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 resid- uated 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 represen- tation 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 rele- vant 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.