Priestley Dualities for Some Lattice-Ordered Algebraic Structures, Including MTL, IMTL and MV-Algebras∗

DOI: 10.2478/s11533-006-0025-6 Research article CEJM 4(4) 2006 600–623

Priestley dualities for some lattice-ordered algebraic structures, including MTL, IMTL and MV-algebras∗

Leonardo Manuel Cabrer†, Sergio Arturo Celani‡,

Departamento de Matem´aticas, Facultad de Ciencias Exactas, Universidad Nacional del Centro, Pinto 399, 7000 Tandil, Argentina

Received 13 July 2005; accepted 13 June 2006

Abstract: In this work we give a duality for many classes of lattice ordered algebras, as Integral Commutative Distributive Residuated Lattices MTL-algebras, IMTL-algebras and MV-algebras (see page 604). These dualities are obtained by restricting the duality given by the second author for DLFI- algebras by means of Priestley spaces with ternary relations (see [2]). We translate the equations that deﬁne some known subvarieties of DLFI-algebras to relational conditions in the associated DLFI-space. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved.

Keywords: Bounded distributive lattices with fusion and implication, Priestley duality, residuated distributive lattices, MTL-algebras, IMTL-algebras, MV-algebras MSC (2000): 03G10, 03B50, 06D35,06D72

1 Introduction

In recent years many varieties of algebras associated to multi-valued logics have been introduced . The majority of these algebras are commutative integral distributive residuated lattices [8] with additional conditions, as for example, the variety of MV-algebras [4], the variety of BL-algebras, and the varieties of MTL-algebras and ITML-algebras [5, 6], recently introduced. All these classes of algebras are bounded distributive lattices with two additional binary operations ◦ and → satisfying special conditions. On the other hand, in [2], the varieties DLF of DLF-algebras, DLI of DLI-algebras and DLFI

∗ The authors are supported by the Agentinian Consejo de Investigaciones Cientificas y Tecnicas (CON- ICET). † E-mail: [email protected] ‡ E-mail: [email protected] L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 601 of DLFI-algebras were introduced and studied. These algebras are bounded distributive lattices with one or two additional binary operations ◦ and → satisfying certain minimal conditions. The variety of DLFI-algebras is a common generalization of many varieties of algebras including residuated lattices [9], relevant algebras [18], MV-algebras, etc. The main objective of [2] was to develop a Priestley duality for these varieties, and in special, to give a topological characterization of the congruences in these varieties. The representation given in [2] is a generalization of the representation developed by A. Urquhart in [18] for Relevant algebras by means of relational Priestley spaces. In this paper we investigate a topological representation for some algebraic structures of fuzzy logic using the results given in [2]. In particular, we shall give a duality for integral commutative distributive residuated lattices, MTL-algebras, IMTL-algebras and MV- algebras. In Section 2 we will recall the Priestley duality given in [2] for DLI-algebras, DLF- algebras and DLFI-algebras. In Section 3 we shall prove that a DLI-algebra or a DLF- algebra satisfies certain identities if and only if its dual space satisfies certain first order conditions. Some of these correspondences are acquaintances in relevance logic and relevant algebras [18], although our presentation is different. In Section 4 we shall give the mentioned Priestley dualities for integral commutative distributive residuated lattices, MTL-algebras, IMTL-algebras and MV-algebras. In section 5 we will discuss how the dualities given by the second author in [2] are related to other known representations and dualities for lattice with different operators.

2 Preliminaries

In this section we recall the Priestley´s duality for bounded distributive lattices (see [13, 14]), the definitions of DLI, DLF,andDLFI-algebras and its Priestley´s style duality given in [2]. We will use the following conventions. Given a set X and a subset Y ⊆ X we will note X\Y = {x ∈ X : x/∈ Y }. When there is no risk of misunderstanding we will note Y c instead of X\Y . Given a bounded distributive lattice A = A, ∨, ∧, 0, 1 ,ifH ⊆ A, we will note F (H)andI(H) to the filter and the ideal generated by H respectively. The collection of filters of A will be denoted by Fi(A). We will note X (A) to the set of prime filters of A. Given a poset X, ≤ , a set Y ⊆ X is called increasing if for every x ∈ Y and every y ∈ X,ifx ≤ y then y ∈ Y . Dually, Y ⊆ X is said to be decreasing if for every x ∈ Y and every y ∈ X,ify ≤ x then y ∈ Y . We will note [Y )((Y ]) to the least increasing (decreasing) set that contains Y. If Y = {y} then we will note [Y )by[y)and(Y ]by (y] , and we will say that a set Z ⊆ X is principal increasing (decreasing) if there exists x ∈ X such that Z =[x)(Z =(x]). The set of all increasing subsets of X will be denoted by Pi(X) and the power set of X by P(X). If X1, ≤1 and X2, ≤2 are posets, a map f : X1 −→ X2 is increasing when for every x, y ∈ X1 such that x ≤1 y it holds that 602 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623

f(x) ≤2 f(y). A Priestley space is a triple X, ≤,τ where X, ≤ is a poset and X, τ is a topological Stone space (compact, Hausdorﬀ and 0-dimensional) that is totally separated in the order, i.e., for every x, y ∈ X such that x ≤ y, there is a clopen (closed and open in τ)increasingsetU ⊆ X such that x ∈ U and y/∈ U.IfX, ≤,τ is a Priestley space, the set of all clopen increasing subsets of X will be denoted by D(X). Since D(X)isa ring of sets, D(X), ∪, ∩, ∅,X is a bounded distributive lattice. We will denote by PR the category whose objects are Priestley spaces and whose arrows are continuous and increasing functions. Given a bounded distributive lattice A, consider the topology τA on X (A) generated by the subbase whose elements are the sets of the form β(a)={P ∈X(A):a ∈ P } and c β(a) = X (A) \ β(a), for each a ∈ A.ThenX (A), ⊆,τA is a Priestley space. The map β : A →D(X (A)) is a bounded lattice isomorphism. For every Priestley space X, ≤,τ , the map ε : X →X(D(X)) given by ε (x)={U ∈D(X):x ∈ U}, for x ∈ X,isan order-isomorphism and a homeomorphisms of topological spaces. Moreover, there is a duality between the algebraic category of bounded distributive lattices and the category PR. For details on Priestley duality see [13]. Now we recall the deﬁnitions of DLI, DLF,andDLFI-algebras and some results given in [2].

Definition 2.1. An algebra A = A, ∨, ∧, →, 0, 1 of type (2, 2, 2, 0, 0) is a Distributive Lattice with Implication (or DLI-algebra) if and only if A, ∨, ∧, 0, 1 is a bounded distributive lattice and the following conditions are satisfied: I1. (a → b) ∧ (a → c) ≈ a → (b ∧ c). I2. (a → c) ∧ (b → c) ≈ (a ∨ b) → c. I3. a → 1 ≈ 1. I4. 0 → a ≈ 1. An algebra A = A, ∨, ∧, ◦, 0, 1 of type (2, 2, 2, 0, 0) is a distributive lattice with Fusion (or DLF-algebra) if and only if A, ∨, ∧, 0, 1 is a bounded distributive lattice and the following conditions are satisfied: F1. a ◦ (b ∨ c) ≈ (a ◦ b) ∨ (a ◦ c). F2. (a ∨ b) ◦ c ≈ (a ◦ c) ∨ (b ◦ c). F3. a ◦ 0 ≈ 0 ◦ a ≈ 0.

An algebra A = A, ∨, ∧, ◦, →, 0, 1 is a distributive lattice with Fusion and Implication (or DLFI-algebra) if and only if A, ∨, ∧, ◦, 0, 1 is a DLF-algebra and A, ∧, ∨, →, 0, 1 is a DLI-algebra.

Let A be a DLFI-algebra. Let us recall that for any a, b, x ∈ A,ifa ≤ b,then x → a ≤ x → b, b → x ≤ a → x, a ◦ x ≤ b ◦ x,andx ◦ a ≤ x ◦ b. We will note by DLF, DLI, DLFI the varieties of DLF, DLI and DLFI-algebras respectively. L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 603

Example 2.2. In any distributive lattice A, ∨, ∧, 0, 1 we can deﬁne at least four binary operations on its domain such that the algebra A = A, ∨, ∧, →, 0, 1 be a DLI-algebra. Indeed, any of the following binary functions →i: A × A → A, for 1 ≤ i ≤ 4, deﬁne structures of DLI-algebra:

a →1 b =1.

⎧ ⎨⎪ 1ifb =1ora =0, → a 2 b = ⎪ ⎩ 0ifb =1and a =0.

⎧ ⎨⎪ 1ifa ≤ b, → a 3 b = ⎪ ⎩ 0ifa b.

⎧ ⎨⎪ 1ifa =0, → a 4 b = ⎪ ⎩ b if a =0.

Example 2.3. As in the case of implication in any distributive lattice A, ∨, ∧, 0, 1 we can deﬁne at least four binary operations on its domain such that the algebra A = A, ∨, ∧, ◦, 0, 1 is a DLF-algebra. The following binary functions ◦i : A × A → A, for 1 ≤ i ≤ 4, deﬁne structures of DLF-algebra:

a ◦1 b =0.

⎧ ⎨⎪ 1ifa =0and b =0, ◦ a 2 b = ⎪ ⎩ 0ifa =0orb =0.

⎧ ⎨⎪ a if b =0, ◦ a 3 b = ⎪ ⎩ 0ifb =0.

⎧ ⎨⎪ b if a =0 , ◦ a 4 b = ⎪ ⎩ 0ifa =0. 604 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623

Example 2.4. Let us consider a relational structure F = X, ≤,R,T ,whereX, ≤ is aposet,andR and T are ternary relations deﬁned on X such that for each S ∈{R, T}, if (x, y, z) ∈ S & x ≤ x & y ≤ y & z ≤ z,then(x,y,z) ∈ S. It is easy to prove that

A (F)=Pi (X) , ∪, ∩, ∗, →∅,X is a DLFI-algebra, where the operations → and ∗ are deﬁned by:

U → V = {x ∈ X : For every y, z ∈ X,((x, y, z) ∈ T and y ∈ U) implies z ∈ V } .

U ∗ V = {z ∈ X : There exists (x, y) ∈ U × V, (x, y, z) ∈ R} .

The algebra A (F) is called the DLFI-algebra associated with F.

Now we shall see diﬀerent examples of DLFI-algebras connected with ordered algebraic structures associated with known multi-valued logics.

Deﬁnition 2.5. An integral commutative residuated lattice,orICR-lattice,isaDLFI- algebra A = A, ∧, ∨, ◦, →, 0, 1 satisfying the following axioms: (R) a ◦ b ≤ c if and only if a ≤ b → c, for any a, b, c ∈ A, (A) a ◦ (b ◦ c) ≈ (a ◦ b) ◦ c. (I) 1 ◦ a ≈ a. (C) a ◦ b ≈ b ◦ a.

The class of ICR-lattices is a variety and we will denote it by ICRL.Forasurvey on residuated lattices in general and further references see [9]. Now, we shall recall the definitions of some known subvarieties of ICRL. Let V be a subvariety of ICRL and let Γ be a finite set of equations in the algebraic language of the ICR-lattices. We shall denote by V+Γ the subvariety of V whose elements satisfies the equations Γ. Let us consider the following subvarieties of ICRL:

MT L = ICRL + {(a → b) ∨ (b → a) ≈ 1}

IMT L = MT L + {a ≈ (a → 0) → 0} .

MV = IMT L + {(a → b) → b ≈ (b → a) → a} .

The variety MT L is the variety of Monoidal T–norm based Logic algebras introduced in [5], the variety IMT L is the variety of Involutive MTL-algebras deﬁned in [6], and MV is the variety of MV-algebras (see [4]). For the rest of the paper we are going to denote the algebraic category whose objects are the members of certain variety the same way that we denote the variety, e.g. we will denote by MT L the variety of MTL-algebras and the algebraic category whose objects are MTL-algebras. L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 605

2.1 Representation and duality

In this subsection we recall the Priestley duality for DLF, DLI,andDLFI-algebras. For the proof of the results of this subsection see [2]. Let A be a DLFI-algebra and let F, G ∈ Fi(A). We deﬁne the following subsets of A: F → G = {a ∈ A :thereexist(f,g) ∈ F × G such that f ≤ g → a}.

F ◦ G = {a ∈ A :thereexist(f,g) ∈ F × G such that f ◦ g ≤ a}

It is easy to see that for every F, G ∈ Fi(A), F ◦ G and F → G ∈ Fi(A).

Deﬁnition 2.6. Let A be a DLFI-algebra. Let us deﬁne in X (A) the ternary relations RA and TA as follows:

(P, Q, D) ∈ TA if and only if P → Q ⊆ D,

(P, Q, D) ∈ RA if and only if P ◦ Q ⊆ D.

Example 2.7. Let A = A, ∧, ∨, 0, 1 be a bounded distributive lattice and let us i consider the DLI-algebras A = A, ∧, ∨, →i, 0, 1 for 1 ≤ i ≤ 4 deﬁned in Exam- 3 ple 2.2.ItholdsTA1 = ∅, TA2 = X (A) ,TA3 = {(P, Q, D):Q ⊆ D} and TA4 = {(P, Q, D):P ⊆ D}.

Example 2.8. Let A = A, ∧, ∨, 0, 1 be a bounded distributive lattice and let us i consider the DLF-algebras A = A, ∧, ∨, ◦i, 0, 1 for 1 ≤ i ≤ 4 deﬁned in Exam- 3 ple 2.3.ItholdsRA1 = ∅, RA2 = X (A) ,RA3 = {(P, Q, D):P ⊆ D} and RA4 = {(P, Q, D):Q ⊆ D}.

The following results are going to be useful in the proof of Theorems 3.1 and 3.5.

Theorem 2.9. Let A be a DLFI-algebra. Let F, G ∈ Fi(A) and P ∈X(A). Then the following propositions hold: (1) If F → G ⊆ P , then there exist Q, D ∈X(A) such that F ⊆ Q, G ⊆ D and Q → D ⊆ P . (2) If F ◦G ⊆ P, then there exist Q, D ∈X(A) such that F ⊆ Q, G ⊆ D and Q◦D ⊆ P .

Theorem 2.10. Let A be a DLFI-algebra, a, b ∈ A and P ∈X(A). Then the following propositions hold: (1) a → b ∈ P if and only if for every Q, D ∈X(A) such that a ∈ Q and P → Q ⊆ D then b ∈ D. (2) a ◦ b ∈ P if and only if there exist Q, D ∈X(A) such that a ∈ Q, b ∈ D and Q ◦ D ⊆ P . 606 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623

Now we are going to recall the deﬁnition of DLI, DLF, DLFI-spaces and their morphisms.

Definition 2.11. Let us consider the relational structure X, ≤,R,τ , where X, ≤,τ is a Priestley space and R is a ternary relation defined on X . (1) We will say that X, ≤,R,τ is a DLI-space if and only if for all U, V ∈D(X), U → V ∈D(X) , and (x, y, z) ∈ R whenever ε (x) → ε (y) ⊆ ε (z), for any x, y, z ∈ X, where → is defined in Example 2.4. (2) We will say that X, ≤,R,τ is a DLF-space if and only if for all U, V ∈D(X), U ∗ V ∈D(X), and (x, y, z) ∈ R, whenever ε (x) ∗ ε (y) ⊆ ε (z), for any x, y, z ∈ X, where ∗ is defined in Example 2.4. If T is a ternary relation defined on X, we will say that X, ≤,R,T,τ is a DLFI- space if and only if X, ≤,R,τ is a DLF-space and X, ≤,T,τ is a DLI-space.

Remark 2.12. Note that if X, ≤,S,τ is a DLF or a DLI-space, for all x, y, z ∈ X, if (x, y, z) ∈ S and x ≤ x, y ≤ y and z ≤ z,then(x,y,z) ∈ S.

Deﬁnition 2.13. An i-morphism, between DLI-spaces X1, ≤1,T1,τ1 and X2, ≤2,T2,τ2 , is a map f : X1 −→ X2 with the following properties: (1) f is increasing and continuous, (2) If (x, y, z) ∈ T1,then(f (x) ,f(y) ,f(z)) ∈ T2, (3) If (f (x) ,y,z) ∈ T2, then there exist y, z ∈ X1 such that (x, y, z) ∈ T1, y ≤ f (y) and f (z) ≤ z. An f-morphism, between DLF-spaces X1, ≤1,R1,τ1 and X2, ≤2,R2,τ2 ,isamap f : X1 −→ X2 with the following properties: (1) f is increasing and continuous, (2) If (x, y, z) ∈ R1, then (f (x) ,f(y) ,f(z)) ∈ R2, (3) If (x ,y,f(z)) ∈ R2, then there exist x, y ∈ X1 such that (x, y, z) ∈ T1, x ≤ f (x) and y ≤ f (y). An fi-morphism, between DLFI-spaces X1, ≤1,R1,T1,τ1 and X2, ≤2,R2,T2,τ2 , is a map f : X1 −→ X2 such that it is an f-morphism between the DLF-spaces X1, ≤1,R1,τ1 and X2, ≤2,R2,τ2 and that is an i-morphism between the DLI-spaces X1, ≤1,T1,τ1 and X2, ≤2,T2,τ2 .

Deﬁnition 2.14. We will call FI to the category which have DLI-spaces as objects and i-morphims as arrows. In similar way we deﬁne FF to be the category of DLI-spaces, and FFI the category of DLFI-spaces.

Let A be a DLFI-algebra and consider the structure:

X (A)=X (A) , ⊆,RA,TA,τA , where RA and TA are the relations given in Deﬁnition 2.6.In[2] the second author proves that X (A)isaDLFI-space, called the DLFI-space associated with A. L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 607

For every DLFI-space X,

D (X)=D (X) , ∩, ∪, ∗, →, ∅,X

is a DLFI-algebra. Moreover, there is a categorical duality between the categories DLFI and FFI.In[2]itisprovedthatDLI is dually equivalent to FI and DLF is dually equivalent to FF.

Remark 2.15. There are others representations and dualities for lattices with diﬀerent additional operators, given by A. Urquhart [18], V. Sofronie-Stokkermans [15, 17], N.G. Martinez [10, 11], and N.G. Martinez & H.A. Priestley [12] to mention some of them. In section 5 we will study the relation between these representations and dualities and the duality described in this section.

3 Some extensions of DLI-algebras and DLF-algebras

In this section we give a translation of some algebraic conditions deﬁned on DLI-algebras or DLF-algebras into ﬁrst-order conditions in its dual space. Notation: During the rest of this paper we will use the following conventions: If ψ and ϕ are formulas in the language of DLI-algebras (DLF) , we denote by ϕ ψ the equation ϕ ∧ ψ ≈ ϕ.IfA is a DLI-algebra (DLF), we will note A ϕ ψ if and only if ϕ ψ is valid in A. We will note ⇔ instead of “if and only if”.

Theorem 3.1. Let A be a DLI-algebra and let X = X (A) be the set of prime ﬁlters of A and let T = TA be the ternary relation given in Deﬁnition 2.6.Then: (1) A a (a → b) → b ⇔∀P, Q, D ∈ X, (P, Q, D) ∈ T implies (Q, P, D) ∈ T . (2) A a → (b → c) b → (a → c) ⇔∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ T and (D, Z, W) ∈ T ) implies (∃K ∈ X, (P, Z, K) ∈ T and (K, Q, W) ∈ T ). (3) A a → b (b → c) → (a → c) ⇔∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ T and (D, Z, W) ∈ T ) implies ∃K ∈ X, ((P, Z, K) ∈ T and (Q, K, W ) ∈ T ). (4) A (a ∧ b) → c a → (b → c) ⇔∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ T and (D, Z, W) ∈ T ) implies ∃K ∈ X, (Q ⊆ K and Z ⊆ K and (P, K, W) ∈ T ). (5) A a → (b → c) (a ∧ b) → c ⇔∀P, Q, D ∈ X, (P, Q, D) ∈ T implies ∃K ∈ X, ((P, Q, K) ∈ T and (K, Q, D) ∈ T ). (6) A (a ∧ b) → c (a → c) ∨ (b → c) ⇔∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ T and (P, Z, W) ∈ T ) implies (∃H, K ∈ X, (Q ⊆ H and Z ⊆ H and (K ⊆ D or K ⊆ W )) and (P, H, K) ∈ T ). (7) A a b → (a ∧ b) ⇔∀P, Q, D ∈ X, (P, Q, D) ∈ T implies (P ⊆ D and Q ⊆ D). (8) A a b → b ⇔∀P, Q, D ∈ X, (P, Q, D) ∈ T implies Q ⊆ D. (9) A a a → a ⇔∀P, Q, D ∈ X, (P, Q, D) ∈ T implies (P ⊆ D or Q ⊆ D). (10) A a b → a ⇔∀P, Q, D ∈ X, (P, Q, D) ∈ T implies P ⊆ D. (11) A a ∧ (a → b) b ⇔∀P ∈ X, (P, P, P) ∈ T . 608 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623

(12) A a → (b ∨ c) (a → b) ∨ (a → c) ⇔∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ T and (P, Z, W) ∈ T ) implies (∃H, K ∈ X, (Q ⊆ H or Z ⊆ H) and K ⊆ D and K ⊆ W and (P, H, K) ∈ T ). (13) A (a → b) ∧ (b → c) a → c ⇔∀P, Q, D ∈ X, (P, Q, D) ∈ T implies ∃K ∈ X, ((P, Q, K) ∈ T and (P, K, D) ∈ T ). (14) A 1 ≈ (a → b)∨(b → a) ⇔∀P, Q, D, Z, W ∈ X, (P, Q, D) ∈ T and (P, Z, W) ∈ T implies Q ⊆ W or Z ⊆ D. (15) A 1 → a a ⇔∀P, Q ∈ X, P ⊆ Q implies ∃D ∈ X, (P, D, Q) ∈ T .

Proof. We will prove 2, 3, 6, 7, 12, 13 and 14. The other conditions are similar and left to the reader. 2. ⇒)LetP, Q, D, Z, W ∈ X such that (P, Q, D) , (D, Z, W) ∈ T , i.e., P → Q ⊆ D and D → Z ⊆ W . We will prove that:

(P → Z) → Q ⊆ W .

Let a ∈ (P → Z) → Q, then there exist p ∈ P, q ∈ Q and z ∈ Z such that:

p ≤ z → (q → a) .

Thus, by hypothesis, p ≤ q → (z → a) ∈ P .SinceP → Q ⊆ D, z → a ∈ D. As D → Z ⊆ W, we have that a ∈ W . Thus, by Theorem 2.9, there exists K ∈ X such that P → Z ⊆ K and K → Q ⊆ W , i.e., (P, Z, K),(K, Q, W) ∈ T . ⇐) Let us suppose that there exist a, b, c ∈ A such that a → (b → c) b → (a → c). Then, a → (b → c) ∈ P and b → (a → c) ∈/ P , for some P ∈ X.ByTheorem2.10 there exist Q, D ∈ X such that P → Q ⊆ D, b ∈ Q and a → c/∈ D.Sincea → c/∈ D, there exist Z, W ∈ X such that D → Z ⊆ W , a ∈ Z,andc/∈ W. Then we have that (P, Q, D) , (D, Z, W) ∈ T , by hypothesis, there exists K ∈ X such that P → Z ⊆ K and K → Q ⊆ W .Sincea → (b → c) ∈ P , we deduce that c ∈ W , which is a contradiction. 3. ⇒)LetP, Q, D, Z, W ∈ X such that P → Q ⊆ D and D → Z ⊆ W . Let us prove that: Q → (P → Z) ⊆ W . If a ∈ Q → (P → Z), there exists x ∈ P → Z such that x → a ∈ Q and there exists z ∈ Z such that z → x ∈ P . By hypothesis, z → x ≤ (x → a) → (z → a) ∈ P .Since P → Q ⊆ D,(z → a) ∈ P. As D → Z ⊆ W , a ∈ W . Therefore, by Theorem 2.9, there exists K ∈ X such that P → Z ⊆ K and Q → K ⊆ W . The proof of the only if part is easy and left to the reader. 6. ⇒)LetP, Q, D, W, Z ∈ X such that P → Q ⊆ D and P → W ⊆ Z. Consider the ﬁlter P → F (Q ∪ W ) and the ideal I = Dc ∩ Zc. We will prove that:

P → F (Q ∪ W ) ∩ I = ∅. L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 609

Suppose the contrary. Then there exist q ∈ Q, w ∈ W and x ∈ I such that (q ∧ w) → x ∈ P . By hypothesis, (q → x) ∨ (w → x) ∈ P. Since P is prime, q → x ∈ P or w → x ∈ P . Thus, x ∈ D or x ∈ Z, which is a contradiction. By Theorem 2.9 and Prime Filter Theorem there exist H, K ∈ X such that P → H ⊆ K, Q ∪ W ⊆ H and K ⊆ D or K ⊆ Z. The proof of the only if part is easy and it is left to the reader. 7. ⇒)LetP, Q, D ∈ X such that P → Q ⊆ D. Consider p ∈ P . By hypothesis p ≤ 1 → (p ∧ 1) = 1 → p,thenp ∈ P → Q. Thus, P ⊆ D. Now consider q ∈ Q.Byhypothesis1≤ q → (1 ∧ q)=q → q ∈ P ,thenq ∈ D.Thus, Q ⊆ D. ⇐) Let us suppose that for some a, b ∈ A, a b → (a ∧ b). Then there exists P ∈ X such that a ∈ P, b → (a ∧ b) ∈/ P. Since b → (a ∧ b) ∈/ P, there exist Q, D ∈ X such that P → Q ⊆ D, b ∈ Q and a ∧ b/∈ D.SinceP ⊆ D and Q ⊆ D, a, b ∈ D.Thusa ∧ b ∈ D, which is a contradiction. 12. ⇒)LetP, Q, D, Z, W ∈ X such that P → Q ⊆ D and P → Z ⊆ W .Letus consider the ﬁlter P → (Q ∩ Z) and the ideal I = I (Dc ∪ W c). We will prove that:

P → (Q ∩ Z) ∩ I = ∅.

Suppose the contrary, then there exist q ∈ Q ∩ Z, d∈ / D and w/∈ W such that

q → (d ∨ w) ∈ P .

So (q → d) ∨ (q → w) ∈ P . Since P is prime, q → d ∈ P or q → w ∈ P .Ifq → d ∈ P and taking into account that P → Q ⊆ D,wehavethatd ∈ D, which is a contradiction. If q → w ∈ P we arrive at a similar contradiction. Thus, P → (Q ∩ Z) ∩ I = ∅. Then there exists K ∈ X such that P → (Q ∩ Z) ⊆ K and K ∩ (Dc ∪ W c)= c K ∩ (D ∩ W ) = ∅, i.e., K ⊆ D ∩ W .SinceQ ∩ Z is a ﬁlter, by Theorem 2.9, there exists H ∈ X such that Q ∩ Z ⊆ H and P → H ⊆ K. ⇐) Suppose that there exist a, b, c ∈ A such that a → (b ∨ c) (a → b) ∨ (a → c), then there exists P ∈ X such that a → (b ∨ c) ∈ P and (a → b) ∨ (a → c) ∈/ P. Thus there exist Q, D, Z, W ∈ X such that P → Q ⊆ D, P → Z ⊆ W , a ∈ Q ∩ Z, b/∈ D and c/∈ W . By the hypothesis, there exist H, K ∈ X such that Q ∩ Z ⊆ H, K ⊆ D ∩ W and P → H ⊆ K.Sincea ∈ H and a → (b ∨ c), b ∨ c ∈ K ⊆ D ∩ W , which is a contradiction. 13. ⇒)LetP, Q, D ∈ X such that P → Q ⊆ D. Let us prove that:

P → (P → Q) ⊆ D.

Let x ∈ P → (P → Q) , then there exists y ∈ P → Q such that y → x ∈ P and there exists q ∈ Q such that q → y ∈ P. Thus (q → y) ∧ (y → x) ≤ q → x ∈ P which implies that x ∈ P → Q ⊆ D. 610 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623

By Theorem 2.9, there exists K ∈ X such that P → Q ⊆ K and P → K ⊆ D. ⇐) Suppose that there exist a, b, c ∈ A such that (a → b) ∧ (b → c) a → c,then there exists P ∈ X such that (a → b) ∧ (b → c) ∈ P and a → c/∈ P .ByTheorem2.10 there exist Q, D ∈ X such that a ∈ Q, c/∈ D and P → Q ⊆ D. By hypothesis there exists D1 ∈ X such that P → Q ⊆ D1 and P → D1 ⊆ D. Since a ∈ Q, b ∈ D1.Then c ∈ D, because b → c ∈ P, which is a contradiction. 14. ⇒)LetP, Q, D, W, Z ∈ X such that P → Q ⊆ D and P → W ⊆ Z. Suppose that there exist a ∈ Q\Z and b ∈ W \D.Since1=(a → b) ∨ (b → a) ∈ P ,ifa → b ∈ P, then b ∈ D, and if b → a ∈ P, then a ∈ Z. In any case we arrive at a contradiction. The proof of the only if part is easy and it is left to the reader.

Using the previous Theorem we can prove the following results.

Corollary 3.2. Let A be a DLI-algebra satisfying the equation:

a (a → b) → b.

Then the following propositions hold: (1) A a → (b → c) b → (a → c) ⇔ A a → b (b → c) → (a → c). (2) A a b → b ⇔ A a b → a ⇔ A a b → (a ∧ b). (3) A a → (b → c) (a ∧ b) → c ⇔ A (a → b) ∧ (b → c) a → c.

Corollary 3.3. Let A be a DLI-algebra satisfying the equation:

a b → b. (a)

Consider the following propositions: (1) A a → (b ∨ c) (a → b) ∨ (a → c). (2) A (a ∧ b) → c (a → c) ∨ (b → c). (3) A 1 ≈ (a → b) ∨ (b → a). Then 1 implies 3 and 2 implies 3.

Proof. First we will prove that 1 implies 3 . Let P, Q1,Q2,D1,D2 ∈X(A) such that P → Q1 ⊆ D1 and P → Q2 ⊆ D2. By hypothesis 1 and item 12 of Theorem 3.1,there exist Q, D ∈X(A) such that Q1 ⊆ Q or Q2 ⊆ Q and D ⊆ D1 ∩D2 and (P, Q, D) ∈ T .By condition (a)anditem8 of Theorem 3.1,wehavethatQ ⊆ D.ThusQ1 ⊆ Q ⊆ D ⊆ D2 or Q2 ⊆ Q ⊆ D ⊆ D1. Finally, by item 14 of Theorem 3.1,(a → b) ∨ (b → a) ≈ 1 is valid in A.

Now we will prove that 2 implies 3.LetP, Q1,Q2,D1,D2 ∈X(A) such that P → Q1 ⊆ D1 and P → Q2 ⊆ D2. By hypothesis 2 and item 6 of Theorem 3.1,thereexist Q, D ∈X(A) such that Q1 ∪ Q2 ⊆ Q and D ⊆ D1 or D ⊆ D2 and P → Q ⊆ D.If D ⊆ D1. By condition (a)anditem8 of Theorem 3.1,wehavethatQ2 ⊆ Q ⊆ D ⊆ D1. If D ⊆ D2 by the same argument we have that Q1 ⊆ D2. Thus by item 14 of Theorem 3.1,(a → b) ∨ (b → a) ≈ 1 is valid in A. L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 611

Theorem 3.4. Let A be a DLI-algebra. Suppose that the following equations are valid in A : (1) a b → b. (2) 1 → a a. (3) a → (b ∨ c) (a → b) ∨ (a → c). Then, A [(a → b) → b] ∧ [(b → a) → a] a ∨ b.

Proof. Suppose that there exist a, b ∈ A such that [(a → b) → b]∧[(b → a) → a] a∨b. Then there exists P ∈X(A), such that [(a → b) → b] ∧ [(b → a) → a] ∈ P , a/∈ P and b/∈ P . By condition 2,1→ a/∈ P and 1 → b/∈ P .ByTheorem2.10,thereexist Q1,Q2,D1,D2 ∈X(A) such that P → Q1 ⊆ D1, P → Q2 ⊆ D2, a/∈ D1 and b/∈ D2. By condition 3 and by item 12 of Theorem 3.1,thereexistQ, D ∈X(A) such that P → Q ⊆ D, Q1 ∩ Q2 ⊆ Q, D ⊆ D1 and D ⊆ D2. By conditions 1, 3 and Theorem 3.3, 1=(a → b) ∨ (b → a) ∈ Q.Ifa → b ∈ Q,since(a → b) → b ∈ P and P → Q ⊆ D, we have that b ∈ D, which is a contradiction. If b → a ∈ Q we arrive at a similar contradiction. Thus, [(a → b) → b] ∧ [(b → a) → a] ≤ a ∨ b.

The following theorem gives a characterization of algebraic conditions in DLF-algebras. These results, and more general ones, has been obtained by Urquhart in [18] for relevant algebras, but here we will give a proof just using the results of [2].

Theorem 3.5. Let A be a DLF-algebra and let X = X (A) be the set of prime ﬁlters of A and R = RA the ternary relation given in Deﬁnition 2.6.Then: (1) A (a ◦ b) ◦ c a ◦ (b ◦ c) ⇔∀P, Q, D, Z, W ∈ X, ((P, Q, D) ∈ R and (D, Z, W) ∈ R) implies ∃K ∈ X, (P, K, W) ∈ R and (Q, Z, K) ∈ R. (2) A a ◦ (b ◦ c) (a ◦ b) ◦ c ⇔∀P, Q, D, Z, W ∈ X. ((P, Q, D) ∈ R and (Z, D, W) ∈ R) implies ∃K ∈ X, (Z, P, K) ∈ R and (K, Q, W) ∈ R. (3) A a ◦ b b ◦ a ⇔∀P, Q, D ∈ X, (P, Q, D) ∈ R implies (Q, P, D) ∈ R. (4) A 1 ◦ a a ⇔∀P, Q, D ∈ X, (P, Q, D) ∈ R implies Q ⊆ D. (5) A a ◦ 1 a ⇔∀P, Q, D ∈ X, (P, Q, D) ∈ R implies P ⊆ D. (6) A a a ◦ 1 ⇔∀P ∈ X, ∃Q ∈ X, (P, Q, P) ∈ R. (7) A a 1 ◦ a ⇔∀P ∈ X, ∃Q ∈ X, (Q, P, P ) ∈ R.

Proof. We will prove 1, 4 and 6, the other are similar and left to the reader. 1. ⇒)LetP, Q, D, Z, W ∈ X such that (P, Q, D) , (D, Z, W) ∈ R, i.e., P ◦ Q ⊆ D and D ◦ Z ⊆ W . We will prove that:

P ◦ (Q ◦ Z) ⊆ W .

Let y ∈ P ◦ (Q ◦ Z), then there exist p ∈ P, q ∈ Q, z ∈ Z such that p ◦ (q ◦ z) ≤ y. By hypothesis, (p ◦ q) ◦ z ≤ y.Sincep ◦ q ∈ D, (p ◦ q) ◦ z ∈ D ◦ Z ⊆ W . By Theorem 2.9, there exists K ∈ X such that Q ◦ Z ⊆ K and P ◦ K ⊆ W, i.e., .there exists K ∈ X such that (Q, Z, K),(P, K, W) ∈ R. 612 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623

⇐) Suppose that there exist a, b, c ∈ A such that (a ◦ b) ◦ c a ◦ (b ◦ c), then there exists W ∈ X such that (a ◦ b) ◦ c ∈ W and a ◦ (b ◦ c) ∈/ W .ByTheorem2.10,thereexist D, Z ∈ X such that a ◦ b ∈ D, c ∈ Z and D ◦ Z ⊆ W and there exist P, Q ∈ X such that a ∈ P , b ∈ Q and P ◦ Q ⊆ D. By hypothesis, we have that there exists K ∈ X such that Q ◦ Z ⊆ K and P ◦ K ⊆ W .Thusb ◦ c ∈ K and a ◦ (b ◦ c) ∈ W, which is a contradiction. 4. ⇒)LetP, Q, D ∈ X such that P ◦ Q ⊆ D. By hypothesis for every q ∈ Q, 1 ◦ q ≤ q ∈ P ◦ Q ⊆ D. ⇐) Let us suppose that there exists a ∈ A such that 1 ◦ a a, then there exists D ∈ X such that 1 ◦ a ∈ D and a/∈ D. By Theorem 2.10,thereexistP, Q ∈ X such that a ∈ Q and P ◦ Q ⊆ D.Thena ∈ Q ⊆ D, which is a contradiction. 6. ⇒)LetP ∈ X. By hypothesis, for every p ∈ P, p ≤ p ◦ 1. Then P ◦{1}⊆P. By Theorem 2.9 there exists Q ∈ X such that P ◦ Q ⊆ P . ⇐) Let suppose that there exists a ∈ A such that a a ◦ 1. There exists P ∈ X such that a ∈ P and a ◦ 1 ∈/ P .ThusP ◦{1} P. Therefore P ◦ Q P for every Q ∈ X, which is a contradiction with the hypothesis.

4 Duality for subvarieties of Commutative Residuated Lattices

Using the results of the previous sections we are going to develop topological dualities for some known algebraic subcategories of ICRL. We will ﬁrst give a duality for ICRL, and subsequently we will extend this duality for MT L, IMT L and MV.

4.1 Integral Commutative Residuated Lattices

Theorem 4.1 (Duality for ICR-lattices). The category ICRL is dually equivalent to the full subcategory of FFI whose spaces satisfy the following conditions: (1) T = R. (2) ∀x, y, z ∈ X, (x, y, z) ∈ T implies (y, x, z) ∈ T . (3) ∀x, y, z, v, w ∈ X, ∃s ∈ X, ((x, y, z) ∈ T and (z,v,w)) ∈ T implies ((x, v, s) ∈ T and (s, y, w) ∈ T ). (4) ∀x ∈ X, ∃y ∈ X, (x, y, x) ∈ T . (5) ∀x, y, z ∈ X, (x, y, z) ∈ T implies y ≤ z.

Proof. Let us consider A a DLFI-algebra. By Theorem 3.5 we have that A satisfies (A), (I)and(C) if and only if its dual DLFI-space X (A) satisfies the conditions from 2 to 5. We are going to prove that A satisfies (R) if and only if TA = RA. L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 613

Suppose ﬁrst that A satisﬁes (R). Consider P, Q ∈X(A), then:

P → Q = {x ∈ A :thereexistp ∈ P and q ∈ Q such that p ≤ q → x}

= {x ∈ A :thereexistp ∈ P and q ∈ Q such that p ◦ q ≤ x}

= P ◦ Q.

Thus, TA = RA. For the converse suppose that TA = RA and let a, b, c ∈ A. By Theorem 2.10 we have that:

a ◦ b ≤ c ⇔ for every P, Q ∈X(A) such that a ∈ P and b ∈ Q, c ∈ P ◦ Q

⇔ for every P, Q ∈X(A) such that a ∈ P and b ∈ Q, c ∈ P → Q

⇔ a ≤ b → c.

Thus, A satisﬁes (R).

The DLFI-spaces satisﬁyng the conditions of Theorem 4.1 will be called ICR-spaces. ∗ The full subcategory of FFI whose objects are ICR-spaces will be noted by ICRL .

4.2 MTL-algebras and IMTL-algebras

Let us recall that the variety MT L of Monoidal T-norm based Logic algebras, or MTL- algebras, introduced in [5], can be deﬁned as the subvariety of integral commutative residuated whose algebras satisfy the prelinearity equation:

MT L = ICRL + {(a → b) ∨ (b → a) ≈ 1} .

Theorem 4.2 (Duality for MTL-algebras). The category MT L is dually equivalent to the full subcategory of ICRL∗ whose spaces satisfy:

∀x, y, z, v, w ∈ X , ((x, y, z) ∈ T and (x, v, w) ∈ T ) implies (y ≤ w or v ≤ z) .

Proof. Immediate from Theorem 4.1,anditem14 of Theorem 3.1.

Now, we shall study the representation for the variety IMT L of IMTL-algebras. The variety IMT L is deﬁned by:

IMT L = MT L + {(a → 0) → 0 ≈ a} .

First we will need some previous results. By item 2 of Theorem 4.1,anditem1 of Theorem 3.1,everyICR-lattice satisﬁes the identity a (a → b) → b, and in particular a (a → 0) → 0. By items 4 and 5 of 614 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623

Theorem 4.1,andbyitems10 and 15 of Theorem 3.1,wehavethateveryICR-lattice satisﬁes the identity: a ≈ 1 → a. (for an algebraic proof of these properties see [9]).

Lemma 4.3 ([6, Proposition 3]). Each MTL-algebra is a subdirect product of linearly ordered MTL-algebras.

Corollary 4.4. The following equation:

(a ∧ b) → c ≈ (a → c) ∨ (b → c) is valid in every MTL-algebra A.

Proof. Since the equation (a ∧ b) → c ≈ (a → c) ∨ (b → c) is valid in every lineraly ordered MTL-algebra, by the previous Lemma we deduce that it is valid in each MTL- algebra.

Remark 4.5. A proof of the previous Corollary can be deduced from the results of Ward and Dilworth in [19], or from the results of J.B. Hart, L. Rafter and C. Tsinakis in [8].

Notation. If T is a ternary relation over a set X, for every x, y ∈ X we are going to note T (x, y)={z ∈ X :(x, y, z) ∈ T } .

Lemma 4.6. Let A be a DLI-algebra and let X = X (A) and T = TA.Then

A 1 → 0 ≈ 0 ⇔∀P ∈ X, ∃Q ∈ X, T (P, Q) = ∅.

Proof. ⇒)LetP ∈ X.Byhypothesis0∈/ P →{1}, then there exist Q, D ∈ X such that P → Q ⊆ D = A, i.e., T (P, Q) = ∅. ⇐) Let suppose that 1 → 0 =0 , then there exist P ∈ X such that 1 → 0 ∈ P .Thus for every Q ∈ X, 0 ∈ P → Q which is a contradiction.

Theorem 4.7. Let A be a DLI-algebra, X = X (A) and T = TA. Suppose that A satisﬁes the following identities: (1) 1 → 0 ≈ 0. (2) (a ∧ b) → c ≈ (a → c) ∨ (b → c). Then the following propositions are equivalent: a. A (a → 0) → 0 a. b. ∀P, Q ∈ X,If(∀D ∈ X, T (P, Q) = ∅ implies T (Q, D) = ∅),thenQ ⊆ P . L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 615

Proof. a ⇒ b). Let P, D ∈ X such that for every Q ∈ X, T (P, Q) = ∅ implies T (Q, D) = ∅, i.e., if P → Q = A, then Q → D = A. Let a ∈ D. We will prove that:

P → [a → 0) = A.

If we suppose that P → [a → 0) = A, there exists Q ∈ X such that a → 0 ∈ Q and P → Q = A.ThusQ → D = A. But a ∈ D and a → 0 ∈ Q, implies that 0 ∈ Q → D, which is a contradiction. Then 0 ∈ A = P → [a → 0), i.e., there exists p ∈ P such that p ≤ (a → 0) → 0 ≤ a.Thusa ∈ P . b ⇒ a). Let a ∈ A.Ifa =0, then (0 → 0) → 0=1→ 0=0≤ 0=a and the result follows. Suppose that there exists a = 0 such that (a → 0) → 0 a. Then there exists P ∈ X such that (a → 0) → 0 ∈ P and a/∈ P . Let us consider the set:

NP = {Q ∈ X : P → Q = A} .

By condition 1 and Lemma 4.6,wehavethatNP = ∅. Now let us consider the set

F = {H ∈ Fi(A):a ∈ H and for every Q ∈NP ,Q→ H = A} , ordered by inclusion. Since (a → 0) → 0 ∈ P ,ifQ ∈Np, then a → 0 ∈/ Q.Thus Q → [a) = A. Consequently [a) ∈F. Let us consider a chain {Hi : i ∈ I} , in F with I = ∅.LetG = {Hi : i ∈ I} . We prove that for each Q ∈NP ,

Q → G = A.

If we suppose the contrary there exist Q ∈NP , q ∈ Q,andh ∈ G such that q ≤ h → 0.

As h ∈ G, there exists i0 ∈ I such that hi0 ∈ Hi0 . So, Q → Hi0 = A, which contradicts the fact that Hi0 ∈F. Thus, G ∈F. By Zorn’s Lemma there exists an element D ∈Fwhich is maximal. We will prove that D ∈ X. Let suppose that there exist b, c ∈ A such that b ∨ c ∈ D but b, c∈ / D Let us consider the ﬁlters Db = F (D ∪{b})andDc = F (D ∪{c}). Since Db,Dc ∈F/ ,thereexistQb,Qc ∈NP such that

Qb → Db = A and Qc → Dc = A.

Since P → Qb = A and P → Qc = A, we have by condition 2 and item 6 of Theorem 3.1, that there exist Q ∈NP such that Qb ∪ Qc ⊆ Q.ThenQ → Db = A = Q → Dc. So there exist q ∈ Q and pb,pc ∈ D, such that q ≤ (pb ∧ b) → 0andq ≤ (pc ∧ c) → 0. Considering p = pb ∧ pc ∈ D, by condition 2 we have:

q ≤ ((p ∧ b) → 0) ∧ ((p ∧ c) → 0) ≤ ((p ∧ b) ∨ (p ∧ c)) → 0 ≤ (p ∧ (b ∨ c)) → 0.

Thus, Q → D = A, which is a contradiction with the fact that D ∈F.ThenD ∈ X and for every Q ∈NP ,Q→ D = A. By hypothesis, a ∈ D ⊆ P, which is a contradiction. 616 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623

Therefore, (a → 0) → 0 ≤ a for every a ∈ A.

Using Theorems 4.7 and 4.2 we can conclude that:

Theorem 4.8 (Duality for IMTL-algebras). The category IMT L is dually equivalent to the full subcategory of ICRL∗ satisfying the following conditions: (1) ∀x, y, z, v, w ∈ X, ((x, y, z) ∈ T and (x, v, w) ∈ T ) implies (y ≤ w or v ≤ z). (2) ∀x, y ∈ X,If(∀z ∈ X, T (x, z) = ∅ implies T (z,y) = ∅) then y ≤ x.

4.3 MV-algebras

Let us recall that MV-algebras are deﬁnable as IMTL-algebras satisfying the identity:

(a → b) → b ≈ (b → a) → a.§

Note that if an MTL-algebra A satisﬁes the identity (a → b) → b ≈ (b → a) → a, then it is an IMTL-algebra. Thus we have that the variety MV-algebras can be deﬁned as:

MV = MT L + {(a → b) → b ≈ (b → a) → a} .

We shall use this characterization to give a duality for MV-algebras.

Theorem 4.9. Let A be a DLI-algebra. If A satisﬁes: (1) (a ∧ b) → c ≈ (a → c) ∨ (b → c) . (2) 1 → a a. Then the next propositions are equivalent: a. A (a → b) → b a ∨ b. b. For every P ∈X(A) , for every b/∈ P, if K ∈X(A) is such that (for every Q ∈X(A) (if b/∈ P → Q implies that b/∈ Q → K)) then K ⊆ P .

Proof. a ⇒ b) Let us suppose that there are P, K ∈X(A)andb/∈ P such that for every Q ∈X(A)(ifb/∈ P → Q then b/∈ Q → K). Let a ∈ K,If(a → b) → b/∈ P ,then there exists Q ∈X(A) such that a → b ∈ Q and b/∈ P → Q.Thusb/∈ Q → K,which is a contradiction with the fact the a ∈ K.Then(a → b) → b ≤ a ∨ b ∈ P .Sinceb/∈ P and P is a prime ﬁlter, we have that a ∈ P .ThusK ⊆ P . b ⇐ a) Let us suppose that there exist a, b ∈ A such that (a → b) → b a ∨ b.Then there exists P ∈X(A) such that (a → b) → b ∈ P and a ∨ b/∈ P . Consider

NP,b = {Q ∈X(A):b/∈ P → Q} , by condition 2 and item 15 of Theorem 3.1, we have that there exists Q ∈X(A)such that P → Q ⊆ P. Therefore NP,b = ∅. Now consider the set

F = {F ∈ Fi(A):a ∈ F and b/∈ Q → F , for every Q ∈NP,b} .

§ see [6] L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 617

Since (a → b) → b ∈ P and b/∈ P → Q, if Q ∈NP,b then a → b/∈ Q.Thuswehave that b/∈ Q → [a) ,i.e. [a) ∈F. By condition 1 and using the same argument of the proof of Theorem 4.7 we have that there exists K ∈X(A) such that a ∈ K and for every Q ∈NP,b, b/∈ Q → K.Thena ∈ K ⊆ P, which is a contradiction. Therefore. (a → b) → b ≤ a ∨ b, for every a, b ∈ A.

It’s known (see [6, pg 275]) that every MTL-algebra A satisﬁes:

a ∨ b ≈ ((a → b) → b) ∧ ((b → a) → a) , it also can be deduced from Theorem 3.4 and the fact that A a (a → b) → b. From that, we have that in every MTL-algebra A, the following statements are equivalent: (1) A (a → b) → b a ∨ b. (2) A (a → b) → b ≈ (b → a) → a. By this observation and the previous Theorem we obtain the following result:

Theorem 4.10 (Duality for MV-algebras). The category of MV-algebras is dually equivalent to the full subcategory of ICRL∗ whose spaces satisfy: (1) For every x, y, z, v, w ∈ X if (x, y, z) , (x, v, w) ∈ T ,theny ≤ w or v ≤ z. (2) For every x ∈ X and every U ∈D(X) such that x/∈ U, if z ∈ X is satisﬁes that (for every y ∈ XT(x, y) ∩ U c = ∅ implies T (y, z) ∩ U c = ∅), then z ≤ x.

5 Relationship with others Representations

There are a large number of representations and dualities for distributive lattices developed by diﬀerent motivations. In this section we will describe how some of these representations or dualities are related to the one studied in this paper. All the dualities considered in this section have a similarity, they are based on Priestley duality for distributive lattices (bounded or unbounded), i.e., they use Priestley spaces and some additional structure to represent the implication (and other operators in the case of Urquhart duality for relevant algebras and Sofronie-Stokkermans duality for lattices with operators).

5.1 Relevant Algebras

The duality developed by the second author in [2] is an extension of the duality given by A. Urquhart in [18]. Urquhart gives a duality for relevant algebras which are DLFI- algebras with an antihomomorphism (Ockham negation) and a distinguished element. Since in relevant algebras the implication is the left residual of the fusion, Urquhart use the same ternary relation to represent the implication (see Theorem 4.1). This ternary relation is exactly the one used by the second author to represent the fusion. In the same paper Urquhart give a general result for translating the validity of some inequalities in 618 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 the algebra to the validity of some ﬁrst order formula over its dual space. This result is developed for inequalities between some special terms constructed just using the fusion connective. We think that it is possible to give more general results of this kind for terms with includes the implication connective, but this is out of the purpose of this paper.

5.2 Lattices with Operators

In [15], Sofronie-Stokkermans develop a duality for a large classes of lattices with diﬀerent kinds of operators, by means of Priestley spaces with relations and functions. Sofronie- Stokkermans extend this duality for more general classes of operators in [17]. In her notation it is possible to describe a DLFI-algebra as a distributive lattice with two binary operations one of type 1, 1 → 1 (fusion) and other of type 1, −1 →−1 (implication). Sofronie-Stokkermans’s representation for the fusion (type 1, 1 → 1) is exactly the same ternary relation that we use. For the implication Sofronie-Stokkermans use a ternary relation R→. We give here an equivalent deﬁniton of R→:

(P, Q, D) ∈ R→ if and only if p → q/∈ D for every p ∈ P and q/∈ Q, It is easy to see that:

(P, Q, D) ∈ R→ if and only if (D, P, Q) ∈ T .

Then R→ and T are interdefinables. In [17], Sofronie-Stokkermans gives a way to check the validity of certain formulas in a class of lattice with operators. This technique requires that the class of lattices and its related frames satisfy certain properties. The first condition is that the class of frames is defined by first order conditions and the second is that the canonicity of the class of lattice with operators considered. In the case of DLI, DLF and DLFI-algebras we describe a large list of equations which are characterized by first order conditions in the related spaces (Theorems 3.1 and 3.5). So for the subvarieties determined by these equations we have that the dual spaces are described by first order conditions. This allow us to think that the results of Sofronie-Stokkermans hold in these varieties. We only need to prove the canonicity of these subvarieties of DLI, DLF and DLFI-algebras. This result was already proved for some of these varieties by the authors and it will appear in a future work, where we will study some logics related to DLI, DLF and DLFI-algebras.

5.3 Implicative Lattices

In [10] N.G. Martinez gives a duality between Implicative lattices and Bounded Priestley spaces with a continuous binary function, and improve its results in [11]. We recall the deﬁnition of implicative lattices. L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 619

Deﬁnition 5.1. An implicative lattice A, ∧, ∨, → is an algebra of type (2, 2, 2) such that A, ∧, ∨ is a distributive lattice and satisﬁes the following equations: (1) a → (b ∧ c) ≈ (a → b) ∧ (a → c). (2) (a ∨ b) → c ≈ (a → c) ∧ (b → c). (3) a → (b ∨ c) ≈ (a → b) ∨ (b → c). (4) (a ∧ b) → c ≈ (a → c) ∨ (b → c).

We will say that A, ∧, ∨, →, 0, 1 is a Bounded implicative lattice if the reduct A, ∧, ∨, → is an implicative lattice and A, ∧, ∨, 0, 1 is a bounded lattice. For every distributive lattice A we will note S (A)=X (A)∪{∅,A} . In [11] Martinez deﬁne for every implicative lattice the following function Φ : S (A) ×S(A) →S(A): Φ(P, Q)= {y : x → y ∈ Q} . x∈P

It follows straightforwardly that if P, Q ∈X(A) , Φ(P, Q)=Q → P (see page 605). Then the ternary relation TA (Deﬁnition 2.6) can be recovered using Φ by

(P, Q, D) ∈ TA ifandonlyifΦ(Q, P ) ⊆ D.

Items 6 and 12 of Theorem 3.1 give conditions over the ternary relation which determine when a DLI-algebra is a bounded implicative lattice. It is interesting to note that, in the other way, there are simple properties of Φ which determine when a bounded implicative lattice is a DLI-algebra. It follows from Deﬁnitions 2.1 and 5.1 that a bounded implicative lattice is a DLI- algebra if and only if the following conditions hold: a → 1 ≈ 1and0→ a ≈ 1. Given a bounded implicative lattice A = A, ∧, ∨, →, 0, 1 , it is easy to prove that:

A a → 1=1 ⇔ Φ(P, Q) = ∅ for every P, Q ∈X(A) . (B) A 0 → a =1 ⇔ Φ(A, Q)=A for every Q ∈X(A) .

If A is a DLI-algebra which satisﬁes items 3 and 4 of Deﬁnition 5.1,thenitiseasy to see that P → Q ∈X(A) for every P, Q ∈X(A) . Thus TA (P, Q) is a principal increasing set. Using the correspondences (B) and the previous observation we have that the function Φ can be described using the ternary relation TA by: ⎧ ⎪ ∅ ⎪ A if P = A and Q = , ⎪ ⎨⎪ A if Q = A and P = ∅, Φ(P, Q)= ⎪ ⎪ ∅ if P = ∅ or Q = ∅, ⎪ ⎩⎪ D if P, Q ∈X(A)andTA (P, Q)=[D).

In [12] N.G. Martinez and H.A. Priestley give a duality between implicative lattices and bounded Priestley spaces with a special function. 620 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623

If A is an implicative lattice they deﬁne μ : D (S (A)) ×S(A) −→ S (A)by:

μ (β (a) ,P)=Pa = {x ∈ A : x → a/∈ P } .

As in (B) we can give some properties of μ which are equivalent to the fact that a bounded implicative lattice A is a DLI-algebra.

A a → 1=1⇔ μ (β (1) ,P)=∅ for every P ∈S(A) \ {∅} , A 0 → a =1⇔ μ (β (0) ,P) = A for every P ∈S(A) \{∅}.

It is easy to see that Pa is the maximal element of S (A) such that a/∈ P → Pa.Thus given a bounded implicative lattice A which is a DLI-algebra we have that if P ∈X(A) , then:

{Q ∈X(A):T (P, Q) ∩ βc (a) = ∅} = {Q : a/∈ P → Q} = {Q : for every x ∈ Q, x → a/∈ P }

=(Pa].

It follows that μ can be deﬁned by ⎧ ⎪ ⎪ ∅ if P = A, ⎨⎪ μ (β (a) ,P)=⎪ A if P = ∅, ⎪ ⎩⎪ c D if P ∈ X (A)and{Q ∈X(A):T (P, Q) ∩ β (a) = ∅} =(D].

Thus clearly the function μ can be obtained from the ternary relation TA. 3 In the other way, if we deﬁne Tµ ⊆ (X (A)) by (P, Q, D) ∈ Tµ if and only if D ∈ {βA (a):a ∈ A and Q Pa} then Tµ = TA.Sincea ∈ P → Q ifandonlyifthereexistx ∈ Q such that x → a ∈ P, if and only if Q Pa,wehavethatP→ Q ⊆ D if and only if a ∈ D (i.e. D ∈ βA (a)) for every a such that Q Pa,.i.e. D ∈ {βA (a):a ∈ A and Q Pa} We can conclude that in the case of bounded implicative lattices which are DLI- algebras the dualities given in [10, 12]andin[2] are interdeﬁnables.

5.4 WH-algebras

In [3] Celani and Jansana introduce the variety of Weakly Heyting algebras, or WH- algebras, and give a Priestley style duality between this algebras and Priestley spaces endowed with a binary relation.

Deﬁnition 5.2. A DLI-algebra A is a WH-algebra if the following conditions hold: (1) (a → b) ∧ (b → c) a → c, (2) a → a ≈ 1. L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623 621

Using items 8 and 13 of Theorem 3.1 we can conclude that a DLI-algebra A is a WH-algebra if and only if TA satisﬁes the following properties: (1) ∀P, Q, D ∈X(A), ∃Z ∈X(A), (P, Q, D) ∈ TA implies that (P, Q, Z) ∈ TA and (P, Z, D) ∈ TA. (2) ∀P, Q, D ∈X(A), (P, Q, D) ∈ TA implies that Q ⊆ D. Given WH-algebra A,in[3] Celani and Jansana deﬁne a binary relation over X (A) as follows

(P, Q) ∈ SA ⇔ for every a, b ∈ A, (a → b ∈ P and a ∈ Q,thenb ∈ Q). (1)

Let A be a WH-algebra. Let TA be the ternary relation deﬁned on X (A)asin

Deﬁnition 2.6, and let us deﬁne a binary relation STA as follows:

∈ ∈ (P, Q) STA if and only if (P, Q, Q) TA.

Then it is easy to see that SA = STA . So, the binary relation SA can be obtained from TA. Now, let us consider the binary relation SA deﬁned in (1). Let us deﬁne a ternary relation TSA as follows: ∈X A 3 ∃ ∈X A ⊆ ⊆ ∈ TSA = (P, Q, D) ( ) : F ( ), Q F D and (P, F) SA .

∈ → → We will see that TA = TSA .Let(P, Q, D) TA.LetF = P Q. Since, a a = 1 for all a ∈ A, Q ⊆ P → Q.ThenQ ⊆ F ⊆ D. We prove that (P, F) ∈ SA. Let a → b ∈ P and a ∈ F . Then there exists q ∈ Q such that q → a ∈ P .So,(q → a)∧(a → b) ≤ q → b ∈ P, i.e. b ∈ Q → P = F Thus, (P, Q, D) ∈ TS . ” A ∈ ∈ A ⊆ ⊆ Suppose that (P, Q, D) TSA , then there exists F X ( ) such that Q F D and (P, F) ∈ SA. Let a, b ∈ A such that a → b ∈ P and a ∈ Q.Thenb ∈ F, and since ∈ ∈ ∈ ∈ (P, F) SA, b F .So,b D. Therefore, (P, Q, D) TA. We conclude that TA = TSA . Since the variety of WH-algebras is a subvariety of DLI and the binary relation deﬁned in [3] can be obtained from the ternary relation used in this paper, we conclude that the duality given in [3] can be deduced from the duality given in [2] using the results of this paper.

6 Conclusions and future work

In this paper we gave a duality for many algebraic categories of algebras related to fuzzy logics. These dualities are an extension of the duality given by the second author in [2]. The method was to translate the equations which define the subvarieties in conditions over the dual spaces. In the case of ICRL, MT L and IMT L these conditions are of first order, but this was not the case for MV. Some open questions, also raised by one of the referees, are: Is there a generalized way to give the translations for some classes of equations? Is it possible to determine when an equation has a first order translation? 622 L.M. Cabrer, S.A. Celani / Central European Journal of Mathematics 4(4) 2006 600–623

The problem of canonicity of almost all the varieties deﬁnable by the equations given in this work is solved by the authors and will be showed in a future paper. It is interesting to note that M. Gehrke and H.A. Priestley in [7] had proved that the variety MV is not canonical. This variety is the only one, in this paper, for which we cannot obtain ﬁrst order conditions to determine its dual spaces.

Acknowledgment

We would like to thank the referees for their observations and suggestions which have contributed to improve this paper.

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