an algebraic approach in a fuzzy logic setting - CiteSeerX

Annals of Mathematics and Artificial Intelligence 35: 197–214, 2002.  2002 Kluwer Academic Publishers. Printed in the Netherlands.

The variety generated by perfect BL-algebras: an algebraic approach in a fuzzy logic setting ∗ Antonio Di Nola a , Salvatore Sessa b,∗∗ , Francesc Esteva c , Lluis Godo c and Pere Garcia c a Dipartimento di Matematica e Informatica, Università di Salerno, Via Allende, 84081 Baronissi,

Salerno, Italy E-mail: [email protected] b Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università di Napoli, via Monteoliveto, 3, 80134 Napoli, Italy E-mail: [email protected] c Institut d’Investigació en Intelligència Artificial (IIIA), Consejo Superior de Investigaciones Científicas (CSIC), Campus Universitat Autònoma de Barcelona s/n, 08193 Bellaterra, Spain E-mail: {esteva; godo; pere}@iiia, csic.es

BL-algebras are the Lindenbaum algebras of the propositional calculus coming from the continuous triangular norms and their residua in the real unit interval. Any BL-algebra is a subdirect product of local (linear) BL-algebras. A local BL-algebra is either locally finite (and hence an MV-algebra) or perfect or peculiar. Here we study extensively perfect BL-algebras characterizing, with a finite scheme of equations, the generated variety. We first establish some results for general BL-algebras, afterwards the variety is studied in detail. All the results are parallel to those ones already existing in the theory of perfect MV-algebras, but these results must be reformulated and reproved in a different way, because the axioms of BL-algebras are obviously weaker than those for MV-algebras.

1.

Introduction

The popularity of the fuzzy logic stands mainly in the applicational aspects. In these numerous applications, the authors use usually linguistic variables, transformed “ad hoc” in fuzzy sets with membership values in [0, 1]. Then the fuzzy sets are combined together using as “conjunction” a continuous triangular norm t (t-norm, for short), as “disjunction” the associated dual conorm and as “implication” the correspondent residuum →. Experts of “Soft Computing” know very well this mechanism, in particular in fuzzy control, based essentially on simple-minded calculus called “fuzzy propositional calculus” which, from a logical point of view, is seemed not fully justified (at least for the success of the results). ∗ This paper is performed under the financial support of an “Integrated Action” Italy–Spain for the years

1999–2000.

∗∗ Corresponding author.

198

A. Di Nola et al. / The variety generated by perfect BL-algebras

Hájek [10] has invented BL-algebras in order to study the “Basic Logic” arising from these applications, in other words he proved that the “Basic Logic” (here understood in “narrow sense”) is that logic which generalizes the three logics, more commonly used in fuzzy set theory: Lukasiewicz logic, Product logic and Gödel logic. The related Lindebaum algebras are the MV-algebras [3], the Product algebras [12] and the Heyting algebras satisfying the additional condition of prelinearity [12]. Since in a BL-algebra a notion of ideal is missing, we are obliged to deal with filters (called also deductive systems [13]). By BL-chain we mean a linearly ordered BL-algebra. Each BL-algebra is a subdirect product of BL-chains [10, lemma 2.3.16]. A BL-algebra is called local if has exactly one maximal filter [13]. Since any BL-chain is local, we can say that any BL-algebra is subdirectly embeddable in a direct product of local BL-algebras [13]. A local BL-algebra is either locally finite or perfect or peculiar [13]. This classification is similar to the one for local MV-algebras [2, theorem 1.1], in which the authors distinguish three classes: locally finite, perfect and singular MV-algebras. In every BL-algebra L, one can consider the largest MV-algebra MV(L) which is subalgebra of L [13]. Obviously, MV(L) = L iff L is an MV-algebra [10, definition 3.2.2]. A locally finite BL-algebra is a locally finite MV-algebra [14, theorem 1]. A local BL-algebra L is perfect iff MV(L) is a perfect MV-algebra [13, proposition 7] whereas a local BL-algebra L such that MV(L) = L is peculiar iff MV(L) = {0, 1} is either singular or locally finite. For the sake of completeness, we present and easily reprove these results. The deepest study shall be made on the perfect BL-algebras. These algebras do not form a variety, but the generated variety P0 shall be studied in detail. We prove that L ∈ P0 iff L is strongly bipartite, i.e., iff L is bipartite by every maximal filter. Furthermore, P0 is an equational variety, i.e., we give a finite scheme of axioms which guarentee when a BL-algebra lies in P0 . We also study the class P of the bipartite BL-algebras, i.e., L ∈ P iff L is bipartite by some maximal filter. Obviously P0 ⊆ P but P does not form a variety. This study is parallel to that one for MV-algebras [1,6]. In the sequel, the integer numbers will be denoted by the symbols m, n, p, etc. 2.

Definitions

Throughout this paper, for literature involving MV-algebras we refer to [4]. A BL-algebra L = (L, ∧, ∨, , →, 0, 1) is an algebra having four binary operations ∧, ∨, , → and two constants 0, 1 such that for all x, y, z ∈ L the following conditions hold: (i) (L , ∧, ∨, 0, 1) is a distributive lattice with universal bounds 0, 1, the ordering “” being defined as x y iff x = x ∧ y;


199

(ii) is an associative, commutative and monotone operation having 1 as neutral element, i.e., x 1 = x, (iii) x ∧ y = x (x → y) (divisibility); (iv) x y z iff x (y → z), i.e., by (iii), (y → z) = ∨{x ∈ L: x y z} (residuation); (v) (x → y) ∨ (y → x) = 1 (prelinearity). Important examples of BL-algebras are the so-called t-algebras [11] L = [0, 1], ∧, ∨, , →, 0, 1 , where ([0, 1], ∧, ∨, 0, 1) is the usual lattice on the real unit interval, “ ” is a continuous t-norm and → is the corresponding residuum [5]. The most used t-algebras in fuzzy set theory are the following [11]: • Gödel algebra: x y = min{x, y},

x→y=

• Goguen algebra:

x y = xy,

x→y=

1 y x

1 y

if x y, otherwise.

if x y, otherwise.

• Lukasiewicz algebra: x y = max{0, x + y − 1},

x → y = min{1, 1 − x + y}.

The properties listed below (where x ∗ = x → 0) are shown to hold in any BL-algebra L (for a proof, see [10, chapter 1]). For all x, y, z ∈ L we have: (vi) x → (y → z) = y → (x → z) = (x y) → z. In particular, we get for z = 0: (vii) x → y ∗ = y → x ∗ = (x y)∗ . (viii) x x ∗ = 0, 0∗ = 1, 1∗ = 0. (ix) If x y, then y ∗ x ∗ . (x) x x ∗∗ , x ∗ = x ∗∗∗ . (xi) (x ∨ y)∗ = x ∗ ∧ y ∗ . (xii) (z → y) (z → x) and x, y z imply x y. In particular, we get:

200


(xiii) (z → x) = (z → y) and x, y z imply x = y. (xiv) x ∨ y = [(x → y) → y] ∧ [(y → x) → x]. A BL-algebra L is an MV-algebra iff x ∗∗ = x for all x ∈ L [10, definition 3.2.2]. Note that an operation of addition is defined in an MV-algebra L by setting x ⊕ y = (x ∗ y ∗ )∗ for all x, y ∈ L. In a general BL-algebra L, in accordance to [14], the order of an element x ∈ L is the smallest integer n such that · · x = 0. x n = x · n times

In this case we put ord(x) = n, otherwise ord(x) = ∞ if no such n exists. By convention, we put x 0 = 1 for any x = 0. We explicitly point out that, in an MV-algebra, the MV-order of an element x is given as the smallest integer n for which nx = x ⊕ · · · ⊕ x = 1. n times

If it exists we write MV-ord(x) = n, otherwise we write MV-ord(x) = ∞. Now, if x = 0, 1/2, 1, ord(x) and MV-ord(x) do not coincide, as it is easily seen in the Lukasiewicz algebra over [0, 1]. Indeed, if 1/3 < x < 1/2, we have ord(x) = 2 (because x 2 = max{0, 2x − 1} = 0) and MV-ord(x) = 3 (because 3x = min{1, 2x + x} = 1). Actually it is easy to check that in an MV-algebra it holds that MV-ord(x) = ord(x ∗ ). Following [10], we recall some general results about filters of a BL-algebra L. A nonempty subset D ⊆ L such that (i) 1 ∈ L; (ii) x ∈ D, (x → y) ∈ D imply y ∈ D is called a filter (or deductive system [13]). This is equivalent to say also D is a filter iff D is nonempty and (x y) ∈ D iff x, y ∈ D. Of course, L and {1} are filters and a filter D is proper if 0 ∈ / D, that is x, x ∗ ∈ D for no x ∈ L. A filter D is prime if, for all x, y ∈ L, (x ∨ y) ∈ D implies x ∈ D or y ∈ D. Equivalently, D is prime iff for all x, y ∈ D, either (x → y) ∈ D or (y → x) ∈ D. Any prime filter can be extended to a maximal filter. A maximal filter is prime. As usually there is a bijective correspondence between filters and congruence relations of L. Thus if D is a filter, we write x ∼ y iff (x → y) (y → x) ∈ D, moreover, the relative quotient algebra L/D, with the induced operations by those ones of L, is a BL-algebra. L/D is a BL-chain iff D is prime and we note that x/D = 1/D iff x ∈ D. Now we recall some results of [13]. A BL-algebra L is local iff, for all x ∈ L, ord(x) < ∞ or ord(x ∗ ) < ∞. Then the class of all local BL-algebras can be partitioned in three subclasses, strictly speaking a local BL-algebra L is


201

(i) perfect iff, for all x ∈ L, ord(x) < ∞ iff ord(x ∗ ) = ∞; (ii) locally finite iff, for all x ∈ L − {1}, ord(x) < ∞; (iii) peculiar iff there exist x, y ∈ L − {0, 1}, such that ord(x) = ∞, ord(y) < ∞, ord(y ∗ ) < ∞. In parallel, following [2], an MV-algebra A is local iff, for all x ∈ A, MV-ord(x) < ∞ or MV-ord(x ∗ ) < ∞. A local MV-algebra A is (i) perfect iff, for all x ∈ A, MV-ord(x) < ∞ iff MV-ord(x ∗ ) = ∞; (ii) locally finite iff, for all x ∈ A \ {0}, MV-ord(x) < ∞; (iii) singular iff there exist x, y ∈ A \ {0}, such that MV-ord(x) = ∞, MV-ord(y) < ∞ and MV-ord(y ∗ ) < ∞ [13]. In any BL-algebra L = (L, ∧, ∨, , →, 0, 1), it is possible to define its greatest MV-subalgebra MV(L) = (MV(L), ∧, ∨, MV , ⊕MV , 0, 1), where MV(L) = {x ∗ : x ∈ L}, via the following operations [13]: x ∗ MV y ∗ = x ∗ y ∗ , ∗ x ∗ ⊕MV y ∗ = x ∗∗ → y ∗ = (x y)∗ = x ∗∗ y ∗∗ . Note that x ∗∗ is the negate of x ∗ for every x ∗ ∈ MV(L). For simplicity, the symbol MV shall be omitted in the above operations. 3.

General results Let D be a proper filter of a BL-algebra L. We put, following [15]:

D ∗ = x ∈ L: x y ∗ for some y ∈ D .

Obviously D ∩ D ∗ = ∅ and D ∪ D ∗ is a subalgebra of L. Now, we put [13]:

D = D ∗ ∩ MV(L) = x ∗ ∈ MV(L): x ∈ D . It is routine to check that D is a proper ideal of MV(L). Proposition 20 of [13] proves that if D is a maximal filter of L, then D is maximal ideal of MV(L). Suppose P is an ideal of MV(L) and define

P = x ∈ L: x ∗ ∈ P . Lemma 1. P is a filter of L. If P is proper, P is proper. Proof. First of all, 1 ∈ P since 0 = 1∗ ∈ P . Let x, y ∈ P . Now (x y)∗ = x ∗ ⊕ y ∗ ∈ P , thus (x y) ∈ P . If y x, x ∈ P , we have y ∗ x ∗ , that is y ∗ ∈ P which implies y ∈ P . This proves that P is a filter of L. If 0 ∈ P , then 0∗ = 1 ∈ P , a contradiction if P is proper.

202


Lemma 2. For any proper filter D of L, we have (D)− ⊇ D. If D is maximal, then (D)− = D. Proof. Let x ∈ D, thus x ∗ ∈ D and hence x ∈ (D)− , then D ⊆ (D)− . By lemma 1, (D)− is a proper filter of L. If D is maximal, obviously we deduce (D)− = D. Example 1. Note that if D is not maximal, the equality (D)− = D, in general, is not true. It suffices to consider the Gödel algebra L and choose D = (1/2, 1], which is not a maximal filter. We have MV(L) = {0, 1} and D = D ∗ = {0} and hence D ⊂ (D)− = L − {0} = (0, 1]. Lemma 3. For any proper ideal P of MV(L), we have (P )− = P . If P is maximal, then P is a maximal filter of L. Proof. If x ∗ ∈ P , then x ∈ P . So x ∗ ∈ ( P )∗ ∩ MV(L) = ( P )− and then P ⊆ ( P )− . Conversely, let x ∗ ∈ ( P )− ⊆ ( P )∗ . Then x ∗ y ∗ for some y ∈ P , that is for some y ∗ ∈ P . Since P is MV-ideal, then x ∗ ∈ P , i.e., ( P )− ⊆ P and hence the thesis. Now let Q ⊇ P with Q proper filter of L and P be maximal in MV(L). Then P = ( P )− ⊆ Q. Since Q is proper, we have P = Q which implies P = (Q)− ⊇ Q by lemma 2. So Q = P , i.e., P is maximal in L. Lemma 4. In any BL-algebra L, we have for all x, y ∈ L: x ∗∗ ∨ y ∗∗ = (x ∨ y)∗∗ . Proof.

Using (vii) and (xiv) of section 2, we get x ∗∗ ∨ y ∗∗ = x ∗∗ → y ∗∗ → y ∗∗ ∧ y ∗∗ → x ∗∗ → x ∗∗ = y ∗ → x ∗ → y ∗∗ ∧ x ∗ → y ∗ → x ∗∗ .

On the other hand, we deduce using (xi) of section 2: ∗ ∗ ∗ = x → y ∗ → x ∗∗ (x ∨ y)∗∗ = x ∗ ∧ y ∗ = x ∗ (x ∗ → y ∗ and by symmetry, ∗ (y ∨ x)∗∗ = y ∗ ∧ x ∗ = y ∗ → x ∗ → y ∗∗ . Then (x ∨ y)∗∗ = (x ∨ y)∗∗ ∧ (y ∨ x)∗∗ = x ∗∗ ∨ y ∗∗ . Now let M be a maximal filter of L. According to [15], let

M(L) = {M: M is a maximal filter of L}.


203

By [14, proposition 15], A/M is a locally finite MV-algebra. This implies that x ∗∗ /M = x/M for all x ∈ L, i.e., (x ∗∗ → x)/M = 1/M for all x ∈ M. Then (x ∗∗ → x) ∈ M for all x ∈ M. Since M is arbitrary, we have proved Theorem 1. In any BL-algebra L, (x ∗∗ → x) ∈ M(L) for any x ∈ L. Let F (L) = {x ∈ L: (x n )∗∗ > x ∗ for all n} and, if A is an MV-algebra, let Rad A be the intersection of all maximal ideals of A. By [4, 3.6.4], a ∈ Rad A iff na a ∗ for all n. Proposition 19 of [13] proves that F (L) is a filter of L and if x ∈ L, then x ∈ F (L) iff x ∗ ∈ Rad MV(L), while F (L) = M(L) by [13, proposition 21]. These results are useful for proving the following statement. Proposition 1. (Rad MV(L))− = M(L). Proof. Let x ∈ L. Then x ∈ (Rad MV(L))− iff x ∗ ∈ Rad MV(L) iff x ∈ F (L) = M(L). Recalling that an MV-algebra A is semisimple iff Rad A = {0}, we have Proposition 2. Let L be a BL-algebra. Then L is a semisimple MV-algebra iff F (L) = {1}. Proof. Let L = MV(L) and Rad MV(L) = {0}. Then x ∗∗ = x for all x ∈ L and let y ∈ F (L). Then y ∗ = 0, i.e., y = y ∗∗ = 1 which implies F (L) = {1}. Conversely, let F (L) = {1}. By theorem 1, we have (x ∗∗ → x) = 1, i.e., x ∗∗ x for all x ∈ L. Then x ∗∗ = x by property (x) of section 2, i.e., L = MV(L) is an MV-algebra. Now if y ∗ ∈ Rad MV(L), we have y = y ∗∗ = 1, i.e., y ∗ = 0 and so L is semisimple. Proposition 2 is also in [14]. Definition 1. Let L be a BL-algebra. Define S(L) = {x ∈ L: x ∗∗ x ∗ }. In [15], the subset sup L = {x ∨ x ∗ : x ∈ L} is defined . We point out Proposition 3. sup L = {x ∈ L: x x ∗ }. Proof. Let y ∈ sup L. Then y = x ∨ x ∗ for some x ∈ L. Hence, by property (xi) of section 2, y ∗ = (x ∨ x ∗ )∗ = x ∗ ∧ x ∗∗ which implies y = x ∨ x ∗ x ∗ x ∗ ∧ x ∗∗ = y ∗ . Conversely, x = x ∨ x ∗ if x x ∗ . Corollary 1. (i) sup L ⊆ S(L);

204


(ii) M(L) ⊆ S(L); (iii) M(L) = {x ∈ L: (x n )∗ < x for all n}. Proof. (i) is obvious since x x ∗ implies x ∗∗ x ∗ . (ii) follows from [13, proposition 21]. Indeed if x ∈ M(L) = F (L), then x ∗∗ n ∗∗ (x ) > x ∗ for all integer n, hence x ∈ S(L). (iii) (See [9].) If (x n )∗ < x for all n, then (x n )∗∗ > x ∗ for all n and then x ∈ F (L) = M(L). Conversely, let x ∈ F (L) = M(L) and suppose (x m )∗ x for some m. / P for some prime filter P by [10, Then [(x m )∗ → x] = 1, so [(x m )∗ → x] ∈ m ∗ lemma 2.3.15]. Then (x → (x ) ) ∈ P and let M be a maximal extension of P . Since x ∈ M, we have (x m )∗ ∈ M. But x m ∈ M, so we obtain the contradiction 0 = (x m ) (x m )∗ ∈ M. Hence (x n )∗ x for all n. Now if (x n )∗ = x for some n, we should have (x n )∗∗ = x ∗ , an absurd to the fact that x ∈ F (L). This proves that (x n )∗ < x for all n, that is the thesis. Lemma 5. In any BL-algebra L, we have for all n and x, y ∈ L: (i) (x ∨ y)n = x n ∨ y n , (ii) (x ∧ y)n = x n ∧ y n . Proof. By Hájek’s representation theorem [10, lemma 2.3.16], it is enough to prove that (i) and (ii) hold in any BL-chain. (i) is surely true because if, for instance, we assume x y, then x n y n for every integer n, thus (x ∨ y)n = y n = x n ∨ y n for all x, y ∈ L. (ii) is proved similarly. 4.

Bipartite BL-algebras

According to [15], we define a BL-algebra to be bipartite (respectively, strongly bipartite) if L = M ∪M ∗ for some (respectively, every) maximal filter M. This definition comes back from MV-algebras theory [7]. In [15, theorems 3 and 4] the next lemma is proved. Lemma 6. L is bipartite (respectively, strongly bipartite) iff sup L ⊆ M for some (respectively, every) maximal filter M. Now we prove the following theorem. Theorem 2. L = M ∪ M ∗ for some maximal filter M iff S(L) ⊆ M. Proof. Let x ∈ S(L) and L be bipartite by M. If x ∈ M ∗ , then x y ∗ for some y ∈ M. This means x ∗ y ∗∗ y, i.e., x ∗ ∈ M. Since x ∗∗ x ∗ , then x ∗∗ ∈ M, a / M. Then x ∈ M, i.e., S(L) ⊆ M. contradiction, because x ∗ x ∗∗ = 0 ∈


205

Conversely, if S(L) ⊆ M, then sup L ⊆ S(L) ⊆ M by corollary 1(i). Then sup L ⊆ M and L is bipartite by lemma 6. Definition 2. Let P be an ideal of MV(L) and put P ◦ = {x ∗∗ ∈ MV(L): x ∗ ∈ P }. We remark immediately that P ◦ = P ∩MV(L) because x ∈ P iff x ∗∗ ∈ P , while if M is a maximal filter of L, we have (M)◦ = (M)− ∩ MV(L) = M ∩ MV(L) by lemma 2. This remark allows to prove the following theorem. Theorem 3. Let L be a BL-algebra. Then L is bipartite iff MV(L) is bipartite. Proof. Let L = M ∪ M ∗ for some maximal filter M. Then MV(L) = (M ∗ ∩ MV(L)) ∪ (M ∩MV(L)) = M ∪(M)◦ , M being a maximal ideal of MV(L) by [13, proposition 20]. Thus MV(L) is bipartite. Conversely, let MV(L) = P ∪ P ◦ for some maximal ideal P of MV(L). We claim / P . Then x ∗ ∈ / P , hence that L = P ∪ P ∗ . Indeed, let x ∈ L and assume that x ∈ ∗ ◦ ∗ ∗∗ x ∈ P , i.e., x ∈ P . Being x x , we then have that x ∈ P ∗ . Theorem 4. The following are equivalent: (i) S(L) is a proper filter of L; (ii) M(L) = S(L); (iii) L is strongly bipartite. Proof. (i) implies (ii). Let x ∈ S(L). Then x n ∈ S(L) since S(L) is a filter. So (x n )∗∗ / M. By [14, proposition 15], (x n )∗ for all n. Let M be a maximal filter be such that x ∈ (x m )∗ ∈ M for some integer m. Then (x m )∗∗ ∈ M and hence 0 = (x m )∗∗ (x m )∗ ∈ M, a contradiction. So S(L) ⊆ M for every maximal filter of L, that is S(L) ⊆ M(L). Corollary 1(ii) guarantees the converse inclusion, so we get (ii). (ii) implies (i). It is evident. (ii) implies (iii). By corollary 1(i), we have sup L ⊆ S(L) = M(L) ⊆ M for every maximal filter M of L. Then lemma 6 gives (iii). (iii) implies (ii). Theorem 2 assumes that S(L) ⊆ M for every maximal filter M, that is S(L) ⊆ M(L). So we get (ii) by corollary 1(ii). Lemma 7. Let Inf MV(L) = {x ∗ ∈ MV(L): x ∗ x ∗∗ } be a proper ideal of MV(L). Then (Inf MV(L))− = S(L). Proof.

x ∈ S(L) iff x ∗ ∈ Inf MV(L) iff x ∈ (Inf MV(L))− .

By lemmas 3 and 7, we also have:

206


Lemma 8. Let S(L) be a proper filter of L. Then S(L) = Inf MV(L). Theorem 5. A BL-algebra L is strongly bipartite iff MV(L) is strongly bipartite. Proof. L is strongly bipartite iff (by theorem 4) S(L) is a proper filter of L iff (by lemmas 7 and 8) Inf MV(L) = Rad MV(L) is a proper ideal of MV(L) iff (by [1, proposition 21]) MV(L) is strongly bipartite. 5.

The variety P0

Lemma 9. In any BL-algebra L, we have for all n and x ∈ L: ∗∗ (i) (x ∗ )n = (x ∗ )n ; (ii) (x ∗∗ )n = (x n )∗∗ . Proof.

A simple induction on the following identities: ∗ ∗∗ ∗ x x ∗ = x ∗∗ ⊕ x ∗∗ = x ∗ x ∗ , ∗∗ ∗ ∗∗ x x ∗∗ = x ∗ ⊕ x ∗ = x x

gives both thesis.

Remark. Lemma 9(ii) is lemma 2 of [13]. The class of strongly bipartite BL-algebras is a variety which we denote by P0 . Indeed, L ∈ P0 iff L satisfies axioms (i)–(v) of section 2 plus the axiom contained in the following theorem 6 or theorem 7. Theorem 6. A BL-algebra L is strongly bipartite iff for all n and x ∈ L: (xv) [(x n )∗∗ ∧ x ∗ ] ∨ [x ∗∗ ∧ (x ∗ )n ] = x ∗ ∧ x ∗∗ . Proof. Let L be strongly bipartite. Then sup L ⊆ M(L) = F (L) by lemma 6, that is x ∨ x ∗ ∈ F (L) for every x ∈ L. This means for all n and x ∈ L: n ∗∗ ∗ x ∨ x ∗ = x ∗ ∧ x ∗∗ , x ∨ x∗ by using property (xi) of section 2. By lemmas 4 and 5, we deduce: n ∗∗ ∗ n ∗∗ ∗ ∨ x ∧ x ∧ x ∗∗ = x ∗ ∧ x ∗∗ , x which, in turn, implies by lemma 9(i): n n ∗∗ ∧ x ∗ ∧ x ∗∗ ∨ x ∗ ∧ x ∗ ∧ x ∗∗ = x ∗ ∧ x ∗∗ . x This gives the thesis since (x n )∗∗ x ∗∗ and (x ∗ )n x ∗ .


207

Conversely, we can read (xv), by lemma 9(ii), as n ∗∗ n ∧ x ∗ ∨ x ∗∗ ∧ x ∗ = x ∗ ∧ x ∗∗ . x By [1, theorem 3.2], MV(L) is strongly bipartite and hence L is strongly bipartite by theorem 4. By using essentially corollary 1(iii), similarly is proved Theorem 7. A BL-algebra L is strongly bipartite iff for all n and x ∈ L n ∗ n ∗ ∨ x ∧ x∗ ∨ x∗ = x ∨ x∗. x Now we characterize the greatest strongly bipartite BL-algebra contained in any BL-algebra L. Let F be the set of all prime filters of L and Max F be the set of all maximal filters of F . For simplicity of notations, we put A = MV(L) and let Max A be the set of all maximal ideals of A. Definition 3. We define for any BL-algebra L:

P ∪ P∗ . L0 = P ∈F

L0 is obviously a subalgebra of L. Following [6, proposition 5.3], we recall that

M ∪ M◦ A0 = M∈Max A

is a strongly bipartite MV-algebra, subalgebra of A. Proposition 4. L0 is a strongly bipartite BL-algebra. / Proof. Let x ∈ L0 , that is x ∈ (P ∪ P ∗ ) for any P ∈ Max F . Suppose that x ∗ ∈ / M, that is x ∈ M with M ∈ F by lemma 3. (M ∪ M ◦ ) for some M ∈ Max A. Then x ∗ ∈ Then x ∈ M ∗ which means x y ∗ for some y ∈ M. Thus y ∗ ∈ M and since x ∗∗ y ∗ , we have x ∗∗ ∈ M, that is x ∈ M ◦ , a contradiction. This proves that x ∗ ∈ A0 . Conversely, / L0 , that is x ∈ / (P ∪ P ∗ ) for some let x ∈ L be such that x ∗ ∈ A0 and assume x ∈ ◦ ∗ / P . Then x ∗ ∈ P = P ∩ A, P ∈ Max F . By [13, proposition 20], P ∈ Max A and x ∈ that is x ∗ ∈ P . Since x x ∗∗ , then x ∈ P ∗ , a contradiction. In other words, we have that x ∈ L0 iff x ∗ ∈ A0 , that is MV(L0 ) = A0 . By [6, proposition 5.3], A0 is strongly bipartite and hence L0 is strongly bipartite by theorem 5. Proposition 5. Let L be subalgebra of L such that L ∈ P0 . Then L ⊆ L0 . Proof. Let A = MV(L ) ⊆ MV(L) = A. A is strongly bipartite by theorem 5 and then A ⊆ A0 by [6, proposition 5.4]. This implies L ⊆ L0 .

208


Proposition 6. For any BL-algebra L, L/M(L) is an MV-algebra isomorphic to A/Rad A. Proof.

By theorem 1, we have for any x ∈ L: x → x ∗∗ x ∗∗ → x = 1 x ∗∗ → x = x ∗∗ → x ∈ M(L)

and this implies that x/M(L) = x ∗∗ /M(L) = (x/M(L))∗∗ , that is the quotient BLalgebra L/M(L) is an MV-algebra. Define the map: x/M(L) ∈ L/M(L) → x ∗∗ /Rad A ∈ A/Rad A. This map is well defined. Indeed, if x/M(L) = y/M(L), we also have x ∗∗ /M(L) = y ∗∗ /M(L), that is (x ∗∗ → y ∗∗ ) ∈ M(L) and (y ∗∗ → x ∗∗ ) ∈ M(L). By property (vii) of section 2, we have (x ∗∗ → y ∗∗ ) = (x ∗∗ y ∗ )∗ ∈ M(L). By proposition 1, (x ∗∗ y ∗ )∗∗ = (x ∗∗ y ∗ ) ∈ Rad A. Similarly, we have that (y ∗∗ x ∗ ) ∈ Rad A. Hence x ∗∗ /Rad A = y ∗∗ /Rad A. Analogously, we see that the above map is one-to-one. Since for all x, y ∈ L ∗ (x y)∗∗ = x ∗ ⊕ y ∗ = x ∗∗ y ∗∗ , we easily deduce that the above map is an MV-isomorphism.

Proposition 7. L ∈ P0 iff L/M(L) is a Boolean algebra. Proof. L is strongly bipartite iff (by theorem 5) MV(L) is strongly bipartite iff (by [6, proposition 5.12]) A/Rad A is a Boolean algebra iff (by proposition 6) L/M(L) is a Boolean algebra. 6.

Perfect and peculiar BL-algebras

By taking in account (see [9]) that ord(x) = MV-ord(x ∗ ) for all x ∈ L, we easily deduce and improve the following results of [13]. Theorem 8. A local BL-algebra is perfect iff MV(L) is perfect. Proof.

For all x ∈ L MV − ord x ∗ = ord(x) < ∞ iff

MV − ord x ∗∗ = ord x ∗ = ∞.

Lemma 10. Let L be a BL-algebra such that MV(L) = {0, 1}. Then L is perfect. Proof. Note that x ∗ = 1 iff x = 0. If x ∈ L − {0}, then ord(x) = MV − ord(x ∗ ) = ∞ and ord(x ∗ ) = MV − ord(x ∗∗ ) = 1 since x ∗ = 0 and x ∗∗ = 1. Theorem 9. Let L be a local BL-algebra such that L = MV(L). Then L is peculiar iff MV(L) = {0, 1} is either singular or locally finite.


209

Proof. Let L be peculiar and assume MV(L) perfect. By theorem 8, L is perfect, a contradiction. Further, MV(L) = {0, 1} otherwise L is perfect by lemma 10. Hence MV(L) is either singular or locally finite by [2, theorem 5.1]. Conversely, let MV(L) = {0, 1} be locally finite or singular. L is not locally finite, otherwise L = MV(L) by [14, theorem 1]; a contradiction. L is not perfect otherwise MV(L) is perfect by theorem 8. So L is necessarily peculiar. We illustrate theorem 9 by the following examples: Example 2. Let t : [0, 1]2 → [0, 1] be the continuous t-norm defined as xty =

max x + y − 1/2, 0

if x, y ∈ [0, 1/2],

x∧y

otherwise.

t is ordinal sum of the t-norm of Lukasiewicz and min with respect to the intervals [0, 1/2] and [1/2, 1]. Its residuum “→” is defined as  if x y,  1 x → y = min{1/2, 1/2 − x + y} if x, y ∈ [0, 1/2] and x > y,   y otherwise. By [11], L = ([0, 1], ∧, ∨, t, →, 0, 1) is a t-algebra which is a BL-chain, hence in local. Note that  if x = 0,  1 ∗ x = 1/2 − x if x ∈ (0, 1/2],   0 otherwise. L is peculiar. Indeed, for any x ∈ [1/2, 1), we have ord(x) = ∞ because x n = x > 0 for all n, while if y ∈ [1/4, 1/3], we have ord(y) = 3 (because y 2 = yty = max{y + y − 1/2, 0} = 2y − 1/2 and y 3 = y 2 ty = max{3y − 1, 0} = 0) and ord(y ∗ ) = ord(1/2 − y) = 2 (because (1/2 − y)2 = (1/2 − y)t (1/2 − y) = max{1/2 − 2y, 0} = 0). Further, MV(L) = [0, 1/2) ∪ {1}, which is locally finite because MV-isomorphic to the locally finite MV-algebra of Lukasiewicz over [0, 1], via the map φ defined as φ(x) = 2x if x ∈ [0, 1/2), φ(1) = 1. Now we give an example of L peculiar and MV(L) singular. Example 3. Let Li = (Li , ∧i , ∨i , i , →i , 0i , 1i ) be two disjoint chains, i = 1, 2 and let L1 +L2 = (L1 ∪L2 , ∧, ∨, , →, 0, 1) be a BL-chain [11] by putting 0 = 01 , 11 = 02 , 1 = 12 and

210


xy

iff

x, y ∈ Li and x i y or if x ∈ L1 , y ∈ L2 .

x i y if x, y ∈ Li , x if x ∈ L1 , y ∈ L2 .  if x y, 1 x → y = x →i y, if x, y ∈ Li ,  y, if x ∈ L2 , y ∈ L1 . x y =

Let N and R be the set of integers and real numbers, respectively. Let F be an ultrafilter of subsets of N containing the cofinite subsets of N and R∗ = RN /F be the ultrapower determined from F . Denote by [0, 1]∗ the unit interval of R∗ , structured in MV-algebra with the same operations which define the Lukasiewicz algebra over [0, 1]. By setting L1 = [0, 1]∗ , L1 is a singular MV-algebra [2]. Let L2 = C be the Chang’s MV-algebra [3] and let L = L1 + L2 . Take x = mc ∈ L. Thus x n = 02 > 0 for all n, i.e., ord(x) = ∞. Now we choose an infinitesimal τ ∈ L1 − {0} and take y = 1/2 + τ , so y ∗ = 1/2 − τ . We have ord(y) = MV − ord(y ∗ ) = 3 (because y ∗ ⊕ y ∗ = min{1, 1 − 2τ } = 1 − 2τ and (y ∗ ⊕ y ∗ ) ⊕ y ∗ = min{1, 3/2 − 3τ } = 1) and ord(y ∗ ) = MV − ord(y) = 2 (because y ⊕ y = min{1, 1 + 2τ } = 1). Then L is peculiar and MV(L) = L1 − {11 } ∪ {1} is singular because MV-isomorphic to L1 . Let L and L be two BL-algebras and f : L → L be a BL-homomorphism, i.e., a map such that f (0) = 0, f (1) = 1 and f preserves the operations “ ” and “→”. Assume L is a homomorphic image of L, that is f (L) = L . If L is perfect, then MV(L) is perfect by theorem 8. Clearly, f (MV(L)) ⊆ MV(L ). Now let y ∗ ∈ MV(L ) and x ∈ L be such that f (x) = y ∗ . Then f (x ∗∗ ) = [f (x ∗ )]∗ = [f (x)]∗∗ = y ∗∗∗ = y ∗ which implies f (MV(L) ⊇ MV(L ). Hence MV(L ) = f (MV(L)) is perfect by [6, proposition 3.6] and L is perfect by theorem 8. If L1 is a subalgebra of the perfect BL-algebra L, then L1 is, clearly, also perfect. Unfortunately, the direct product of two perfect BL-algebras is, generally, not perfect (it suffices to consider C × C, where C is the Chang MV-algebra [3]). Then the class of perfect BL-algebras, although closed by homomorphic images and subalgebras, is not a variety. We can consider the related generating variety which coincides with P0 because of the following statement. Theorem 10. Let L ∈ P0 . Then L is subdirect product of perfect BL-chains. Proof. Let D be a prime filter of L, so L/D is a BL-chain, hence local. Let M be the maximal extension of D. Let x ∈ L be such that ord(x/D) < ∞, i.e., x m /D = 0/D for some integer m. If x ∈ M, then x m ∈ M. Now (x m )∗ = (x m )∗ 1 = (x m → 0) (0 → / M. x m ) ∈ D ⊆ M, thus 0 = (x m ) (x m )∗ ∈ M, a contradiction. Then x ∈ Since L is a strongly bipartite, we have L = M ∪ M ∗ . So x ∈ M ∗ , that is x y ∗ for some y ∈ M. Then x ∗ y ∗∗ y, hence x ∗ ∈ M. Then x ∗ /D ∈ M/D and this implies ord(x ∗ /D) = ∞.


211

Conversely, let ord(x ∗ /D) = ∞. Since L/D is local, we necessarily have ord(x/D) < ∞. Then L/D is a perfect BL-chain and the thesis is proved because, by Hájek [10, lemma 2.3.13] representation theorem, L is the subdirect product of the perfect BL-chains L/D, where D varies on the set of prime filters of L. By generalizing theorem 6, we show the following. Theorem 11. A BL-algebra L ∈ P0 iff for all x ∈ L, ∗ 2 ∗ 2 2 ∗ x2 = x∗ Proof. We have that L is strongly bipartite iff (by theorem 5) MV(L) is strongly bipartite iff (by [6, theorem 5.11]) (x ∗ ⊕ x ∗ )2 = (x ∗ )2 ⊕ (x ∗ )2 for all x ∈ L iff (by lemma 9(i)) ((x 2 )∗ )2 = ((x ∗ )2 )∗∗ ⊕ ((x ∗ )2 )∗∗ for all x ∈ L iff ((x 2 )∗ )2 = [((x ∗ )2 )∗ ((x ∗ )2 )∗ ]∗ for all x ∈ L iff ((x 2 )∗ )2 = ((((x ∗ )2 )∗ )2 )∗ for all x ∈ L iff (by lemma 9(i)) ∗ ∗ 2 ∗∗ ∗ 2 ∗ 2 2 2 = x∗ = x∗ x2 for all x ∈ L. This completes the proof.

Now we examine the generators of the variety P0 but we first need some simple results about BL-chains. Lemma 11. Let L be a BL-chain and u = 0 be an idempotent of L (that is u u = u). Then u∗ = 0. Proof.

By [11, lemma 1], we have u∗ = u → 0 = 0.

Lemma 12. Let L be a BL-chain and x ∈ L − {0}. Then 1 = x ∗ < u for every idempotent u ∈ L − {0}. Proof. Let u x ∗ for some idempotent u = 0. By lemma 11, then 0 = u∗ x ∗∗ x, that is x = 0, a contradiction. Following [11, definition 6], a BL-chain L is saturated if each cut (X, Y ) of L (in the sense of [11, definition 5] is separated from an idempotent u ∈ L, i.e., x u y for all x ∈ X and y ∈ Y . By [11, theorem 3], any BL-chain L can be isomorphically embedded into a saturated BL-chain L∞ = L ∪ I , where I is a family of idempotents, inserted “ad hoc” in the structure of L, such that every u ∈ I separates each given cut of L, not separated from no idempotent of L. Theorem 12. Let L be a perfect BL-chain. Then L∞ is a perfect BL-chain.

212


Proof. By lemma 11, we have u∗ = 0 for every idempotent u ∈ L∞ − {0}. Now let x ∈ L − {0}. By lemma 12, x ∗ = x →L∞ 0 ∈ L − I , hence x ∗ = x →L∞ 0 = x →L 0. This means that MV(L) = MV(L∞ ), thus MV(L∞ ) is a perfect MV-algebra. Then L∞ is a perfect BL-chain by theorem 8. By theorem 10, the variety P0 has an infinite number of generators which are the perfect BL-chains. By theorem 12, we can choose perfect saturated BL-chains as generators of P0 . 7.

A classification of bipartite BL-algebras The class P of bipartite BL-algebras is closed for subalgebras. Indeed, we have

Theorem 13. Let L1 be a BL-subalgebra of a bipartite BL-algebra L. Then L1 is bipartite. Proof. Clearly, MV(L1 ) ⊆ MV(L) and MV(L) is bipartite by theorem 3. By [7, theo rem 4.1], MV(L1 ) is bipartite and hence L1 is bipartite by theorem 3. Lemma 13. Let L, Li , i ∈ I , be BL-algebras such that L = !i∈I Li . If Lh is bipartite for some h ∈ I , then L is bipartite. Proof. Clearly, MV(L) = !i∈I MV(Li ) and MV(Lh ) is bipartite by theorem 3. By [7, theorem 4.5], MV(L) is bipartite and L is bipartite by theorem 3. Then the following holds. Theorem 14. P is closed under formation of direct products. / P and L2 ∈ P. By lemma 12, we get Now let L = L1 × L2 , where L1 ∈ L ∈ P and let pr1 : L → L1 be the first canonical projection, which is obviously a BL-homomorphism. Now pr1 L = L1 , thus the class P is not closed for homomorphic images, even if the above theorems 13 and 14 assure that P is closed for subalgebras and direct products. P is not therefore a variety. However, using the results of [1], we can classify the elements of P in the below manner. We recall that a BL-algebra [8] L satisfying the axiom (x y)∗ = x ∗ ∨ y ∗ is said strict BL-algebra [8] (SBL-algebra, for short). In other words, L is an SBLalgebra iff MV(L) is a Boolean algebra [13, proposition 9]. An SBL-algebra L is strongly bipartite because a Boolean algebra is as well. In particular, if MV(L) = {0, 1}, then L is a perfect SBL-algebra by lemma 10.


213

We define bimaximal any BL-algebra L such that MV(L) is semisimple and bipartite with two only maximal ideals. In other words, MV(L) is subdirect product of the minimal Boolean algebra B2 = {0, 1} and the Lukasiewicz MV-algebra over [0, 1], that is, MV(L) semisimple does not imply L semisimple as it is proved in the following. Example 4. Let L = B2 × G, where G is the Gödel BL-algebra over [0, 1]. Then MV(L) = {(0, 0), (0, 1), (1, 0), (1, 1)} is (isomorphic to) the Boolean algebra B4 with four elements and M1 = {(0, 0), (0, 1)} and M2 = {(0, 0), (1, 0)} are the only maximal ideals of MV(L). MV(L) is semisimple. Now M 1 = {(1, x): x ∈ [0, 1]} and M 2 = {(0, x) ∪ (1, x): x ∈ (0, 1]} are, by lemma 1, the unique maximal filters of L, which is an SBL-algebra. Since M 1 ∩ M 2 = {(1, x): x ∈ (0, 1]} = F (L), then L is not a semisimple MV-algebra by proposition 2. Now let L ∈ P and we define L to be essentially preboolean if there exist x, x ∗ , y, y ∈ L such that ord(x) = ord(x ∗ ) = ord(y) = ord(y ∗ ) = ∞ and x ∗∗ ⊕ y ∗∗ ∈ S(L) − {1}. Consequently, we have MV − ord(x ∗ ) = MV − ord(x ∗∗ ) = MV − ord(y ∗ ) = MV − ord(y ∗∗ ) = ∞ and (x ∗ y ∗ ) = (x ∗∗ ⊕y ∗∗ )∗ (x ∗∗ ⊕y ∗∗ )∗∗ = (x ∗ y ∗ )∗∗∗ = (x ∗ y ∗ )∗ , that is (x ∗ y ∗ ) ∈ Inf MV(L) − {0}. Then MV(L) is essentially preboolean in the sense of [1, definition 4.6]. ∗

Theorem 15. Let L ∈ P. Then L is either a perfect SBL-algebra or is an SBL-algebra such that MV(L) = B4 or is an SBL-algebra such that MV(L) = B2 , B4 , or is a local perfect BL-algebra or is bimaximal or is essentially preboolean. Proof. The claim is true if MV(L) = B2 or MV(L) = B4 . Hence assume MV(L) = B2 , B4 . By theorem 3, MV(L) is bipartite. By [7, theorem 4.3], MV(L) is either a Boolean algebra or is a local perfect MV-algebra or is bimaximal or is essentially preboolean. Bearing in mind theorem 8 and the above definitions, then we get the thesis. Theorem 15 gives a full description of all the elements of the class P. References [1] R. Ambrosio and A. Lettieri, A classification of bipartite MV-algebras, Math. Japon. 38 (1993) 111–117. [2] L.P. Belluce, A. Di Nola and A. Lettieri, Local MV-algebras, Rend. Circ. Mat. Palermo 42 (1993) 347–361. [3] C.C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958) 467–490. [4] R. Cignoli, M.L. D’Ottaviano and D. Mundici, Algebraic Foundations of Many-Valued Reasoning (Kluwer, Dordrecht, 1999). [5] R. Cignoli, F. Esteva, L. Godo and A. Torrens, Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft Computing 4 (2000) 106–112. [6] A. Di Nola and A. Lettieri, Perfect MV-algebras are categorically equivalent to Abelian %-groups Studia Logica 53 (1994) 417–432.

214


[7] A. Di Nola, F. Liguori and S. Sessa, Using maximal ideals in the classification of MV-algebras, Portugal. Math. 50 (1993) 87–102. [8] F. Esteva, L. Godo, P. Hájek and M. Navara, Residuated fuzzy logic with an involutive negation, Arch. Math. Logic 39 (2000) 103–124. [9] G. Georgescu, Private comunication. [10] P. Hájek, Metamathematics of Fuzzy Logic (Kluwer, Dordrecht, 1998). [11] P. Hájek, Basic fuzzy logic and BL-algebras, Soft Computing 2 (1998) 124–128. [12] P. Hájek, L. Godo and F. Esteva, A complete many-valued logic with product-conjuction, Arch. Math. Logic 35 (1996) 191–208. [13] S. Sessa and E. Turunen, Local BL-algebras, Multi-Valued Logic 6 (2001) 229–249. [14] E. Turunen, BL-algebras and fuzzy logic, Mathware and Soft Computing 1 (1999) 49–61. [15] E. Turunen, Boolean deductive systems of BL-algebras, Arch. Math. Logic, to appear.