Modules with fusion and implication based over distributive lattices: Representation and Duality
aa r X i v : . [ m a t h . L O ] J u l MODULES WITH FUSION AND IMPLICATION BASED OVERDISTRIBUTIVE LATTICES: REPRESENTATION AND DUALITY
ISMAEL CALOMINO* AND WILLIAM J. ZULUAGA BOTERO**
Abstract.
In this paper we study the class of modules with fusion and implication based overdistributive lattices , or
FIDL-modules , for short. We introduce the concepts of FIDL-subalgebraand FIDL-congruence as well as the notions of simple and subdirectly irreducible FIDL-modules.We give a bi-sorted Priestley-like duality for FIDL-modules and moreover, as an application ofsuch a duality, we provide a topological bi-spaced description of the FIDL-congruences. Thisresult will allows us to characterize the simple and subdirectly irreducible FIDL-modules. Introduction
Bounded distributive lattices with additional operators occur often as algebraic models of non-classical logics. This is the case of Boolean algebras which are the algebraic semantics of classicallogic, Heyting algebras which model intuitionistic logic, BL-algebras which correspond to algebraicsemantics of basic propositional logic ([10]), MTL-algebras which are the algebraic semantics ofthe basic fuzzy logic of left-continuous t-norms ([8, 2]), Modal algebras which model propositionalmodal logics ([5, 1]), to name a few. In all these cases, the binary operations ∨ and ∧ modellogical disjunction and conjunction and the additional operations are usually interpretations ofother logical connectives such as the modal necessity ( (cid:3) ) or modal possibility ( ♦ ), or various typesof implication. All these operations has as a common property: the preservation of some part ofthe lattice structure, for example, the necessity modal operator satisfies the conditions (cid:3) (cid:3) ( x ∧ y ) = (cid:3) ( x ) ∧ (cid:3) ( y ), or the possibility modal operator ♦ ♦ ( x ∨ y ) = ♦ ( x ) ∨ ♦ ( y ).In some sense, the aforementioned may suggest that these ideas can be treated as a moregeneral phenomenon which can be studied by employing tools of universal algebra. Some papers inwhich this approach is used are [9] and [15]. Nevertheless, in an independent way, a more concretetreatment of the preservation of the lattice structure by two additional connectives in a distributivelattice leads to the introduction of the class of distributive lattices with fusion and implication in[3], which encompasses all the algebraic structures mentioned before.The aim of this paper is to introduce the class of modules with fusion and implication basedover distributive lattices , for short, FIDL-modules . The FIDL-modules generalize both distributivelattices with fusion and implication and modal distributive lattices, giving a different approach tostudy these structures. A bi-sorted Priestley-like duality is developed for FIDL-modules, extendingthe dualities given in [3] for distributive lattices with fusion and implication and in [14] for algebrasof relevant logics. This duality enables us to describe the congruences of a FIDL-module and alsoto give a topological characterization of the simple and subdirectly irreducible FIDL-modules.The paper is organized as follows. In Section 2 we give some definitions and introduce thenotations which are needed for the rest of the paper. In Section 3 we introduce the class ofmodules with fusion and implication based over distributive lattices, or simply FIDL-modules.
Mathematics Subject Classification.
Primary 06D50, 06D05; Secondary 06D75.
Key words and phrases.
Distributive lattice, module, Priestley-like duality.This work was supported by the CONICET under Grant PIP 112-201501-00412.
Also the concept of FIDL-subalgebra is developed and studied. In Section 4 we study the notionof FIDL-homomorphism and we exhibit a representation theorem for FIDL-modules by means ofrelational structures. In Section 5 we use the representation theorem and together with a suitableextension of the Priestley duality, we obtain a duality for FIDL-modules as certain topologicalbi-spaces. Finally, in Section 6 we introduce the notion of congruence of FIDL-modules and asan application of the duality, we obtain a topological bi-spaced description for the simple andsubdirectly irreducible FIDL-modules. 2.
Preliminaries
Given a poset h X, ≤i , a subset U ⊆ X is said to be increasing ( decreasing ), if for every x, y ∈ X such that x ∈ U ( y ∈ U ) and x ≤ y , then y ∈ U ( x ∈ U ). The set of all increasing subsetsof X is denoted by P i ( X ). For each Y ⊆ X , the increasing (decreasing) set generated by Y is[ Y ) = { x ∈ X : ∃ y ∈ Y ( y ≤ x ) } (( Y ] = { x ∈ X : ∃ y ∈ Y ( x ≤ y ) } ). If Y = { y } , then we will write[ y ) and ( y ] instead of [ { y } ) and ( { y } ], respectively.Given a bounded distributive lattice A = h A, ∨ , ∧ , , i , a set F ⊆ A is called filter if 1 ∈ F , F is increasing, and if a, b ∈ F , then a ∧ b ∈ F . The filter generated by a subset X ⊆ A is the setFig A ( X ) = { x ∈ A : ∃ x , . . . , x n ∈ X such that x ∧ . . . ∧ x n ≤ x } . If X = { a } , then Fig A ( { a } ) = [ a ). Denote by Fi( A ) the set of all filters of A . A proper filter P is prime if for every a, b ∈ A , a ∨ b ∈ P implies a ∈ P or b ∈ P . We write X ( A ) the set of all primefilters of A . Similarly, a set I ⊆ A is called ideal if 0 ∈ I , I is decreasing, and if a, b ∈ I , then a ∨ b ∈ I . Then the ideal generated by a subset X ⊆ A is the setIdg A ( X ) = { x ∈ A : ∃ x , . . . , x n ∈ X such that x ≤ x ∨ . . . ∨ x n } . In particular, if X = { a } , then Idg A ( { a } ) = ( a ]. Denote by Id( A ) the set of all ideals of A .Let β A : A → P i ( X ( A )) be the map defined by β A ( a ) = { P ∈ X ( A ) : a ∈ P } . Then the family β A [ A ] = { β A ( a ) : a ∈ A } is closed under unions, intersections, and contains ∅ and A , i.e., it is abounded distributive lattice. Moreover, β A establishes an isomorphism between A and β A [ A ].A Priestley space is a triple h X, ≤ , τ i where h X, ≤i is a poset and h X, τ i is a compact totallyorder-disconnected topological space. A morphism between Priestley spaces is a continuous andmonotone function between them. If h X, ≤ , τ i is a Priestley space, then the family of all clopenincreasing sets is denoted by C ( X ), and it is well known that C ( X ) is a bounded distributive lattice.The Priestley space of a bounded distributive lattice A is the triple hX ( A ) , ⊆ A , τ A i , where τ A isthe topology generated by taking as a subbase the family { β A ( a ) : a ∈ A }∪{ β A ( a ) c : a ∈ A } , where β A ( a ) c = X ( A ) − β A ( a ). Therefore, A and C ( X ( A )) are isomorphic. If h X, ≤ , τ i is a Priestleyspace, then the map ǫ X : X → X ( C ( X )) defined by ǫ X ( x ) = { U ∈ C ( X ) : x ∈ U } , for every x ∈ X ,is a homeomorphism and an order-isomorphism. On the other hand, if Y is a closed set of X ( A ),then the relation θ ( Y ) = { ( a, b ) ∈ A × A : β A ( a ) ∩ Y = β A ( b ) ∩ Y } (2.1)is a congruence of A and the correspondence Y → θ ( Y ) establishes an anti-isomorphism betweenthe lattice of closed subsets of X ( A ) and the lattice of congruences of A .If h : A → B is a homomorphism between bounded distributive lattices A and B , then the map h ∗ : X ( B ) → X ( A ) defined by h ∗ ( P ) = h − ( P ), for each P ∈ X ( B ), is a continuous and monotonefunction. Conversely, if h X, ≤ X , τ X i and h Y, ≤ Y , τ Y i are Priestley spaces and f : X → Y is acontinuous and monotone function, then the map f ∗ : C ( Y ) → C ( X ) defined by f ∗ ( U ) = f − ( U ),for each C ( Y ), is a homomorphism between bounded distributive lattices. Furthermore, there is aduality between the algebraic category of bounded distributive lattices with homomorphisms andthe category of Priestley spaces with continuous and monotone functions ([13, 6, 7]). IDL-MODULES: REPRESENTATION AND DUALITY 3 FIDL-modules
In this section we present the class of modules with fusion and implication based over distributivelattices , or
FIDL-modules , for short. These structures can be considered as bi-sorted distributivelattices endowed with two operations which preserve some of the lattice structure. We introducethe notion of FIDL-subalgebra and we exhibit a characterization of those in terms of some relations.
Definition 1.
Let A , B be two bounded distributive lattices. A structure h A , B , f i is called aFDL-module, if f : A × B → A is a function such that for every x, y ∈ A and every b, c ∈ B thefollowing conditions hold:(F1) f ( x ∨ y, b ) = f ( x, b ) ∨ f ( y, b ),(F2) f ( x, b ∨ c ) = f ( x, b ) ∨ f ( x, c ),(F3) f (0 , b ) = 0,(F4) f ( x,
0) = 0.A structure h A , B , i i is called an IDL-module, if i : B × A → A is a function such that for every x, y ∈ A and every b, c ∈ B the following conditions hold:(I1) i ( b, x ∧ y ) = i ( b, x ) ∧ i ( b, y ),(I2) i ( b ∨ c, x ) = i ( b, x ) ∧ i ( c, x ),(I3) i ( b,
1) = 1.Moreover, a structure M = h A , B , f, i i is called a FIDL-module, if h A , B , f i is a FDL-module and h A , B , i i is an IDL-module. Remark 1.
Let M be a FIDL-module. Then the function f determines and it is determined bya unique family F B = { f b : A → A | b ∈ B } of unary operations on A such that for every x, y ∈ A and every b, c ∈ B the following conditions hold:(F1’) f b ( x ∨ y ) = f b ( x ) ∨ f b ( y ),(F2’) f b ∨ c ( x ) = f b ( x ) ∨ f c ( x ),(F3’) f b (0) = 0,(F4’) f ( x ) = 0.Analogously, the function i determines and it is determined by a unique family I B = { i b : A → A | b ∈ B } of unary operations on A such that for every x, y ∈ A and every b, c ∈ B the followingconditions hold:(I1’) i b ( x ∧ y ) = i b ( x ) ∧ i b ( y ),(I2’) i b ∨ c ( x ) = i b ( x ) ∧ i c ( x ),(I3’) i b (1) = 1.Hence the FIDL-module M is equivalent to the structure h A , F B , I B i . Therefore, along this paperwe will use the families F B and I B and its corresponding functions f and i indistinctly.The following are important examples of FIDL-modules. Example An algebra h A , ◦ , →i is a bounded distributive lattice with fusion and implication ([3, 2]), if A is a bounded distributive lattice and ◦ and → are binary operations defined on A such that for all x, y, z ∈ A the following conditions hold:(1) x ◦ ( y ∨ z ) = ( x ◦ y ) ∨ ( x ◦ z ),(2) ( x ∨ y ) ◦ z = ( x ◦ z ) ∨ ( y ◦ z ),(3) x ◦ ◦ x = 0,(4) x → x → y ) ∧ ( x → z ) = x → ( y ∧ z ),(6) ( x → z ) ∧ ( y → z ) = ( x ∨ y ) → z . ISMAEL CALOMINO AND WILLIAM J. ZULUAGA BOTERO
Notice that if M is a FIDL-module such that B = A and we consider the functions x ◦ f y = f ( x, y )and x → i y = i ( x, y ), then h A , ◦ , →i is a bounded distributive lattice with fusion and implication.Moreover, if M satisfies the condition f ( x, y ) ≤ z if and only if x ≤ i ( y, z ), then the structure h A , ◦ , →i is a residuated lattice ([11]). Example Recall that an algebra h A , (cid:3) , ♦ i is a modal distributive lattice , or (cid:3)♦ -lattice , if A is a bounded distributive lattice and (cid:3) and ♦ are unary operations defined on A such that forevery x, y ∈ A we have (cid:3) (cid:3) ( x ∧ y ) = (cid:3) ( x ) ∧ (cid:3) ( y ), ♦ ♦ ( x ∨ y ) = ♦ ( x ) ∨ ♦ ( y )([5, 1, 4]). If M is a FIDL-module and B = { , } , we can consider the functions ♦ f ( x ) = f ( x, (cid:3) i ( x ) = i (1 , x ) such that h A , ♦ , (cid:3) i is a (cid:3)♦ -lattice. Example Let h A , →i be a Heyting algebra, where A is its bounded lattice reduct. Let X be anon-empty set and let A X = h A X , ∨ , ∧ , , i be the bounded distributive lattice of functions from X to A with the operations defined pointwise. Then, by following the notation of Remark 1, if weconsider the families of functions F A = { f a : A X → A X | a ∈ A } and I A = { i a : A X → A X | a ∈ A } defined for every a ∈ A by f a ( g )( x ) = a ∧ g ( x ) and i a ( g )( x ) = a → g ( x ), respectively, it is the casethat h A X , F A , I A i is a FIDL-module.The following results are inspired by [3]. Proposition 3.1.
Let M be a FIDL-module. Then for every x, y ∈ A and every b, c ∈ B , if x ≤ y and b ≤ c , then f ( x, b ) ≤ f ( y, c ) and i ( c, x ) ≤ i ( b, y ) .Proof. Since y = y ∨ x and c = b ∨ c , then by (F1) and (F2) of Definition 1 f ( y, c ) = f ( y ∨ x, b ∨ c ) = f ( y, b ) ∨ f ( y, c ) ∨ f ( x, b ) ∨ f ( x, c ) ≥ f ( x, b ) , i.e., f ( x, b ) ≤ f ( y, c ). Analogously, as x = x ∧ y , by (I1) and (I2) of Definition 1 we have i ( c, x ) = i ( b ∨ c, x ∧ y ) = i ( b, x ) ∧ i ( b, y ) ∧ i ( c, x ) ∧ i ( c, y ) ≤ i ( b, y )and i ( c, x ) ≤ i ( b, y ). (cid:3) Let M be a FIDL-module. Let G ∈ Fi( A ) and H ∈ Fi( B ). We define the following subsets: f ( G, H ) = { x ∈ A : ∃ ( g, h ) ∈ G × H such that f ( g, h ) ≤ x } and i ( H, G ) = { x ∈ A : ∃ ( h, g ) ∈ H × G such that g ≤ i ( h, x ) } . Proposition 3.2.
Let M be a FIDL-module. If G ∈ Fi( A ) and H ∈ Fi( B ) , then f ( G, H ) , i ( H, G ) ∈ Fi( A ) .Proof. We prove that f ( G, H ) ∈ Fi( A ). It is clear that 1 ∈ f ( G, H ) and f ( G, H ) is increasing.If x, y ∈ f ( G, H ), then there exist ( g, h ) , (ˆ g, ˆ h ) ∈ G × H such that f ( g, h ) ≤ x and f (ˆ g, ˆ h ) ≤ y .Since G and H are filters, ¯ g = g ∧ ˆ g ∈ G and ¯ h = h ∧ ˆ h ∈ H . By Proposition 3.1, f (¯ g, ¯ h ) ≤ x and f (¯ g, ¯ h ) ≤ y . So, f (¯ g, ¯ h ) ≤ x ∧ y and x ∧ y ∈ f ( G, H ). Then f ( G, H ) ∈ Fi( A ). The proof for i ( H, G ) ∈ Fi( A ) is similar. (cid:3) Theorem 3.3.
Let M be a FIDL-module. Let G ∈ Fi( A ) , H ∈ Fi( B ) and P ∈ X ( A ) . Then: (1) If f ( G, H ) ⊆ P , then there exist Q ∈ X ( A ) and R ∈ X ( B ) such that G ⊆ Q , H ⊆ R and f ( Q, R ) ⊆ P . (2) If i ( H, G ) ⊆ P , then there exist R ∈ X ( B ) and Q ∈ X ( A ) such that H ⊆ R , G ⊆ Q and i ( R, Q ) ⊆ P . Also called in [12] distributive lattices with join and meet-homomorphisms . IDL-MODULES: REPRESENTATION AND DUALITY 5
Proof.
We prove only (1) because the proof of (2) is analogous. Let us consider the family J = { ( K, W ) ∈ Fi( A ) × Fi( B ) : G ⊆ K, H ⊆ W and f ( K, W ) ⊆ P } . Since (
G, H ) ∈ J , then J 6 = ∅ . Observe that the union of a chain of elements of J is also in J . So,by Zorn’s Lemma, there is a maximal element ( Q, R ) ∈ J . We see that ( Q, R ) ∈ X ( A ) ×X ( B ). Let x, y ∈ A be such that x ∨ y ∈ Q . Suppose that x, y / ∈ Q . Consider the filters F x = Fig A ( Q ∪ { x } )and F y = Fig A ( Q ∪ { y } ). Then Q ⊂ F x and Q ⊂ F y , and since ( Q, R ) is maximal in J , it followsthat f ( F x , R ) * P and f ( F y , R ) * P , i.e., there is z ∈ f ( F x , R ) such that z / ∈ P and there is t ∈ f ( F y , R ) such that t / ∈ P . Then there exist ( f , r ) ∈ F x × R and ( f , r ) ∈ F y × R such that f ( f , r ) ≤ z and f ( f , r ) ≤ t . So, there are q , q ∈ Q such that q ∧ x ≤ f and q ∧ y ≤ f .We take q = q ∧ q ∈ Q and r = r ∧ r ∈ R . By Proposition 3.1, we have f ( q ∧ x, r ) ≤ z and f ( q ∧ y, r ) ≤ t . Thus, f ( q ∧ x, r ) ∨ f ( q ∧ y, r ) = f (( q ∧ x ) ∨ ( q ∧ y ) , r ) = f ( q ∧ ( x ∨ y ) , r ) ≤ z ∨ t. As q, x ∨ y ∈ Q , then q ∧ ( x ∨ y ) ∈ Q and z ∨ t ∈ f ( Q, R ). On the other hand, since f ( Q, R ) ⊆ P ,we have z ∨ t ∈ P . As P is prime, z ∈ P or t ∈ P which is a contradiction. Then Q ∈ X ( A ).The proof for R ∈ X ( B ) is similar. It follows that there exist Q ∈ X ( A ) and R ∈ X ( B ) such that G ⊆ Q , H ⊆ R and f ( Q, R ) ⊆ P . (cid:3) Let M be a FIDL-module. We define the following relations R M ⊆ X ( A ) × X ( B ) × X ( A ) and T M ⊆ X ( B ) × X ( A ) × X ( A ) by( Q, R, P ) ∈ R M ⇐⇒ f ( Q, R ) ⊆ P, (3.1)and ( R, P, Q ) ∈ T M ⇐⇒ i ( R, P ) ⊆ Q. (3.2) Lemma 3.1.
Let M be a FIDL-module. Let x ∈ A , b ∈ B and P ∈ X ( A ) . Then: (1) f ( x, b ) ∈ P if and only if there exist Q ∈ X ( A ) and R ∈ X ( B ) such that ( Q, R, P ) ∈ R M , x ∈ Q and b ∈ R . (2) i ( b, x ) ∈ P if and only if for every R ∈ X ( B ) and every Q ∈ X ( A ) , if ( R, P, Q ) ∈ T M and b ∈ R , then x ∈ Q .Proof. (1) Suppose f ( x, b ) ∈ P . We see that f ([ x ) , [ b )) ⊆ P . If y ∈ f ([ x ) , [ b )), then thereexists ( g, h ) ∈ [ x ) × [ b ) such that f ( g, h ) ≤ y . So, x ≤ g and b ≤ h , and by Proposition 3.1, f ( x, b ) ≤ f ( g, h ) ≤ y . Since P is a filter, y ∈ P and f ([ x ) , [ b )) ⊆ P . So, by Theorem 3.3, thereexist Q ∈ X ( A ) and R ∈ X ( B ) such that [ x ) ⊆ Q , [ b ) ⊆ R and f ( Q, R ) ⊆ P , i.e., x ∈ Q , b ∈ R and ( Q, R, P ) ∈ R M . Conversely, if there exist Q ∈ X ( A ) and R ∈ X ( A ) such that f ( Q, R ) ⊆ P , x ∈ Q and b ∈ R , because ( x, b ) ∈ Q × R , we have f ( x, b ) ∈ f ( Q, R ) and f ( x, b ) ∈ P .(2) Suppose i ( b, x ) ∈ P . Let R ∈ X ( B ) and Q ∈ X ( A ) be such that i ( R, P ) ⊆ Q and b ∈ R .Then ( b, i ( b, x )) ∈ R × P and x ∈ i ( R, P ). So, x ∈ Q . Reciprocally, suppose i ( b, x ) / ∈ P . Weprove that i ([ b ) , P ) ∩ ( x ] = ∅ . Otherwise, there is y ∈ i ([ b ) , P ) such that y ∈ ( x ]. Thus, thereexists ( z, p ) ∈ [ b ) × P such that p ≤ i ( z, y ). Since y ≤ x and b ≤ z , by Proposition 3.1, we have i ( z, y ) ≤ i ( b, x ). Then p ≤ i ( b, x ) and i ( b, x ) ∈ P , which is a contradiction. So, i ([ b ) , P ) ∩ ( x ] = ∅ and since i ([ b ) , P ) ∈ Fi( A ), by the Prime Filter Theorem there exists Q ∈ X ( A ) such that i ([ b ) , P ) ⊆ Q and x / ∈ Q . Then, by Theorem 3.3, there exist R ∈ X ( B ) and ˆ P ∈ X ( A ) such that[ b ) ⊆ R , P ⊆ ˆ P and i ( R, ˆ P ) ⊆ Q . It is clear that i ( R, P ) ⊆ i ( R, ˆ P ). Summarizing, there exist R ∈ X ( B ) and Q ∈ X ( A ) such that ( R, P, Q ) ∈ T M , b ∈ R and x / ∈ Q , which contradicts thehypothesis. Therefore, i ( b, x ) ∈ P . (cid:3) Now, we introduce the concept of subalgebra of a FIDL-module.
ISMAEL CALOMINO AND WILLIAM J. ZULUAGA BOTERO
Definition 2.
Let M be a FIDL-module. Let ˆ A be a bounded sublattice of A and ˆ B a boundedsublattice of B .(S1) A structure h ˆ A , ˆ B , f i is called a FDL-subalgebra of h A , B , f i , if for every ˆ x ∈ ˆ A and everyˆ b ∈ ˆ B , we have f (ˆ x, ˆ b ) ∈ ˆ A .(S2) A structure h ˆ A , ˆ B , i i is called an IDL-subalgebra of h A , B , i i , if for every ˆ x ∈ ˆ A and everyˆ b ∈ ˆ B , we have i (ˆ b, ˆ x ) ∈ ˆ A .Moreover, a structure ˆ M = h ˆ A , ˆ B , f, i i is called a FIDL-subalgebra of M , if h ˆ A , ˆ B , f i is a FDL-subalgebra and h ˆ A , ˆ B , i i is an IDL-subalgebra.We conclude this section with a characterization of FIDL-subalgebras by means of the relationsdefined in (3.1) and (3.2). Theorem 3.4.
Let M be a FIDL-module. Let ˆ A be a bounded sublattice of A and ˆ B a boundedsublattice of B . Then: (1) h ˆ A , ˆ B , f i is a FDL-subalgebra of h A , B , f i if and only if for all P, Q, Q ∈ X ( A ) and forall R ∈ X ( B ) , if ( Q , R , P ) ∈ R M and P ∩ ˆ A ⊆ Q , then there exist Q ∈ X ( A ) and R ∈ X ( B ) such that Q ∩ ˆ A ⊆ Q , R ∩ ˆ B ⊆ R and ( Q , R , Q ) ∈ R M . (2) h ˆ A , ˆ B , i i is an IDL-subalgebra of h A , B , i i if and only if for all P, Q, Q ∈ X ( A ) and forall R ∈ X ( B ) , if ( R , Q, Q ) ∈ T M and P ∩ ˆ A ⊆ Q , then there exist Q ∈ X ( A ) and R ∈ X ( B ) such that Q ∩ ˆ A ⊆ Q , R ∩ ˆ B ⊆ R and ( R , P, Q ) ∈ T M .We conclude that the structure ˆ M = h ˆ A , ˆ B , f, i i is a FIDL-subalgebra of M if and only if verifiesthe conditions (1) and (2) .Proof. (1) Let P, Q, Q ∈ X ( A ) and R ∈ X ( B ) be such that ( Q , R , P ) ∈ R M and P ∩ ˆ A ⊆ Q .Then, f ( Q , R ) ⊆ P . Consider the filters F Q = Fig A ( Q ∩ ˆ A ) and F R = Fig B ( R ∩ ˆ B ). Itfollows that f ( F Q , F R ) ⊆ Q . Indeed, if x ∈ f ( F Q , F R ), then there exists ( g, h ) ∈ F Q × F R such that f ( g, h ) ≤ x . So, there is q ∈ Q ∩ ˆ A such that q ≤ g and there is r ∈ R ∩ ˆ B such that r ≤ h . By Proposition 3.1, f ( q , r ) ≤ f ( g, h ) ≤ x . So, f ( q , r ) ∈ f ( Q , R ) and f ( q , r ) ∈ P .On the other hand, since h ˆ A , ˆ B , f i is a FDL-subalgebra, f ( q , r ) ∈ ˆ A . Thus, f ( q , r ) ∈ Q and x ∈ Q . Therefore, f ( F Q , F R ) ⊆ Q and by Theorem 3.3, there exist Q ∈ X ( A ) and R ∈ X ( B )such that Q ∩ ˆ A ⊆ Q , R ∩ ˆ B ⊆ R and ( Q , R , Q ) ∈ R M .Conversely, suppose there exist ˆ x ∈ ˆ A and ˆ b ∈ ˆ B such that f (ˆ x, ˆ b ) / ∈ ˆ A . We prove thatFig A ([ f (ˆ x, ˆ b )) ∩ ˆ A ) ∩ ( f (ˆ x, ˆ b )] = ∅ . Otherwise, there is y ∈ A such that y ∈ Fig A ([ f (ˆ x, ˆ b )) ∩ ˆ A ) and y ≤ f (ˆ x, ˆ b ). Then there exists z ∈ [ f (ˆ x, ˆ b )) ∩ ˆ A such that z ≤ y . It follows that f (ˆ x, ˆ b ) = z . Since z ∈ ˆ A , we have f (ˆ x, ˆ b ) ∈ ˆ A which is a contradiction. Then Fig A ([ f (ˆ x, ˆ b )) ∩ ˆ A ) ∩ ( f (ˆ x, ˆ b )] = ∅ andby the Prime Filter Theorem there exists Q ∈ X ( A ) such that [ f (ˆ x, ˆ b )) ∩ ˆ A ⊆ Q and f (ˆ x, ˆ b ) / ∈ Q . Itis easy to see that [ f (ˆ x, ˆ b )) ∩ Idg A ( Q c ∩ ˆ A ) = ∅ . Then there exists P ∈ X ( A ) such that f (ˆ x, ˆ b ) ∈ P and P ∩ Idg A ( Q c ∩ ˆ A ) = ∅ , i.e., P ∩ ˆ A ⊆ Q . Since f (ˆ x, ˆ b ) ∈ P , by Lemma 3.1 there exist Q ∈ X ( A ) and R ∈ X ( B ) such that f ( Q , R ) ⊆ P , ˆ x ∈ Q and ˆ b ∈ R . So ( Q , R , P ) ∈ R M .By assumption, there exist Q ∈ X ( A ) and R ∈ X ( B ) such that Q ∩ ˆ A ⊆ Q , R ∩ ˆ B ⊆ R and f ( Q , R ) ⊆ Q . Thus, ( Q , R , Q ) ∈ R M . Since ˆ x ∈ ˆ A and ˆ b ∈ ˆ B , we have ˆ x ∈ Q and ˆ b ∈ R .Hence, f (ˆ x, ˆ b ) ∈ f ( Q , R ) and f (ˆ x, ˆ b ) ∈ Q , which is a contradiction. Therefore, f (ˆ x, ˆ b ) ∈ ˆ A andwe conclude that h ˆ A , ˆ B , f i is a FDL-subalgebra.(2) Let P, Q, Q ∈ X ( A ) and R ∈ X ( A ) be such that ( R , Q, Q ) ∈ T M and P ∩ ˆ A ⊆ Q . So i ( R , Q ) ⊆ Q . We see that Fig A ( i (Fig B ( R ∩ ˆ B ) , P ) ∩ ˆ A ) ⊆ Q . IDL-MODULES: REPRESENTATION AND DUALITY 7 If x ∈ Fig A ( i (Fig B ( R ∩ ˆ B ) , P ) ∩ ˆ A ), then there is y ∈ i (Fig B ( R ∩ ˆ B ) , P ) ∩ ˆ A such that y ≤ x .So, there are r ∈ R ∩ ˆ B and p ∈ P such that p ≤ i ( r, y ). Thus, i ( r, y ) ∈ P . On the other hand, as y ∈ ˆ A , r ∈ ˆ B and h ˆ A , ˆ B , i i is an IDL-subalgebra, i ( r, y ) ∈ ˆ A . Then i ( r, y ) ∈ P ∩ ˆ A and i ( r, y ) ∈ Q .It follows that y ∈ i ( R , Q ) and y ∈ Q . Then x ∈ Q . Now, let us consider the family J = { F ∈ Fi( A ) : i (Fig B ( R ∩ ˆ B ) , P ) ⊆ F and F ∩ ˆ A ⊆ Q } . Then
J 6 = ∅ and by Zorn’s Lemma there exists an maximal element Q ∈ J . We prove that Q ∈ X ( A ). Let x, y ∈ A be such that x ∨ y ∈ Q and suppose x, y / ∈ Q . We take the filters F x = Fig A ( Q ∪ { x } ) and F y = Fig A ( Q ∪ { y } ). Then F x ∩ ˆ A * Q and F y ∩ ˆ A * Q , i.e., thereexist z ∈ F x ∩ ˆ A and t ∈ F y ∩ ˆ A such that z, t / ∈ Q . So, there are q , q ∈ Q such that q ∧ x ≤ z and q ∧ y ≤ t . Then ( q ∧ q ) ∧ ( x ∨ y ) ≤ z ∨ t and z ∨ t ∈ Q . Since ˆ A is a sublattice, z ∨ t ∈ Q ∩ ˆ A and z ∨ t ∈ Q , which is a contradiction because Q is prime and z ∨ t / ∈ Q . Hence, Q ∈ X ( A ).As i (Fig B ( R ∩ ˆ B ) , P ) ⊆ Q and F ∩ ˆ A ⊆ Q , by Theorem 3.3 there exists R ∈ X ( B ) such that R ∩ ˆ B ⊆ R and ( R , P, Q ) ∈ T M .Reciprocally, suppose there exist ˆ x ∈ ˆ A and ˆ b ∈ ˆ B such that i (ˆ b, ˆ x ) / ∈ ˆ A . In order to prove ourclaim, first we show that Idg A (( i (ˆ b, ˆ x )] ∩ ˆ A ) ∩ [ i (ˆ b, ˆ x )) = ∅ . If there is y ∈ Idg A (( i (ˆ b, ˆ x )] ∩ ˆ A ) suchthat i (ˆ b, ˆ x ) ≤ y , then there exists z ∈ ( i (ˆ b, ˆ x )] ∩ ˆ A such that y ≤ z . Thus, i (ˆ b, ˆ x ) = z and i (ˆ b, ˆ x ) ∈ ˆ A ,which is a contradiction. Then Idg A (( i (ˆ b, ˆ x )] ∩ ˆ A ) ∩ [ i (ˆ b, ˆ x )) = ∅ and consequently from the PrimeFilter Theorem, there exists P ∈ X ( A ) such that i (ˆ b, ˆ x ) ∈ P and Idg A (( i (ˆ b, ˆ x )] ∩ ˆ A ) ∩ P = ∅ . Itis easy to prove that ( i (ˆ b, ˆ x )] ∩ Fig A ( P ∩ ˆ A ) = ∅ . Then again by the Prime Filter Theorem, thereis Q ∈ X ( A ) such that P ∩ ˆ A ⊆ Q and i (ˆ b, ˆ x ) / ∈ Q . By Lemma 3.1, there exist R ∈ X ( B ) and Q ∈ X ( A ) such that i ( R , Q ) ⊆ Q , ˆ b ∈ R and ˆ x / ∈ Q . So, by hypothesis, there exist Q ∈ X ( A )and R ∈ X ( B ) such that Q ∩ ˆ A ⊆ Q , R ∩ ˆ B ⊆ R and i ( R , P ) ⊆ Q . Thus, ( R , P, Q ) ∈ T M .As ˆ b ∈ R , we have ˆ b ∈ R . On the other hand, as i (ˆ b, ˆ x ) ∈ P , we have ˆ x ∈ i ( R , P ) and ˆ x ∈ Q .Then ˆ x ∈ Q ∩ ˆ A and ˆ x ∈ Q which is a contradiction. Hence, i (ˆ b, ˆ x ) ∈ ˆ A and h ˆ A , ˆ B , i i is anIDL-subalgebra. (cid:3) Representation for FIDL-modules
The main purpose of this section is to show a representation theorem for FIDL-modules in termsof certain relational structures consisting of bi-posets endowed with two relations.We start by defining a category whose objects are FIDL-modules. So, we need to describefirst, the notion of homomorphism between FIDL-modules. Recall that for every pair of functions α : A → ˆ A and γ : B → ˆ B we can consider the map α × γ : A × B → ˆ A × ˆ B which is defined by( α × γ ) ( x, y ) = ( α ( x ) , γ ( y )). Definition 3.
Let M = h A , B , f, i i and ˆ M = h ˆ A , ˆ B , ˆ f , ˆ i i be two FIDL-modules. We shall say thata pair ( α, γ ) : M → ˆ M is a FIDL-homomorphism, if α : A → ˆ A and γ : B → ˆ B are homomorphismsbetween bounded distributive lattices and the following diagrams commute: A × B f / / α × γ (cid:15) (cid:15) A α (cid:15) (cid:15) ˆ A × ˆ B ˆ f / / ˆ A B × A i / / γ × α (cid:15) (cid:15) A α (cid:15) (cid:15) ˆ B × ˆ A ˆ i / / ˆ A ISMAEL CALOMINO AND WILLIAM J. ZULUAGA BOTERO
Remark 2.
Notice that from Remark 1, the diagrams of Definition 3 are commutative if and onlyif for every b ∈ B , the following diagrams commute: A f b / / α (cid:15) (cid:15) A α (cid:15) (cid:15) ˆ A ˆ f γ ( b ) / / ˆ A A i b / / α (cid:15) (cid:15) A α (cid:15) (cid:15) ˆ A ˆ i γ ( b ) / / ˆ A We stress that for the rest of the paper we will use the functions ˆ f γ ( b ) and ˆ i γ ( b ) as well as thenotation of Definition 3 indistinctly. Example Let M be a FIDL-module. Let C be a bounded distributive lattice and h : C → B be a lattice homomorphism. If we define the functions ˆ f : A × C → A by ˆ f ( x, c ) = f ( x, h ( c )) andˆ i : C × A → A by ˆ i ( c, x ) = i ( h ( c ) , x ), then the structure N = h A , C , ˆ f , ˆ i i is a FIDL-module andthe pair ( id A , h ) : N → M is a FIDL-homomorphism.Let M = h A , B , f, i i , ˆ M = h ˆ A , ˆ B , ˆ f , ˆ i i and ¯ M = h ¯ A , ¯ B , ¯ f , ¯ i i be FIDL-modules. Consider theFIDL-homomorphisms ( α, γ ) : M → ˆ M and ( δ, λ ) : ˆ M → ¯ M . Then we define the composition( δ, λ ) ( α, γ ) : M → ¯ M as the pair ( δα, λγ ). It is clear that the FIDL-homomorphisms betweenFIDL-modules are closed by composition and that such a composition is associative. Moreover,we may define the identity of M as the pair ( id A , id B ). So, we obtain that the class FIMod ofFIDL-modules as objects and FIDL-homomorphisms as morphisms is a category.The following technical result will be useful later.
Lemma 4.1.
Let M = h A , B , f, i i and ˆ M = h ˆ A , ˆ B , ˆ f , ˆ i i be two FIDL-modules and ( α, γ ) : M → ˆ M a FIDL-homomorphism. Then the following conditions are equivalent: (1) ( α, γ ) is an isomorphism in the category FIMod , (2) α and γ are isomorphisms of bounded distributive lattices.Proof. (1) ⇒ (2) Immediate.(2) ⇒ (1) Let us assume that α and γ are isomorphism of bounded distributive lattices. Wewill show that the pair (cid:0) α − , γ − (cid:1) : ˆ M → M is a FIDL-homomorphism. Since α − and γ − areisomorphisms of bounded distributive lattices, only remains to check the commutativity of thefollowing diagram: ˆ A × ˆ B ˆ f / / α − × γ − (cid:15) (cid:15) ˆ A α − (cid:15) (cid:15) A × B f / / A By hypothesis, we have ˆ f ( α × γ ) = αf . So, α − ˆ f ( α × γ ) = f and f (cid:0) α − × γ − (cid:1) = α − ˆ f ( α × γ ) (cid:0) α − × γ − (cid:1) = α − ˆ f (cid:0) id ˆ A × ˆ B (cid:1) = α − ˆ f , i.e., f (cid:0) α − × γ − (cid:1) = α − ˆ f . Similarly, the commutativity of the following diagram is easily verified:ˆ B × ˆ A ˆ i / / γ − × α − (cid:15) (cid:15) ˆ A α − (cid:15) (cid:15) B × A i / / A IDL-MODULES: REPRESENTATION AND DUALITY 9
By definition of the composition in
FIMod , it follows that ( α, γ ) − = (cid:0) α − , γ − (cid:1) and ( α, γ ) is anisomorphism in the category FIMod . (cid:3) Now, we introduce the class of relational structures needed to develop our representation theoremas well as the notion of morphisms between them.
Definition 4.
Let h X, ≤ X i and h Y, ≤ Y i be two posets. A structure h X, Y, ≤ X , ≤ Y , R i is called aF-frame, if R ⊆ X × Y × X is a relation such that:if ( x, y, z ) ∈ R, ¯ x ≤ X x, ¯ y ≤ Y y and z ≤ X ¯ z, then (¯ x, ¯ y, ¯ z ) ∈ R. (4.1)A structure h X, Y, ≤ X , ≤ Y , T i is called an I-frame, if T ⊆ Y × X × X is a relation such that:if ( y, x, z ) ∈ T, ¯ y ≤ Y y, ¯ x ≤ X x and z ≤ X ¯ z, then (¯ y, ¯ x, ¯ z ) ∈ T. (4.2)Moreover, a structure F = h X, Y, ≤ X , ≤ Y , R, T i is called a FI-frame, if h X, Y, ≤ X , ≤ Y , R i is aF-frame and h X, Y, ≤ X , ≤ Y , T i is an I-frame. Definition 5.
Let F and ˆ F be two FI-frames. We shall say that a pair ( g, h ) : F → ˆ F is a FI-morphism, if g : X → ˆ X and h : Y → ˆ Y are morphisms between posets and the following conditionshold:(M1) If ( x, y, z ) ∈ R , then ( g ( x ) , h ( y ) , g ( z )) ∈ ˆ R .(M2) If (¯ x, ¯ y, g ( z )) ∈ ˆ R , then there exist x ∈ X and y ∈ Y such that ( x, y, z ) ∈ R , ¯ x ≤ ˆ X g ( x )and ¯ y ≤ ˆ Y h ( y ).(N1) If ( x, y, z ) ∈ T , then ( h ( x ) , g ( y ) , g ( z )) ∈ ˆ T .(N2) If (¯ x, g ( y ) , ¯ z ) ∈ ˆ T , then there exist x ∈ Y and z ∈ X such that ( x, y, z ) ∈ T , ¯ x ≤ ˆ Y h ( x )and g ( z ) ≤ ˆ X ¯ z .The composition of FI-morphisms is defined component-wise. It is clear from Definition 5, thatsuch a composition is closed and associative and for every FI-Frame F , the identity arrow is givenby the pair ( id X , id Y ). We write FIFram for the category of FI-frames and FI-morphisms.The following result is similar to Lemma 4.1 and will be useful at the moment of proving themain theorem of this section.
Lemma 4.2.
Let F and ˆ F be two FI-frames and ( g, h ) : F → ˆ F a FI-morphism. Then thefollowing conditions are equivalent: (1) ( g, h ) is an isomorphism in the category FIFram , (2) g and h are isomorphisms of posets.Proof. (1) ⇒ (2) Immediate.(2) ⇒ (1) Since g and h are isomorphisms of posets, then there exist g − and h − . It is clearthat ( g, h ) − = ( g − , h − ). We need to check that ( g − , h − ) : ˆ F → F is a FI-morphism. We prove(M1). Let (¯ x, ¯ y, ¯ z ) ∈ ˆ X × ˆ Y × ˆ X such that (¯ x, ¯ y, ¯ z ) ∈ ˆ R and suppose that ( g − (¯ x ) , h − (¯ y ) , g − (¯ z )) / ∈ R . Due to (¯ x, ¯ y, g ( g − (¯ z ))) ∈ ˆ R and ( g, h ) is a FI-morphism, there exist x ∈ X and y ∈ Y suchthat ( x, y, g − (¯ z )) ∈ R , ¯ x ≤ ˆ X g ( x ) and ¯ y ≤ ˆ Y h ( y ). On the other hand, since g − and h − aremonotone, we have g − (¯ x ) ≤ X x , h − (¯ y ) ≤ Y y and g − (¯ z ) ≤ X g − (¯ z ). It follows, by (4.1), that( g − (¯ x ) , h − (¯ y ) , g − (¯ z )) ∈ R which is a contradiction. We prove (M2). Let ( x, y, g − (¯ z )) ∈ R .As x = g − ( g ( x )) and y = h − ( h ( y )), then ( g − ( g ( x )) , h − ( h ( y )) , g − (¯ z )) ∈ R . Since ( g, h ) is aFI-morphism, ( g ( x ) , h ( y ) , ¯ z ) ∈ ˆ R . Conditions (N1) and (N2) can be verified analogously. (cid:3) It is the moment to show how we build our representation. Let F = h X, Y, ≤ X , ≤ Y , R, T i be aFI-frame. Then it follows that hP i ( X ) , ∪ , ∩ , ∅ , X i and hP i ( Y ) , ∪ , ∩ , ∅ , Y i are bounded distributivelattices. Let U ∈ P i ( X ) and V ∈ P i ( Y ), and let us to consider the following subsets of X : f F ( U, V ) = { z ∈ X : ∃ ( x, y ) ∈ U × V such that ( x, y, z ) ∈ R } (4.3) and i F ( V, U ) = { y ∈ X : ∀ x ∈ Y, ∀ z ∈ X ((( x, y, z ) ∈ T and x ∈ V ) implies z ∈ U ) } . (4.4)It is easy to prove that f F ( U, V ) , i F ( V, U ) ∈ P i ( X ).The proof of the following two results are routine so the details are left to the reader. Lemma 4.3.
Let F be a FI-frame. Then the structure M F = hP i ( X ) , P i ( Y ) , f F , i F i is a FIDL-module, where f F and i F are given by 4.3 and 4.4, respectively. Lemma 4.4.
Let M be a FIDL-module. Then the structure F M = hX ( A ) , X ( B ) , ⊆ A , ⊆ B , R M , T M i is a FI-frame, where R M and T M are given by 3.1 and 3.2, respectively. Lemma 4.5.
Let M and ˆ M be two FIDL-modules. If ( α, γ ) : M → ˆ M is a FIDL-homomorphism,then ( α ∗ , γ ∗ ) : F ˆ M → F M is a FI-morphism.Proof. We start by proving (M1). Let ( ˆ Q, ˆ R, ˆ P ) ∈ R ˆ M . In order to prove ( α ∗ ( ˆ Q ) , γ ∗ ( ˆ R ) , α ∗ ( ˆ P )) ∈ R M , let y ∈ f ( α ∗ ( ˆ Q ) , γ ∗ ( ˆ R )). Then there exist ( a, b ) ∈ α ∗ ( ˆ Q ) × γ ∗ ( ˆ R ) such that f ( a, b ) ≤ A y . Since α is monotone and ( α, γ ) is a FIDL-homomorphism, we have α ( f ( a, b )) = ˆ f ( α ( a ) , γ ( b )) ≤ ˆ A α ( a ).Since α ( a ) ∈ P and γ ( b ) ∈ R , then α ( y ) ∈ ˆ f ( ˆ Q, ˆ R ). So, our assumption allows us to concludethat α ( y ) ∈ ˆ P . Hence f ( α ∗ ( ˆ Q ) , γ ∗ ( ˆ R )) ⊆ X ( A ) α ∗ ( ˆ P ). The proof of (N1) is similar. Now we prove(M2). Let us assume that ( Q, R, α ∗ ( ˆ P )) ∈ R M . Note that, such an assumption allows us to saythat f ( a, b ) ∈ α ∗ ( ˆ P ) for every ( a, b ) ∈ Q × R . Then, since ( α, γ ) is a FIDL-homomorphism, itfollows that ˆ f ( α ( a ) , γ ( b )) ∈ P . From (1) of Lemma 3.1, there exist ˆ Q ∈ X ( ˆ A ) and ˆ R ∈ X ( ˆ B ) suchthat α ( a ) ∈ Q , γ ( b ) ∈ R such that ( ˆ Q, ˆ R, ˆ P ) ∈ R ˆ M . Hence (M2) holds. Finally, for proving (N2),let us assume that ( R, α ∗ ( ˆ P ) , Q ) ∈ T M . Consider F = Fig ˆ B ( γ ( R )) and I = Idg ˆ A ( α ( Q c )). Wesee that ˆ i ( F, ˆ P ) ∩ I = ∅ . Assume the contrary, then there exist y ∈ ˆ A , a ∈ ˆ P , b ∈ ˆ B , r ∈ R and q / ∈ Q such that a ≤ ˆ A ˆ i ( b, y ), γ ( r ) ≤ ˆ B b and y ≤ ˆ A α ( q ). Since ( α, γ ) is a FIDL-homomorphism,by Proposition 3.1, we obtain that a ≤ ˆ A ˆ i ( γ ( r ) , α ( q )) = α ( i ( r, q )). Hence i ( r, q ) ∈ α ∗ ( ˆ P ) so q ∈ i ( R, α ∗ ( ˆ P )) and therefore q ∈ Q , which is a contradiction. Then ˆ i ( F, ˆ P ) ∩ I = ∅ and bythe Prime Filter Theorem, there exist ˆ Q ∈ X ( ˆ A ) such that ˆ i ( F, ˆ P ) ⊆ X ( ˆ A ) ˆ Q and ˆ Q ∩ I = ∅ .On the other hand, from Theorem 3.3, there exists ˆ R ∈ X ( ˆ B ) such that ˆ i ( ˆ R, ˆ P ) ⊆ X ( ˆ A ) ˆ Q and R ⊆ X ( B ) γ ∗ ( ˆ R ). As α ( Q c ) ⊆ X ( ˆ A ) I we get that ˆ Q ∩ α ( Q c ) = ∅ . It is not hard to see that the latteris equivalent to say that α ∗ ( ˆ Q ) ⊆ X ( A ) Q . (cid:3) Lemma 4.6.
Let F and ˆ F be two FI-frames. If ( g, h ) : F → ˆ F is a FI-morphism, then ( g ∗ , h ∗ ) : M ˆ F → M F is a FIDL-homomorphism.Proof. Let M F and M ˆ F be the FIDL-modules that arise from Lemma 4.3. In order to simplifynotation, in this proof we write f and i instead of f F . Similarly, we write ˆ f and ˆ i instead of f ˆ F and i ˆ F . This is with the aim of setting our proof within the context of Remark 2. It is clearthat g ∗ : P i ( ˆ X ) → P i ( X ) and h ∗ : P i ( ˆ Y ) → P i ( Y ) are homomorphisms of bounded distributivelattices so, in order to prove our claim we proceed to check that for every U ∈ P i ( ˆ Y ), the following IDL-MODULES: REPRESENTATION AND DUALITY 11 diagrams P i ( ˆ X ) g ∗ (cid:15) (cid:15) ˆ f U / / P i ( ˆ X ) g ∗ (cid:15) (cid:15) P i ( X ) f h ∗ ( U ) / / P i ( X ) P i ( ˆ X ) g ∗ (cid:15) (cid:15) ˆ i U / / P i ( ˆ X ) g ∗ (cid:15) (cid:15) P i ( X ) i h ∗ ( U ) / / P i ( X )commute. For the diagram of the left, let V ∈ P i ( ˆ X ) and z ∈ g ∗ ( ˆ f U ( V )). So, there exist x ′ ∈ U and y ′ ∈ V such that ( x ′ , y ′ , g ( z )) ∈ ˆ R . Since ( g, h ) is a FI-morphism then from (M2) of Definition5, there exist x ∈ Y and y ∈ X such that x ′ ≤ ˆ Y h ( x ), y ′ ≤ ˆ X g ( y ) and ( x, y, z ) ∈ R . Hence x ∈ h ∗ ( U ), y ∈ g ∗ ( V ) and ( x, y, z ) ∈ R . That is to say, z ∈ f h ∗ ( U ) ( g ∗ ( V )). The other inclusion isstraightforward. For the diagram of the right, let V ∈ P i ( ˆ X ) and suppose that y ∈ i h ∗ ( U ) ( g ∗ ( V )).So, for every x ∈ Y and z ∈ X , if ( x, y, z ) ∈ T and h ( x ) ∈ U , then g ( z ) ∈ V . We recallthat for showing y ∈ g ∗ (ˆ i U ( V )) we need to prove that for every x ′ ∈ ˆ Y and z ′ ∈ ˆ X such that( x ′ , g ( y ) , z ′ ) ∈ ˆ T and x ′ ∈ U , then z ′ ∈ V . Indeed, since ( g, h ) is a FI-morphism, from (N2) ofDefinition 5, there exist x ∈ Y and z ∈ X such that ( x, y, z ) ∈ T , x ′ ≤ ˆ Y h ( x ) and g ( z ) ≤ ˆ X z ′ .As U ∈ P i ( Y ), it follows that h ( x ) ∈ U and since ( x, y, z ) ∈ T from assumption, then we obtain g ( z ) ∈ V and z ′ ∈ V . The remaining inclusion is easy. (cid:3) Observe that Lemmas 4.3 and 4.6 allow to define a functor G : FIFram → FIMod op as follows: F 7→ M F ( g, h ) ( g ∗ , h ∗ ) . On the other hand, from Lemmas 4.4 and 4.5, we can define a functor F : FIMod op → FIFram asfollows:
M 7→ F M ( α, γ ) ( α ∗ , γ ∗ ) . We conclude this section by proving our representation theorem for FIDL-modules.
Theorem 4.1. G is a left adjoint of F and the counit is an isomorphism.Proof. We start by showing that G is a left adjoint of F . Let F = h X, Y, ≤ X , ≤ Y , R, T i be a FI-frame, M = h A , B , f, i i a FIDL-module and ( g, h ) : F → F M a FI-morphism. Then g : X → X ( A )and h : Y → X ( B ) are maps of posets satisfying the conditions of Definition 5. From Stone’srepresentation theorem, there exist a unique pair of lattice homomorphisms g : A → P i ( X ) and h : B → P i ( Y ) defined by g ( a ) = { y ∈ X : a ∈ g ( y ) } and h ( b ) = { x ∈ Y : b ∈ h ( x ) } . In order toprove that ( g, h ) : M → M F is a FIDL-homomorphism, we need to show the commutativity of thefollowing diagrams A f b / / g (cid:15) (cid:15) A g (cid:15) (cid:15) P i ( X ) f F h ( b ) / / P i ( X ) A i b / / g (cid:15) (cid:15) A g (cid:15) (cid:15) P i ( X ) i F h ( b ) / / P i ( X )for every b ∈ B . We prove g ( f b ( a )) = f F h ( b ) ( g ( a )), for every a ∈ A . So to check that g ( f b ( a )) ⊆ f F h ( b ) ( g ( a )), let z ∈ g ( f b ( a )). Thus, f b ( a ) ∈ g ( z ). By Lemma 3.1, there exist Q ∈ X ( B ) and E ∈ X ( A ) such that b ∈ Q , a ∈ E and ( Q, E, g ( z )) ∈ R M . From condition (M2), there exist x ∈ Y and y ∈ X such that Q ⊆ X ( B ) h ( x ), E ⊆ X ( A ) g ( x ) and ( x, y, z ) ∈ R . Hence, x ∈ h ( b ) and y ∈ g ( a ). Therefore z ∈ f F h ( b ) ( g ( a )). Conversely, if z ∈ f F h ( b ) ( g ( a )), then there exist x ∈ h ( b ) and y ∈ g ( a ) such that ( x, y, z ) ∈ R . So, ( h ( x ) , g ( y ) , g ( z )) ∈ R M from (M1) and consequently, f ( h ( x ) , g ( y )) ⊆ g ( z ). By Lemma 3.1, f b ( a ) ∈ g ( z ), or equivalently, z ∈ g ( f b ( a )).Now we prove that g ( i b ( a )) = i F h ( b ) ( g ( a )), for every a ∈ A . Let z ∈ g ( i b ( a )). If ( x, y, z ) ∈ T and a ∈ h ( x ), then from condition (N1) we have i ( h ( x ) , g ( y )) ⊆ X ( A ) g ( z ). Since i b ( a ) ∈ g ( z ),from Lemma 3.1, we can conclude that a ∈ g ( z ) and z ∈ i F h ( b ) ( g ( a )). On the other hand, assumethat z ∈ i F h ( b ) ( g ( a )). In order to prove that z ∈ g ( i b ( a )), let R ∈ X ( B ) and Q ∈ X ( A ) such that( R, g ( y ) , Q ) ∈ T M and b ∈ R . So, i ( R, g ( y )) ⊆ X ( A ) Q . From condition (N2), then there exist x ∈ Y and z ∈ X such that ( x, y, z ) ∈ T , R ⊆ X ( B ) h ( x ) and g ( z ) ⊆ X ( B ) Q . So, from assumptionwe have a ∈ g ( z ). Hence, by Lemma 3.1, z ∈ g ( i b ( a )).For the last part, note that for each frame F = h X, Y, ≤ X , ≤ Y , R, T i , the counit of the adjunction G ⊣ F is determined by the pair of monotone maps ǫ X : X → X ( P i ( X )) and ǫ Y : Y → X ( P i ( Y )),which are defined by ǫ X ( y ) = { V ∈ P i ( X ) : y ∈ V } and ǫ Y ( x ) = { U ∈ P i ( Y ) : x ∈ U } . It is clearfrom Stone’s representation theorem, that the maps ǫ X and ǫ Y are isomorphisms of posets. So,from Lemma 4.2, the result follows. (cid:3) Topological duality
In this section we prove a duality for FIDL-modules by using some of the results of Section 4together with a suitable extension of Priestley duality for distributive lattices. The dual objectsare certain topological bi-spaces endowed with two relations satisfying some particular properties.
Definition 6.
A structure U = h X, Y, ≤ X , ≤ Y , τ X , τ Y , R, T i is called an Urquhart space, if thefollowing conditions hold:(1) h X, ≤ X , τ X i and h Y, ≤ Y , τ Y i are Priestley spaces,(2) R ⊆ X × Y × X and T ⊆ Y × X × X ,(3) For every U ∈ C ( Y ) and every V ∈ C ( X ), we have f ( V, U ) , i ( U, V ) ∈ C ( X ),(4) For every x ∈ Y and every y, z ∈ X , if f ( ǫ X ( y ) , ǫ Y ( x )) ⊆ ǫ X ( z ), then ( y, x, z ) ∈ R ,(5) For every x ∈ Y and every y, z ∈ X , if i ( ǫ Y ( x ) , ǫ X ( y )) ⊆ ǫ X ( z ), then ( x, y, z ) ∈ T . Lemma 5.1. If U is an Urquhart spaces, then it is a FI-frame.Proof. Let U = h X, Y, ≤ X , ≤ Y , R, T i be an Urquhart space. We need to check that R and T satisfyconditions (4.1) and (4.2) of Definition 4, respectively. Since both proofs are similar, we only provethat R satisfies (4.1). Suppose ( y, x, z ) ∈ R , y ′ ≤ X y , x ′ ≤ Y x and z ≤ X z ′ . Because of ǫ X and ǫ Y are monotonous, then ǫ X ( y ′ ) ⊆ ǫ X ( y ) and ǫ Y ( x ′ ) ⊆ ǫ Y ( x ). So, from Proposition 3.1, weobtain f ( ǫ X ( y ′ ) , ǫ Y ( x ′ )) ⊆ f ( ǫ X ( y ) , ǫ Y ( x )). If P ∈ f ( ǫ X ( y ′ ) , ǫ Y ( x ′ )), then there exist S ∈ C ( X )and Q ∈ C ( Y ) such that y ∈ S , x ∈ Q and f ( S, Q ) ⊆ P . As U is an Urquhart space then f ( S, Q ) ∈ C ( X ), and due to ( y, x, z ) ∈ R from hypothesis, we obtain z ∈ f ( S, Q ). Thus, z ∈ P .Because P ∈ C ( X ) and z ≤ X z ′ , we conclude that f ( ǫ Y ( x ′ ) , ǫ X ( y ′ )) ⊆ ǫ X ( z ′ ). Therefore, from (4)of Definition 6, we get ( y ′ , x ′ , z ′ ) ∈ R . (cid:3) Definition 7.
Let U = h X, Y, ≤ X , ≤ Y , τ X , τ Y , R, T i and ˆ U = h ˆ X, ˆ Y , ≤ ˆ X , ≤ ˆ Y , τ ˆ X , τ ˆ X ˆ R, ˆ T i beUrquhart spaces. We shall say that a pair ( g, h ) : U → ˆ U is an U-map, if g : X → ˆ X and h : Y → ˆ Y are monotonous and continuous maps, and satisfy the conditions (M1), (M2), (N1) and (N2) ofDefinition 5.We denote by USp the category of Urquhart spaces and U-maps.Let M be a FIDL-module and F : FIMod op → FIFram the functor of Theorem 4.1. Noticethat F ( M ) = F M is an Urquhart space. The latter assertion lies in the following facts which areimmediate from Priestley duality: (1) hX ( A ) , ⊆ A , τ A i and hX ( B ) , ⊆ B , τ B i are Priestley spaces; (2)Since C ( X ( A )) = { β A ( a ) : a ∈ A } and C ( X ( B )) = { β B ( b ) : b ∈ B } then, for every U ∈ C ( X ( B )) IDL-MODULES: REPRESENTATION AND DUALITY 13 and every V ∈ C ( X ( A )), there exist a ∈ A and b ∈ B such that f ( V, U ) = f ( β A ( a ) , β B ( b )) = β A ( f ( a, b )) and i ( U, V ) = i ( β B ( b ) , β A ( a )) = β A ( i ( b, a )). Hence f ( V, U ) , i ( U, V ) ∈ C ( X ( A )); (3)Since ǫ X ( A ) ( P ) = P and ǫ X ( B ) ( Q ) = Q , for every P ∈ X ( A ) and every Q ∈ X ( B ), it followsthat conditions (4) and (5) of the Definition 6 hold. In addition, if ( α, γ ) : M → ˆ M is a FIDL-homomorphism between two FIDL-modules M and ˆ M , it is also clear from Priestley duality that F ( α, γ ) = ( α ∗ , λ ∗ ) is an U-map between Urquhart spaces.On the other hand, if U is an Urquhart space, then from Priestley duality the structure M U = hC ( X ) , C ( Y ) , f U , i U i is a FIDL-module. These facts allows us to define an assignment J : USp → FIMod op as follows: U 7→ M U ( g, h ) ( g ∗ , h ∗ ) , where f U and i U are the operations defined in (4.3) and (4.4), respectively. Such an assignment isclearly functorial. Notice that as an straight application of Priestley duality, it follows that J isthe inverse functor of F . Since this is routine, we leave to the reader the details of the proof of thefollowing result. Theorem 5.1.
The categories
FIMod and
USp are dually equivalent. Congruences of FIDL-modules
In this section we introduce the concept of congruence in the class of FIDL-modules and weshow how through the duality of Section 5 we can provide a characterization of these in terms ofcertain pairs of closed subsets of the associated Urquhart space. This result will allows us to givea topological bi-spaced characterization of the simple and subdirectly irreducible FIDL-modules.
Definition 8.
Let M be a FIDL-module. Let θ A be a congruence of A and θ B a congruence of B .(C1) A pair ( θ A , θ B ) ⊆ A × B is called a FDL-congruence of h A , B , f i , if for every ( a, c ) ∈ θ A and every ( b, d ) ∈ θ B , we have ( f ( a, b ) , f ( c, d )) ∈ θ A .(C2) A pair ( θ A , θ B ) ⊆ A × B is called an IDL-congruence of h A , B , i i , if for every ( a, c ) ∈ θ A and every ( b, d ) ∈ θ B , we have ( i ( b, a ) , i ( d, c )) ∈ θ A .Moreover, a pair ( θ A , θ B ) ⊆ A × B is called a FIDL-congruence of M , if ( θ A , θ B ) is a FDL-congruence and an IDL-congruence.If M is a FIDL-module, then we write Con f ( M ) for the set of all FDL-congruences, Con i ( M )for the set of all IDL-congruences, and Con ( M ) for the set of all FIDL-congruences of M . It isnot hard to see that Con ( M ) is an algebraic lattice.We now proceed to introduce the topological notions required for our characterization. Let U be an Urquhart space. We define the following subsets of X and Y : • For every x, z ∈ X and every y ∈ Y , we have R ( y, z ) = { x ∈ X : x is maximal in X and ( x, y, z ) ∈ R } ,R ( x, z ) = { y ∈ Y : y is maximal in Y and ( x, y, z ) ∈ R } ,T ( x, z ) = { y ∈ Y : y is maximal in Y and ( y, x, z ) ∈ T } ,T ( y, x ) = { z ∈ X : z is minimal in X and ( y, x, z ) ∈ T } . • For every x, z ∈ X , we have M ax ( R − ( z )) = { ( x, y ) ∈ X × Y : x ∈ R ( y, z ) and y ∈ R ( x, z ) } , D ( x ) = { ( y, z ) ∈ Y × X : y ∈ T ( x, z ) and z ∈ T ( y, x ) } . Definition 9.
Let U be an Urquhart space. Let Z be a closed set of X and Z a closed set of Y .(CL1) A pair ( Z , Z ) ⊆ X × Y is called a R -closed set of U , if for every z ∈ Z , we have M ax ( R − ( z )) ⊆ Z × Z .(CL2) A pair ( Z , Z ) ⊆ X × Y is called a T -closed set of U , if for every x ∈ Z , we have D ( x ) ⊆ Z × Z .Moreover, a pair ( Z , Z ) ⊆ X × Y is called a strongly closed set of U , if ( Z , Z ) is both a R -closedset and a T -closed set.If U is an Urquhart space, then we write C f ( U ) for the set of all R -closed sets of U , C i ( U ) forthe set of all T -closed sets of U , and C s ( U ) for the set of all strongly closed sets of U . Theorem 6.1.
Let M be a FIDL-module and F M be the Urquhart space associated of M . Weconsider the correspondence ( Z , Z ) → ( θ ( Z ) , θ ( Z )) for every Z ∈ C ( X ( A )) and every Z ∈C ( X ( B )) , where θ ( − ) is given by (2.1). Then: (1) There exists an anti-isomorphism between C f ( F M ) and Con f ( M ) . (2) There exists an anti-isomorphism between C i ( F M ) and Con i ( M ) . (3) There exists an anti-isomorphism between C s ( F M ) and Con ( M ) .Proof. Since (3) is clearly a straight consequence of (1) and (2), we only prove such items.(1) Let us assume that ( Z , Z ) is a R M -closed set of F M . We prove that ( θ ( Z ) , θ ( Z )) is aFDL-congruence. Let ( x, y ) ∈ θ ( Z ) and ( b, c ) ∈ θ ( Z ). If P ∈ β A ( f ( x, b )) ∩ Z , then f ( x, b ) ∈ P .By Lemma 3.1, there exist Q ∈ X ( A ) and R ∈ X ( B ) such that f ( Q, R ) ⊆ P , x ∈ Q and b ∈ R .Using Zorn’s Lemma, it is easy to prove that there are Q ′ ∈ X ( A ) and R ′ ∈ X ( B ) maximals suchthat ( Q ′ , R ′ ) ∈ M ax ( R − M ( P )). Since ( Z , Z ) is a R M -closed set of F M by assumption, then itfollows that Q ′ ∈ Z and R ′ ∈ Z . Thus, f ( Q ′ , R ′ ) ⊆ P , y ∈ Q ′ and c ∈ R ′ . Then, by Lemma3.1, we have f ( y, c ) ∈ P . So, P ∈ β A ( f ( y, c )) ∩ Z . The other inclusion is similar. Therefore( f ( x, b ) , f ( y, c )) ∈ θ ( Z ).For the converse, let ( θ ( Z ) , θ ( Z )) be a FDL-congruence and suppose that the pair ( Z , Z )is not a R M -closed set of F M . Then, there exist P ∈ Z and ( Q, R ) ∈ X ( A ) × X ( B ) such that( Q, R ) ∈ M ax ( R − M ( P )) and ( Q, R ) / ∈ Z × Z . Suppose that Q / ∈ Z . Since Z is a closed set of X ( A ), then there exist a, b ∈ A such that a ∈ Q , b / ∈ Q and ( a ∧ b, a ) ∈ θ ( Z ). Let us consider thefilter Fig A ( Q ∪ { b } ). As Q ∈ R M ( R, P ), then Q is maximal and f (Fig A ( Q ∪ { b } ) , R ) * P . So,there exist q ∈ Q and r ∈ R such that f ( q ∧ b, r ) / ∈ P . Since ( θ ( Z ) , θ ( Z )) is a FDL-congruence, itfollows that ( f ( a ∧ b ∧ q, r ) , f ( a ∧ q, r )) ∈ θ ( Z ). Now, since a ∧ q ∈ Q , then f ( a ∧ q, r ) ∈ f ( Q, R ) ⊆ P .Hence f ( a ∧ b ∧ q, r ) ∈ P . Notice that f ( a ∧ b ∧ q, r ) ≤ f ( b ∧ q, r ), therefore f ( b ∧ q, r ) ∈ P , whichis a contradiction. Then Q ∈ Z . The proof of R ∈ Z is similar. So, ( Z , Z ) is a R M -closed.(2) Assume that ( Z , Z ) is a T M -closed set of F M . Let ( x, y ) ∈ θ ( Z ) and ( b, c ) ∈ θ ( Z ).We will see that ( θ ( Z ) , θ ( Z )) is an IDL-congruence of A . Let P ∈ X ( A ). Suppose that P ∈ β A ( i ( b, x )) ∩ Z and P / ∈ β A ( i ( c, y )) ∩ Z , i.e., i ( b, x ) ∈ P and i ( c, y ) / ∈ P . By Lemma 3.1, thereexist R ∈ X ( B ) and Q ∈ X ( A ) such that i ( R, P ) ⊆ Q , c ∈ R and y / ∈ Q . Note that from Zorn’sLemma it is not hard to see that there are R ′ ∈ X ( B ) and Q ′ ∈ X ( A ) such that ( R ′ , Q ′ ) ∈ D ( P ).Since ( Z , Z ) is a T M -closed set, then R ′ ∈ Z and Q ′ ∈ Z . Due to R ⊆ R ′ , we have c ∈ R ′ andbecause ( b, c ) ∈ θ ( Z ), it follows that b ∈ R ′ . On the other hand, since i ( b, x ) ∈ P , i ( R ′ , P ) ⊆ Q ′ and b ∈ R ′ , by Lemma 3.1 we have x ∈ Q ′ . Then ( x, y ) ∈ θ ( Z ), y ∈ Q ′ ⊆ Q and y ∈ Q , which isa contradiction. We conclude that ( θ ( Z ) , θ ( Z )) is an IDL-congruence.Conversely, we assume ( θ ( Z ) , θ ( Z )) is an IDL-congruence. Suppose that ( Z , Z ) is not a T M -closed set of F M . Then there exist P ∈ Z , Q ∈ X ( A ) and R ∈ X ( B ) such that ( R, Q ) ∈ D ( P )and ( R, Q ) / ∈ Z × Z . If R / ∈ Z , then since Z is a closed set of X ( B ) there exist b, c ∈ B suchthat b ∈ R , c / ∈ R and ( b ∧ c, b ) ∈ θ ( Z ). Let us consider Fig B ( R ∪ { c } ). Since R ∈ T M ( P, Q ),then i (Fig B ( R ∪ { c } ) , P ) * Q , i.e., there exists x ∈ A such that x / ∈ Q and p ≤ i ( r ∧ c, z ), for IDL-MODULES: REPRESENTATION AND DUALITY 15 some r ∈ R and p ∈ P . Hence i ( r ∧ c, z ) ∈ P and by Proposition 3.1, i ( r ∧ b ∧ c, z ) ∈ P . On theother hand, since ( θ ( Z ) , θ ( Z )) is a congruence, we obtain that ( i ( r ∧ b ∧ c, z ) , i ( r ∧ b, z )) ∈ θ ( Z )and i ( r ∧ b, z ) ∈ P . So, i ( R, P ) ⊆ Q , r ∧ b ∈ R and by Lemma 3.1 we have z ∈ Q , which is acontradiction. If Q / ∈ Z , then there exist x, y ∈ A such that x ∈ Q , y / ∈ Q and ( x ∧ y, x ) ∈ θ ( Z ).Let us consider I = Idg A ( Q c ∪ { y } ). Observe that I ∩ i ( R, P ) = ∅ , because otherwise from thePrime Filter Theorem, there would exists H ∈ X ( A ) such that i ( R, P ) ⊆ H , H ⊆ Q and x / ∈ H which is absurd since Q is minimal. Thus, there exist a ∈ A such that a ≤ q ∨ x and p ≤ i ( r, a ), forsome q / ∈ Q , r ∈ R and p ∈ P . So, by Proposition 3.1, p ≤ i ( r, a ) ≤ i ( r, q ∨ x ) and i ( p, q ∨ x ) ∈ P .Therefore, since ( θ ( Z ) , θ ( Z )) is a congruence, it follows that ( i ( r, q ∨ ( x ∧ y )) , i ( r, q ∨ x )) ∈ θ ( Z ).Hence i ( r, q ∨ ( x ∧ y )) ∈ P . Since i ( R, P ) ⊆ Q and r ∈ R , then by Lemma 3.1 we get that q ∨ ( x ∧ y ) ∈ Q , which is a contradiction because Q is prime. Then ( Z , Z ) is a T M -closed set. (cid:3) Let {M k } k ∈ K be a family of FIDL-modules, with M k = h A k , B k , f k , i k i . Then Y k ∈ K M k = * Y k ∈ K A k , Y k ∈ K B k , f, i + has a FIDL-module structure, where f ( a, b )( k ) = f k ( a ( k ) , b ( k )) and i ( b, a )( k ) = i k ( b ( k ) , a ( k )), forevery k ∈ K . Let π A k : Q k ∈ K A k → A k and π B k : Q k ∈ K B k → B k be the projection homomorphisms.Note that the pair ( π A k , π B k ) is a FIDL-homomorphism, for every k ∈ K . It is no hard to see that Q k ∈ K M k together with the family { ( π A k , π B k ) } k ∈ K is in fact the categorical product of {M k } k ∈ K .Let ( α, γ ) be a FIDL-homomorphism. We say that ( α, γ ) is a 1-1 FIDL-homomorphism if α and γ are 1-1, and similarly, we say that ( α, γ ) is a onto FIDL-homomorphism if α and γ are onto.If M is a FIDL-module, then we introduce the following concepts: • We will say that M is a subdirect product of a family {M k } k ∈ K of FIDL-modules, if thereexists a 1-1 FIDL-homomorphism( α, γ ) : M → Y k ∈ K M k such that ( π A k α, π B k γ ) is an onto FIDL-homomorphism, for every k ∈ K . • We will say that M is subdirectly irreducible if for every family of FIDL-modules {M k } k ∈ K and 1-1 FIDL-homomorphism ( α, γ ) : M → Y k ∈ K M k there exists a k ∈ K such that ( π A k α, π B k γ ) is an isomorphism of FIDL-modules. • We will say that M is simple if the lattice of the FIDL-congruences has only two elements.The following result is immediate from Theorem 6.1. Corollary 6.1.1.
Let M be a FIDL-module. Then M is subdirectly irreducible if and only if M is trivial or there exists a minimal non-trivial FIDL-congruence in M . If U is an Urquhart space, then from Theorem 6.1 it is clear that C s ( U ) is an algebraic lattice.So, if Z × Z ⊆ X × Y , let cl C s ( Z , Z ) be the smallest element of C s ( U ) which contains Z × Z .Let ( x, y ) ∈ X × Y . If there is no place to confusion, we write cl C s ( x, y ) instead of cl C s ( { x } , { y } ). Proposition 6.2.
Let M be a FIDL-module and F M be the Urquhart space associated of M .Then M is simple if and only if cl C s ( P, Q ) = X ( A ) × X ( B ) , for every ( P, Q ) ∈ X ( A ) × X ( B ) .Proof. Since M is simple if and only Con ( M ) = { (∆ A , ∆ B ) , ( ∇ A , ∇ B ) } , then by Theorem 6.1this is equivalent to C s ( F M ) = { ( ∅ , ∅ ) , ( X ( A ) , X ( B )) } and the result follows. (cid:3) Theorem 6.3.
Let M be a FIDL-module and F M be the Urquhart space associated of M . Then M is subdirectly irreducible but no simple if and only if the set J = { ( P, Q ) ∈ X ( A ) × X ( B ) : cl C s ( P, Q ) = ( X ( A ) , X ( B )) } is a non-empty open set distinct from ( X ( A ) , X ( B )) .Proof. Let us assume that M is subdirectly irreducible. Then Con ( M ) − { (∆ A , ∆ B ) } has aminimum element. From Theorem 6.1, C s ( F M ) − ( X ( A ) , X ( B )) has a maximum element. Let( Z , Z ) be such an element. Then Z and Z are non-empty. We prove that J = ( Z , Z ) − ( X ( A ) , X ( B )). On the one hand, if ( P, Q ) / ∈ ( Z , Z ), then ( Z , Z ) ⊆ ( Z , Z ) ∪ cl C s ( P, Q ). So itmust be that cl C s ( P, Q ) = ( X ( A ) , X ( B )), because if it is not the case, then ( Z , Z ) it would not bethe maximum of C s ( F M ) − ( X ( A ) , X ( B )), which is a contradiction. On the other hand, if ( P, Q ) ∈J ∩ ( Z , Z ), then cl C s ( P, Q ) = ( X ( A ) , X ( B )) = ( Z , Z ), which is absurd from assumption. Weconclude the proof by noticing that if J is a non-empty open set distinct from ( X ( A ) , X ( B )), thenit is easy to see that J − ( X ( A ) , X ( B )) is the maximum of C f ( F M ) − ( X ( A ) , X ( B )). Then theresult is an immediate consequence of Theorem 6.1. (cid:3) References [1] BLACKBURN, P.—DE RIJKE, M.—VENEMA, Y.:
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