with
its homomorphic extension v : T(M) → N. Thus, we have v = g ◦ g , where
g : T(M) → T(M)/θv is a natural morphism and g : T(M)/θv → N is an
embedding.
Let P ⇒ (t ≈ t ) be an PM -implication such that P = PM . If P(s, s ) v (s) ≈N v (s ) for some s, s ∈ T (M), then obviously P ⇒ (t ≈ t )N,v = 1.
Thus, let P (s, s ) ≤ v (s) ≈N v (s ) for all s, s ∈ T (M). Consider a mapping
h : M → T (M)/θv defined by h(a) = [a]θv for every a ∈ M. Since
idM (s) ≈M idM (s ) = P (s, s ) ≤ v (s) ≈N v (s ),
for all s, s ∈ T (M), it follows that
a ≈M b = idM (a) ≈M idM (b) ≤ v (a) ≈N v (b)
= θv (a, b) = [a]θv ≈T(M)/θv [b]θv
164
R. Bělohlávek, V. Vychodil
for all a, b ∈ M. That is, h is an ≈-morphism. Moreover, for any n-ary f M ∈
F M and arbitrary elements a1 , . . . , an ∈ M let f M (a1 , . . . , an ) = b. Then,
[f (a1 , . . . , an )]θ = [b]θv and it follows that
v
h f M (a1 , . . . , an ) = h(b) = [b]θv = [f (a1 , . . . , an )]θ v
= f T(M)/θv [a1 ]θv , . . . , [an ]θv = f T(M)/θv h(a1 ), . . . , h(an ) .
Altogether, h : M → T(M)/θv is a morphism. Clearly, g = idM ◦ h.
Since K is sur-reflective, we have h ◦ g = rM ◦ h , where h : RK (M) → N,
see Fig. 2. Thus, v = g ◦ g = idM ◦ h ◦ g = idM ◦ rM ◦ h , which implies
K
M
(P ⇒ (t ≈ t )) = rM idM (t) ≈RK (M) rM idM (t )
≤ h rM idM (t) ≈N h rM idM (t )
= P ⇒ (t ≈ t ) .
= v (t) ≈N v (t ) = t ≈ t N,v
N,v
K (P ⇒ (t ≈ t )) ≤ P ⇒ (t ≈ t ) , i.e. N ∈ Mod( K ).
Therefore, M
M
N
K )”: Let N ∈
K
“K ⊇ M Mod(M
M Mod(M ). It suffices to show that the sur→ RK (N) is an embedding, since then N ∼
reflection rN : N
= RK (N), i.e. N ∈ K.
K ) implies N ∈ Mod( K ). Hence, we can consider
Evidently, N ∈ M Mod(M
N
a valuation idN : N → N and its homomorphic extension idN : T(N ) → N. Taking
into account N ∈ Mod(NK ), it follows that
rN idN (t) ≈ RK (N) rN idN (t ) = NK (PN ⇒ (t ≈ t ))
= t ≈ t = id (t) ≈N id (t ).
≤ PN ⇒ (t ≈ t )
N,idN
N
N,idN
Thus, rN : N → RK (N) is an embedding, i.e. N ∈ K.
N
Now we face the following problem. Given a sur-reflective class K, we
K of P -implications such that
have shown that there is a class of L-sets M
M
K
K = M Mod(M ). For every M we use a separate proper family of premises
PM . In addition to that, we deal with a proper class of L-sets since M ranges over a
proper class of all L-algebras of type F . Thus, Lemma 7 itself does not yield that
K is a P-implicational class.
In ordinary case, this problem has been solved by J. Adámek, see [1]. In what
follows, we adopt Adámek’s approach to get the desired result. The key point of
N
v
g
T(M )
idM
g
h
T(M )/θv
h
M
RK (M)
rM
Fig. 2. Scheme for the proof of Lemma 7
Fuzzy Horn logic II
165
[1] is that one can show that every sur-reflective class (in [1] called a quasivariety)
is definable by a set of implications using the so-called Vopěnka’s Principle. Moreover, it has been shown that assuming the negation of Vopěnka’s Principle, there
is always a sur-reflective class which cannot be defined by implications. For our
purpose, we use a principle concerning classes of L-algebras.
Principle 1. Given any proper class K of L-algebras of the same type, there are
distinct L-algebras M, N ∈ K such that M can be embedded into N.
Theorem 5. Vopěnka’s Principle implies Principle 1.
Proof. Let K be a proper class of L-algebras of type F . We use K to construct
a proper class Kc of ordinary first-order structures corresponding to L-algebras
from K and then apply Vopěnka’s Principle for K to show that there are distinct
L-algebras M, N ∈ K and an embedding h : M → N.
Let R = {≈a | a ∈ L} be a set of binary relation
symbols. For each M ∈ K,
we can consider a first-order structure Mc = Mc , R Mc , F Mc of type R, F, σ ,
where Mc = M, F Mc = F M (i.e. the functional parts of M and Mc coincide),
c
c
M
the a-cut of ≈M
, i.e. u ≈M
and each ≈M
a is a v iff u ≈ v ≥ a. Thus, we
M
c
have u ≈M v =
a | u ≈a v (u, v ∈ M). Clearly, Kc is a proper class of
first-order structures. Hence, by Vopěnka’s Principle, there are Mc , Nc ∈ Kc such
embeddedinto Nc . That is, there is a mapping
that Mc can be (isomorphically)
h : Mc → Nc such that h f Mc (u1 , . . . , un ) = f Nc h(u1 ), . . . , h(un ) (f ∈ F ,
Nc
c
u1 , . . . , un ∈ Mc ) and u ≈M
a v iff h(u) ≈a h(v) (u, v ∈ Mc , a ∈ L). As a
consequence,
N
c
c
u ≈M v =
a | u ≈M
a | h(u) ≈N
a v =
a h(v) = h(u) ≈ h(v),
showing that h is an embedding of L-algebras M, N.
We need to generalize the notion of a directed family of L-algebras [1, 9, 25].
Definition 5. Let κ be an infinite cardinal. A family {Mi | i ∈ I } = ∅ of L-algebras
of type F is called a κ-directed family, if for every J ⊆ I , |J | < κ there exists an
index i ∈ I such that Mj ∈ Sub(Mi ) for all j ∈ J .
Lemma 8. Let κ be an infinite cardinal. Then
(i) every κ-directed family is a directed family;
(ii) every directed family is an ω-directed family.
Proof. (i): Easy, just putting i ≤ j iff Mi ∈ Sub(Mj ), I, ≤ is a directed set and
{Mi | i ∈ I } is a directed family.
(ii): Obvious.
Lemma 8 (i) justifies the following definition.
Definition 6. Given a κ-directed
family
∈ κ I }, theκdirect union of
{Mi | i i∈I Mi , F i∈I Mi , is said to
{Mi | i ∈ I }, denoted by κi∈I Mi =
M
,
≈
i
i∈I
be a κ-direct union of a κ-directed family {Mi | i ∈ I }. An abstract class K of
L-algebras is said to be closed under κ-direct unions, if for every κ-directed
family {Mi ∈ K | i ∈ I } we have κi∈I Mi ∈ K.
166
R. Bělohlávek, V. Vychodil
The following is easy to see.
Lemma 9. Suppose K is an abstract class of L-algebras which is closed under
κ-direct unions. Then K is closed under κ -direct unions for all κ > κ. Therefore,
if U(K) ⊆ K, then K is closed under κ-direct unions for arbitrary κ.
Lemma 10. Let κ be any infinite cardinal. Every L-algebra is isomorphic to a
κ-direct union of a κ-directed family of κ-generated L-algebras.
Proof. As in [9] we get that { M M | M ∈ IM }, where IM = {M ⊆ M | |M | <
κ}, is a κ-directed family
of subalgebras of M. Since every finitely generated subalgebra of M is in { M M | M ∈ IM }, we can use the same arguments as in [9] to
obtain that M is isomorphic to the κ-direct union of { M M | M ∈ IM }.
Lemma 11. Let L∗ be a complete residuated lattice with a truth stresser ∗ defined
by (7). Suppose K is an abstract class of L-algebras, κ is an infinite cardinal,
P = LT (X)×T (X) , |X| = κ. Then K = Mod() for some L-set of Pκ -implications iff K is a sur-reflective class which is closed under κ-direct unions.
Proof. “⇒:” Let K = Mod() for some L-set of Pκ -implications. From
Lemma 2 and Theorem 4 it follows that K is sur-reflective class. Thus, it suffices
to show that K is closed under κ-direct unions.
Let {Mi ∈ K | i ∈ I } be a κ-directed family
of L-algebras. We have to show
that (P ⇒ (t ≈ t )) ≤ P ⇒ (t ≈ t )κ M for every Pκ -implication P ⇒
i∈I i
(t ≈ t ). Takea valuation v : X → i∈I Mi and its homomorphic extension
v : T(X) → κi∈I Mi . Put Y = var(P ) ∪ var(t) ∪ var(t ). For every x ∈ Y we
can
choose an index ix ∈ I
such that v(x) ∈ Mix . Let us have an index set J =
ix | v(x) ∈ Mix and x ∈ Y . Since |Y | < κ, it follows that |J | < κ, i.e.
there is i ∈
I such that v(x) ∈ Mi for every x ∈ Y . Consequently, P ⇒ (t ≈ t )κ M ,v =
i
i∈I
P ⇒ (t ≈ t )
. Hence, κ Mi ∈ Mod() = K.
Mi ,v
i∈I
“⇐”: Let K be a sur-reflective class which is closed under κ-direct unions.
Put = Implκ (K). We claim that K = Mod(Implκ (K)). Trivially, K ⊆
Mod(Implκ (K)). Thus, it remains to check the converse inequality. Doing so, it
is sufficient to show that every κ-generated L-algebra from Mod(Implκ (K)) belongs to K. Indeed, due
10, every M ∈ Mod(Impl
κ (K)) is isomorphic
to Lemma
to a κ-direct union of M M | M ⊆ M, |M | < κ ⊆ Mod(Implκ (K)) and K is
assumed to be closed under κ-direct unions.
So, let us have a κ-generated M ∈ Mod(Implκ (K)). Since K is sur-reflective, M
has a sur-reflection rM : M → RK (M) in K. We will show that rM is an embedding.
By contradiction, suppose there are b, b ∈ M such that b ≈M b rM (b) ≈RK (M)
rM (b ).
Let M , |M | < κ, denote the set of generators of M. For a subset of variables
Y ⊆ X, |Y | = |M | we can consider a surjective valuation v : Y → M and its surjective homomorphic extension v : T(Y ) → M. Define an L-set P ∈ LT (X)×T (X)
by
for s, s ∈ T (Y ),
v (s) ≈M v (s )
P (s, s ) =
0
otherwise.
Fuzzy Horn logic II
167
Since |Y | < κ, it follows that P ∈ Pκ . The surjectivity of v yields that there are
terms t, t ∈ T (Y ), where v (t) = b, v (t ) = b . Hence,
P ⇒ (t ≈ t )
= t ≈ t M,v = v (t) ≈M v (t )
M,v
= b ≈M b rM (b) ≈RK (M) rM (b ).
Since M ∈ Mod(Implκ (K)), we have
Implκ (K) (P ⇒ (t ≈ t )) rM (b) ≈RK (M) rM (b ).
≤
Thus,
there
a valuation w : Y →
is an L-algebra N ∈ K and
N , where P (s,Rs )(M)
s ≈ s holds for all terms s, s ∈ T (Y ), and t ≈ t N,w rM (b) ≈ K
N,w
rM (b ).
On the other hand, we clearly have θv ⊆ θw . Thus, there is a morphism
g : M → N such that w = v ◦ g, see Fig. 3. Since N ∈ K, there is a morphism
g : RK (M) → N, where g = rM ◦ g . As a consequence, w = v ◦ rM ◦ g .
Moreover,
t ≈ t = w (t) ≈N w (t ) = g (rM (v (t))) ≈N g (rM (v (t )))
N,w
= g (rM (b)) ≈N g (rM (b )) ≥ rM (b) ≈RK (M) rM (b )
which is a contradiction. Altogether, K = Mod() for = Implκ (K).
Remark 5. Note that the “⇐” part of the original proof of Lemma 11 for ordinary
algebras (see [1]) differs from that one presented above. In [1], the author presents
a direct construction of an algebra in K, while we proceed by contradiction and
use properties of sur-reflections. From the viewpoint of theory of L-algebras, the
original construction pertains only to trivial ≈M ’s and is thus not applicable for
general L-algebras.
Analogously as for L-sets of P-implications and accordingly to the relationship
between L-set of P-implications and sets of weighted P-implications we denote
the class of all models of a class of weighted P-implications by Mod(), i.e.
Mod() = M | a ≤ P ⇒ (t ≈ t )M for all P ⇒ (t ≈ t ), a ∈ .
Lemma 12. Let L∗ be a complete residuated lattice with globalization, K be a
sur-reflective class of L-algebras of type F . Then there
is a classK of weighted
implications with weighted premises such that K = M Mod(M
) = Mod(),
where M ranges over all L-algebras of type F .
v
T(Y )
w
M
g
N
rM
RK (M)
g
Fig. 3. Scheme for the proof of Lemma 11
168
R. Bělohlávek, V. Vychodil
K (P ⇒ (t ≈ t )) = a
Proof. Let be a class, where P ⇒ (t ≈ t ), a ∈ iff M
K) =
for some L-algebra M of type F . One can easily verify that M Mod(M
Mod(). The rest follows from Lemma 7.
Lemma 13. Let L∗ be a complete residuated lattice with a truth stresser ∗ defined
by (7). Assuming Principle 1, for every sur-reflective class K of L-algebras there
exists an L-set of P-implications such that K = Mod().
Proof. We will show by contradiction that if K were not definable by any L-set
of P-implications then Principle 1 would be violated. Thus, assume K to be a surreflective class such that K = Mod() for every L-set of P-implications, where
P is an arbitrary proper family of premises. Lemma 12 yields that K is definable
by a class of weighted implications with weighted premises.
For every infinite cardinal κ we can consider a proper family of premises Pκ
such that P = LT (X)×T (X) , |X| = κ. Clearly, for every Pκ there is only a set of
weighted Pκ -implications in . Moreover, from Lemma 11 it follows that K cannot be closed under κ-direct unions. That is, for every infinite cardinal κ there is
an
L-algebra Mκ ∈
K such that Mκ is a κ-direct union of some κ-directed family
Mκ,i ∈ K | i ∈ Iκ . Let us define an ordinal sequence of L-algebras formed
of such
Mκ ’s. Put N0 = Mω . For every ordinal α let Nα be Mκ such that κ > | β<α Nβ |.
Observe that |Mκ | < κ implies Mκ,i ∈ K for some index i ∈ Iκ , which contradicts Mκ,i ∈ K. Hence, for every Mκ , we have |Mκ | ≥ κ. It immediately follows
that |Nα | > |Nβ | for every β < α. Thus, for β < α there cannot be an injective
mapping sending elements of Nα to Nβ . Since every embedding is injective, Nα
cannot be embedded into Nβ . On the other hand, Nβ cannot be embedded into Nα
either. Indeed, suppose h : Nβ → Nα is an embedding. Since Nα is a κ-direct
union and |Nβ | < κ by definition, for every a ∈ Nβ there is an index ia ∈ Iκ
such that h(a) ∈ Mκ,ia . Moreover, | ia | a ∈ Nβ | < κ, i.e., there is some k ∈ Iκ
such that h(Nβ ) ⊆ Mκ,k . Thus, Nβ ∈ K is a subalgebra of Mκ,k ∈ K, which is a
contradiction.
To sum up, the class of all Nα ’s is proper and there is no Nα which can be
embedded into another Nβ (for α = β), i.e. Principle 1 is violated.
The following theorem summarizes the equivalent characterizations of
P-implicational classes being defined using globalization.
Theorem 6. Assume Principle 1. Suppose L∗ is a complete residuated lattice with
a truth stresser ∗ defined by (7). Then for any abstract class K of L-algebras the
following are equivalent:
(i) K is a P-implicational class,
(ii) K is closed under S and P,
(iii) K = SP(K),
(iv) K = SP(K ) for some abstract class K of L-algebras,
(v) K is a sur-reflective class,
(vi) K = Mod() for some class of weighted implications,
(vii) K = Mod(Impl(K)),
(viii) K = Mod(Impl(K )) for some abstract class K of L-algebras.
Fuzzy Horn logic II
169
Proof. “(i) ⇒ (ii)”: Consequence of Lemma 2.
“(ii) ⇒ (iii)”: Clearly, SP(K) = S(K) = K.
“(iii) ⇒ (iv)”: Trivial.
“(iv) ⇒ (v)”: Evidently, S(K) = SSP(K ) = SP(K ) = K. Analogously,
P(K) = PSP(K ) ⊆ SPP(K ) = SP(K ) = K. Now, apply Theorem 4.
“(v) ⇒ (vi)”: Apply Lemma 12.
“(vi) ⇒ (vii)”: Clearly, Mod() is sur-reflective since it is an abstract class
closed under S, P (routine to check as in Theorem 2). From Theorem 13 it follows that K = Mod( ) for some L-set of Pκ -implications. Therefore, we get
K = Mod( ) = Mod(Impl(Mod( ))) = Mod(Impl(K)).
“(vii) ⇒ (viii)”: Trivial.
“(viii) ⇒ (i)”: By definition.
A class K of L-algebras is called semivariety if it is an abstract class closed
under formations of subalgebras, direct products, and direct unions. Using analogous arguments as in ordinary case, one can show that USP is a closure operator on
abstract classes of L-algebras and USP(K) is the semivariety generated by K. Now
the following characterization is a consequence of Lemma 4, Theorem 4, Lemma 9,
and Lemma 11.
Theorem 7. Suppose L∗ is a complete residuated lattice with a truth stresser ∗
defined by (7) and let K be an abstract class of L-algebras. Then K is a P-finitary
implicational class for some proper family of premises P iff K is a semivariety. From now on, we shall assume Principle 1 when necessary.
4. Quasivarieties
Finally, we study P-Horn classes. Quasivarieties of algebras have been widely
studied. There are various approaches to quasivarieties. Our investigation is based
mainly on the approach described in [12], but we comment on some other approaches as well.
Definition 7. A class K of L-algebras is called a quasivariety if it is an abstract
class closed under formations of subalgebras, direct products, and direct limits of
direct families of L-algebras from K.
Remark 6. By Theorem 4, quasivarieties are sur-reflective classes closed under
direct limits of direct families.
Since every quasivariety K is a sur-reflective class, we can consider a sur-reflection of M in K. We will use mainly sur-reflections of finitely presented L-algebras.
Thus, we will adopt the following convention. For a finitely presented L-algebra
T(X)/θ (R) let RK (X, R) denote the sur-reflection of T(X)/θ (R) in K, the corresponding epimorphism will be usually denoted simply by r.
In ordinary case, sur-reflections of finitely presented algebras play an important
role in the theory of quasivarieties since every quasivariety can be reconstructed by
direct limits of such sur-reflections. In what follows we focus on this fact in fuzzy
setting.
170
R. Bělohlávek, V. Vychodil
Definition 8. Let K be an abstract class of L-algebras of type F . An L-algebra
M of type F is said to satisfy the QF condition w.r.t. K, if every morphism h :
T(X)/θ (R) → M, where T(X)/θ (R) is a finitely presented L-algebra factorizes through RSP(K) (X, R), i.e. h = r ◦ h for a sur-reflection r : T(X)/θ (R) →
RSP(K) (X, R) of T(X)/θ (R) in SP(K) and a morphism h sending elements of
RSP(K) (X, R) to M.
Lemma 14. Suppose L is a finite residuated lattice. Let K be an abstract class of
L-algebras of type F and let M be an L-algebra satisfying the QF condition w.r.t.
K. Then M ∼
= R, where R is a direct limit of a direct family which consists of
sur-reflections RSP(K) (X, R) of certain finitely presented L-algebras.
Proof. By adoption of the argument from ordinary case (see [25]) to fuzzy setting,
we give only a sketch.
Let M satisfy QF w.r.t. K. M is isomorphic to a direct limit lim T(Yi )/θ (Si )
of a direct family hij : T(Yi )/θ (Si ) → T(Yj )/θ (Sj ) | i ≤ j of finitely presented
L-algebras, see [24].
One can show that for sur-reflections ri : T(Yi )/θ (Si ) → RSP(K) (Yi , Si )
(i
∈
I ) there are morphisms
gij : RSP(K) (Yi , Si ) → RSP(K) (Yj , Sj ) such that
RSP(K) (Yi , Si ) | i ∈ I together with such gij ’s is a weak direct family of L-algebras. Moreover, finiteness of L implies that this family is in fact a direct family.
Now it suffices to check that M ∼
= lim RSP(K) (Yi , Si ) which can be shown using
the QF condition and direct limit property, see [24].
Note the restriction on finiteness of L in Lemma 14. It is used to guarantee
that a certain weak direct family is a direct family (see the proof of Lemma 14).
Since every M of a quasivariety K satisfies the QF condition w.r.t. K, we have the
following corollary.
Corollary 2. Suppose L is a finite residuated lattice. Let K be a quasivariety of
L-algebras, and let M ∈ K. Then M ∼
= R, where R is a direct limit of a direct
family of certain sur-reflections RK (X, R) of finitely presented algebras. Hence,
given a class K = {RK (X, R) | X, R are finite} it follows that K = IL(K ).
Remark 7. Analogously as in ordinary case, using QF condition and properties of
direct limits, for finite L it can be shown that LSP is a closure operator on abstract
classes of L-algebras and LSP(K) is the quasivariety generated by K, generalizing
the result of T. Fujiwara, see [12].
Finally, we characterize quasivarieties as P-Horn classes.
Lemma 15. Suppose L∗ is a finite residuated lattice with a truth stresser ∗ defined
by (7). If K is a quasivariety then K = Mod(Horn(K)).
Proof. Let P = LT (Y )×T (Y ) , where Y is a denumerable set of variables. K ⊆
Mod(Horn(K)) is obvious. It remains to show the converse inclusion.
For the sake of brevity, put K = Mod(Horn(K)). From Lemma 6 it follows
that K is a quasivariety. Thus, every M ∈ K is a isomorphic to direct limit of
Fuzzy Horn logic II
171
some sur-reflections RK (X, R) of finitely presented L-algebras, see Corollary 2.
Hence, it suffices to show that every sur-reflection RK (X, R) of a finitely presented
L-algebra T(X)/θ (R) belongs to K. Then clearly, since K is an abstract class closed
under L, we obtain M ∈ K.
Let T(X)/θ (R), where X, R are finite (without loss of generality, we can assume
X ⊆ Y ), and let r : T(X)/θ (R) → RK (X, R), r : T(X)/θ (R) → RK (X, R) be
the sur-reflections of T(X)/θ (R) in K, K . Since K ⊆ K it follows that RK (X, R) ∈
K . As a consequence, the sur-reflection r : T(X)/θ (R) → RK (X, R) factorizes
through RK (X, R), i.e. there is a morphism g : RK (X, R) → RK (X, R) such that
r = r ◦ g . We finish the proof by demonstrating that g is an embedding, since
then RK (X, R) ∈ S(K) = K.
We proceed by contradiction. So, let there be elements b, b such that
b ≈RK (X,R) b g (b) ≈RK (X,R) g (b ).
Moreover, X, R are finite, i.e. we can consider a finite L-set of premises P ∈
LT (X)×T (X) such that
for R(s, s ) > 0,
θ (R)(s, s )
P (s, s ) =
0
otherwise.
Define a valuation v of X in RK (X, R) by v(x) = r [x]θ(R) for every x ∈ X.
Hence,
for the homomorphic extension v : T(X) → RK (X, R) we have v (s) =
r [s]θ(R) for all terms s ∈ T (X). Clearly,
P (s, s ) ≤ θ (R)(s, s ) = [s]θ(R) ≈T(X)/θ(R) s θ(R)
≤ r [s]θ(R) ≈RK (X,R) r s θ(R)
= v (s) ≈RK (X,R) v (s ) = s ≈ s R (X,R),v
K
holds for
there are terms t, t ∈ T (X) such
all s, s ∈ T (X). Since
r is surjective,
that r [t]θ(R) = b, and r t θ(R) = b . Thus,
P ⇒ (t ≈ t )
R
K (X,R),v
Therefore, P ⇒ (t ≈ t )R
= t ≈ t R (X,R),v = v (t) ≈RK (X,R) v (t )
K
= r [t]θ(R) ≈RK (X,R) r t θ(R)
= b ≈RK (X,R) b g (b) ≈RK (X,R) g (b ).
K (X,R)
g (b) ≈RK (X,R) g (b ). Moreover, we have
RK (X, R) ∈ K = Mod(Horn(K)). Thus,
Horn(K) (P ⇒ (t ≈ t )) g (b) ≈RK (X,R) g (b ).
Hence, there is N ∈ K, and w : X → N such that P (s, s ) ≤ s ≈ s N,w
for all s, s ∈ T (X) but t ≈ t N,w g (b) ≈RK (X,R) g (b ). Observe that for
a homomorphic extension w : T(X) → N we have R ⊆ P ⊆ θw , and thus
θ (R) ⊆ θw . As a consequence, there is a morphism h : T(X)/θ (R) → N with
172
R. Bělohlávek, V. Vychodil
w
T(X)
hθ(R)
N
h
T(X)/θ(R)
h
RK (X, R) K
r
r
g
RK (X, R)
Fig. 4. Scheme for the proof of Theorem 15
w = hθ(R) ◦ h. Since N ∈ K, h factorizes through RK (X, R), i.e. there is a
morphism h : RK (X, R) → N such that h = r ◦ h , see Fig. 4. Now it readily
follows that w = hθ(R) ◦ h = hθ(R) ◦ r ◦ h = hθ(R) ◦ r ◦ g ◦ h . Thus, we have
t ≈ t = w (t) ≈N w (t )
N,w
= (hθ(R) ◦ r ◦ g ◦ h )(t) ≈N (hθ(R) ◦ r ◦ g ◦ h )(t )
= (r ◦ g ◦ h ) [t]θ(R) ≈N (r ◦ g ◦ h ) t θ(R)
≥ (r ◦ g ) [t]θ(R) ≈RK (X,R) (r ◦ g ) t θ(R)
= g (b) ≈RK (X,R) g (b )
which contradicts t ≈ t N,w g (b) ≈RK (X,R) g (b ). Hence, g is an embedding, RK (X, R) ∈ K.
Remark 8. Note that the proof of the bivalent version of Lemma 15 as presented
in [25] is not correct. In [25], it is claimed that if M is isomorphic to a direct limit
lim T(Yi )/θ (Si ) of finitely presented algebras, and if M ∈ Mod(Horn(K)), then
every T(Yi )/θ (Si ) ∈ Mod(Horn(K)). This is not true as it is demonstrated by the
following counterexample.
Let us have a term algebra T(X), where X is finite. Clearly, T(X) is finitely
presented. Take M ∈ K, where K is a variety such that T(X) ∈ K. Moreover, algebra M is isomorphic to a direct limit lim T(Yi )/θ (Si ) of a direct family {T(Yi )/θ (Si ) | i ∈ I } of finitely presented algebras. We can assume T(X) ∈
{T(Yi )/θ (Si ) | i ∈ I } (if T(X) ∈ {T(Yi )/θ (Si ) | i ∈ I }, T(X) can be added to
{T(Yi )/θ (Si ) | i ∈ I } using morphisms gi : T(X) → T(Yi )/θ (Si ) defined by
gi (t) = [t]θ(Si ) for every i ∈ I with X ⊆ Yi —this can be made without
loss of generality, see [24]). Using the argument from [25], one can conclude
{T(Yi )/θ (Si ) | i ∈ I } ⊆ Mod(Horn(K)) = K. That is, the term algebra T(X)
would be a member of K—a contradiction.
Hence, for finite L∗ with globalization we have the following characterization.
Theorem 8. Suppose L∗ is a finite residuated lattice equipped with ∗ defined by (7)
and let K be an abstract class of L-algebras. Then K is a P-Horn class for some
proper family of premises P iff K is a quasivariety.
Fuzzy Horn logic II
173
≈Mi
ai
ai
bi
bi
ai
1
i
0
0
ai
i
1
0
0
bi
0
0
1
0
bi
0
0
0
1
Fig. 5. L-equality from Example 2
Remark 9. It is well known that in ordinary case, the collections of all varieties,
quasivarieties, semivarieties, and sur-reflective classes are pairwise distinct. This
applies to the fuzzy case as well. Namely, suppose L∗ is a complete residuated
lattice equipped with ∗ defined by (7). Let be an L-set of P-implications given
by
= {x ≈ y, a i x ≈ y, 1 | a ∈ L, a = 0} .
Evidently, Mod() consists of all L-algebras (of the given type) with crisp L-equalities. Thus, Mod() is a quasivariety which is not a variety since Mod() is not
closed under homomorphic images. To see that quasivarieties and semivarieties of
L-algebras are distinct, take an ordinary semivariety K which is not a quasivariety (such K exists). Consider a class K of L-algebras such that M ∈ K results
from M ∈ K by equipping M with the crisp L-equality. Then K is a semivariety
of L-algebras which is not a quasivariety (observe that K is closed under any of
I, S, P, U, L iff K is closed under the corresponding crisp operator). In a similar
way one can get a sur-reflective class of L-algebras which is not a semivariety.
Let us comment on the restriction of finiteness of L present in our characterization of quasivarieties. This restriction does not pertain sur-reflective classes and
semivarieties. In case of quasivarieties, finiteness of L was used to ensure that
a weak direct family of L-algebras is a direct family. One can ask whether it is
possible to work with unrestricted weak direct families and arbitrary complete residuated lattices instead. The following counterexample gives a negative answer by
showing that P-Horn classes are not closed under direct limits of arbitrary weak
direct families.
Example 2. Take L∗ , where L is a structure of truth values on the unit interval [0, 1],
and ∗ is defined by (7). Let F = {f1 , f2 , g1 , g2 } be a type of L-algebras such that
f1 , f2 , g1 , g2 are nullary function symbols (constants). Consider a proper family
of premises P = LT (∅)×T (∅) and P-Horn clause
f1 ≈ f2 , 1 i g1 ≈ g2 .
Moreover, let be an L-set of P-Horn clauses such that
Supp() = {f1 ≈ f2 , 1 i g1 ≈ g2 } .
Thus, Mod() is a P-Horn class of L-algebras.
Now we introduce a weak direct family
of L-algebras. Let I, ≤ with I = [0, 1)
be a directed index set and let Mi = Mi , ≈Mi , f1Mi , f2Mi , g1Mi , g2Mi , i ∈ I be
174
R. Bělohlávek, V. Vychodil
L-algebras of type F such that Mi = ai , ai , bi , bi , f1Mi = ai , f2Mi = ai ,
Mi
g1Mi = bi , g2Mi = bi , and
≈ is an L-equality
defined in Fig. 5. In addition to
that, we define a family hij : Mi → Mj | i ≤ j of morphisms by hij (ai ) = aj ,
hij (ai ) = aj , hij (bi ) = bj , hij (bi ) = bj . It is immediate that this defines a weak
direct family of L-algebras which is not a direct family.
Clearly, f1 ≈ f2 Mi = i < 1 for every i ∈ I , thus we have
f1 ≈ f2 , 1 i g1 ≈ g2 Mi = 1.
Hence, Mi ∈ Mod() for all i ∈ I . On the other hand, for the direct limit lim Mi
it follows that θ∞ (ai , ai ) = 1, and θ∞ (bi , bi ) = 0. That is,
f1lim Mi = [ai ]θ∞ = ai θ = f2lim Mi , g1lim Mi = [bi ]θ∞ = bi θ = g2lim Mi ,
∞
i.e.
f1lim Mi
≈lim Mi
f2lim Mi
∞
= 1 while
g1lim Mi
≈lim Mi
g2lim Mi
= 0. Thus,
f1 ≈ f2 , 1 i g1 ≈ g2 lim Mi = g1 ≈ g2 lim Mi = 0.
As a consequence, lim Mi ∈ Mod() since (f1 ≈ f2 , 1 i g1 ≈ g2 ) > 0. In
other words, a P-Horn class Mod() is not closed under direct limits of arbitrary
weak direct families.
∗
∗
∗
Now we present a characterization of P-Horn classes as abstract classes of
L-algebras closed under formations of subalgebras and reduced products. It is worth
to add that as in Lemma 15, we need to suppose a finite structure of truth values.
We start by closedness of P-implicational classes under safe reduced products.
Lemma 16. Let L∗ be a complete residuated lattice with an implicational truth
stresser ∗ . Suppose an L-set of P-Horn clauses is given. Then Mod() is closed
under safe reduced products.
Q
Proof. Since every safe reduced product F Mi (Mi ∈ K) is isomorphic to a direct limit lim MZ of certain direct family {MZ ∈ Mod() | Z ∈ F } (see [24]),
the closedness of Mod() under safe reduced products is a consequence of
Lemma 5.
Before delving into the converse problem, note that P-Horn classes are not
closed under arbitrary reduced products as shown by the following counterexample.
Example 3. Take L∗ such that L = [0, 1], and ∗ is defined by (7). Consider the
same type of L-algebras and as in Example 2. Let N be an index set and for
every i ∈ N let Mi = Mi , ≈Mi , f1Mi , f2Mi , g1Mi , g2Mi be defined the same way
as in Example 2 except that for ai , ai we put ai ≈Mi ai = ai ≈Mi ai = 1 − 1i .
Clearly,
Q Mi ∈ Mod() for every i ∈ N. Let F be a Fréchet filter over N. Put
M = i∈N Mi . For every X ∈ F we have
[[f1M ≈ f2M ]]X = i∈X f1M (i) ≈Mi f2M (i)
= i∈X f1Mi ≈Mi f2Mi = i∈X ai ≈Mi ai < 1.
Fuzzy Horn logic II
175
Thus, F is not safe w.r.t. {Mi | i ∈ N} since
θF (f1M , f2M ) = X∈F [[f1M ≈ f2M ]]X = lim 1 − n1 = 1.
n→∞
As a consequence,
f1 ≈ f2 , 1 i g1 ≈ g2 QF Mi = g1 ≈ g2 QF Mi = θF (g1M , g2M )
= X∈F [[g1M ≈ g2M ]]X = X∈F i∈X bi ≈Mi bi = 0,
Q
showing F Mi ∈ Mod().
Lemma 17. Suppose L∗ is a residuated lattice with a truth stresser ∗ defined by (7).
Let K be an abstract class of L-algebras which is closed under subalgebras and
safe reduced products. If every filter F is K-safe, then K = Mod() for an L-set
of P-Horn clauses, where P = LT (X)×T (X) .
Proof. Observe that K is closed under direct products
sinceQF = {I } is safe with
Q
respect to an arbitrary family {Mi | i ∈ I }, and F Mi ∼
= i∈I Mi . As a consequence, K is a sur-reflective class. Thus, K = Mod(Impl(K)) due to Theorem 6.
We claim that K = Mod(), where = Horn(K). Trivially, K ⊆ Mod(Horn(K)),
i.e. we have to check the converse inclusion. We will proceed by contradiction.
Let M ∈ Mod(Horn(K)) and M ∈ Mod(Impl(K)). That is,
P ⇒ (t ≈ t ) Impl(K) (P ⇒ (t ≈ t ))
(9)
M
for some P-implication P ⇒ (t ≈ t ). P ⇒ (t ≈ t ) induces a family
P ⇒ (t ≈ t ) | P ∈ Fin(P )
of P-Horn clauses, where Fin(P ) denotes
the set of all finite
restrictionsof P .
) ≤ P ⇒ (t ≈ t ) . Thus,
(t
≈
t
Take any P ∈ Fin(P ). Clearly, P ⇒
M
M
from (9) it follows that P ⇒ (t ≈ t )M Impl(K) (P ⇒ (t ≈ t )). Since
M ∈ Mod(Horn(K)), we have
Horn(K) (P ⇒ (t ≈ t )) Impl(K) (P ⇒ (t ≈ t )).
That is, for every P ∈ Fin(P ) there
vP of X in NP such that P (s, s ) ≤
and t ≈ t N ,v Impl(K) (P
is
NP ∈ K and a valuation
an L-algebra
s ≈ s for all terms s, s ∈ T (X)
N ,v P
P
⇒ (t ≈ t )). In the following, we conP
P
struct certain safe reduced product of a family NP | P ∈ Fin(P ) to obtain a
contradiction. Let us introduce a proper filter over Fin(P
). First, we can consider
a family Ps,s = P ∈ Fin(P ) | P (s, s ) = P (s, s ) for every
s, s ∈ T (X). Evi
dently, ∅ = Ps,s ⊆ Fin(P ) for all s, s ∈ T (X). Put J = Ps,s | s, s ∈ T (X) .
Obviously, for every s1 , s1 , . . . , sn , sn ∈ T (X) there is a finite restriction P ∈
Fin(P
) such that P (si ,
si ) = P (si , si ) for all i = 1, . . . , n. As a consequence,
Psi ,si | i = 1, . . . , n = ∅ showing that J has the finite intersection property.
This enables us to define a proper filter F over Fin(P ) to be a filter generated by J .
176
R. Bělohlávek, V. Vychodil
A family
vP : X → NP | P ∈ Fin(P ) of valuations induces a valuation v :
Q
X → P ∈Fin(P ) NP such that v(x)(P ) = vP (x) for all x ∈ X, P ∈ Fin(P ). By
Q
standard argument, there is a valuation w of X in F NP such that w(x) = [v(x)]θF
for every x ∈ X. Thus, for s, s ∈ T (X) we have
Q
Q
s ≈ s Q
= w (s) ≈ F NP w (s ) = v (s) θ ≈ F NP v (s ) θ
N
,w
F
F
F P
= θF v (s), v (s ) = Z∈F [[v (s) ≈ v (s )]]Z
= Z∈F P ∈Z v (s)(P ) ≈NP v (s )(P )
= Z∈F P ∈Z vP (s) ≈NP vP (s ).
Recall that P (s, s ) ≤ s ≈ s N ,v = vP (s) ≈NP vP (s ) holds for every
P
P
P ∈ Fin(P ) and all s, s ∈ T (X). Moreover, Ps,s ∈ F . It readily follows that
s ≈ s Q
≥ [[v (s) ≈ v (s )]]Ps,s = P ∈P vP (s) ≈NP vP (s )
N
,w
s,s
F P
≥ P ∈P P (s, s ) = P ∈P P (s, s ) = P (s, s ),
s,s
s,s
i.e. P (s, s ) ≤ s ≈ s Q N ,w for all terms s, s ∈ T (X). Moreover, since F is
F P
K-safe by the assumption, we have
t ≈ t Q
= [[v (t) ≈ v (t )]]Z0 = P ∈Z0 vP (t) ≈NP vP (t )
N ,w
F
P
for some Z0 ∈ F . In addition to that,
v (t) ≈NP v (t ) = t ≈ t P
P
NP ,vP Impl(K) (P ⇒ (t ≈ t ))
for all P ∈ Z0 . Putting previous facts together, we have
P ⇒ (t ≈ t )Q
= t ≈ t Q N ,w
F NP ,w
F P
= P ∈Z0 t ≈ t N ,v Impl(K) (P ⇒ (t ≈ t )).
P
P
(10)
Since every
Q NP belongs to K which is supposed to be closed under safe reduced
products, F NP belongs to K. As a consequence, P ⇒ (t ≈ t )Q N ,w ≥
F P
Impl(K) (P ⇒ (t ≈ t )) which contradicts (10).
An interesting point to stress is that Lemma 17 requires safeness of a filter even
without invoking the connection to direct limits. If L is finite, then every filter is
safe with respect to any family of L-algebras of the same type. As a corollary, we
have the following theorem.
Theorem 9. Suppose L∗ is a finite residuated lattice with a truth stresser ∗ defined
by (7). Then K is an abstract class of L-algebras closed under subalgebras and
reduced products iff K is a P-Horn class.
Acknowledgements. Supported by grant no. B1137301 of the Grant Agency of the Academy of Sciences of Czech Republic and by instititutional support, research plan MSM
6198959214.
Fuzzy Horn logic II
177
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