Algebras with implication and fusion: a different point of view

This work uses well-known results on tensor products of lattices and semilattices developed by Fraser and Grätzer et al., and the duality for bounded distributive lattices introduced by Cignoli et al., in order to develop dual categorical equivalences involving bounded distributive lattices with fusion and implication, respectively. We show that these equivalences are essentially those developed by Cabrer and Celani as part of the PhD thesis of the former.


Introduction
In recent years, many varieties of algebras associated with many-valued logics have been introduced. Most of these algebras are commutative, integral, bounded distributive residuated lattices ( [14]), as for example, the variety of MV -algebras ( [6]), the variety of MTL-algebras ( [7]), the variety of ITMLalgebras ( [10]), the variety of bounded implicative lattices ( [16]), and the variety of WH -algebras ( [5]). All these algebras are bounded distributive lattices with two additional binary operations (a fusion and an implication) satisfying special additional conditions.
During the decades of the 70s and 80s, many authors studied tensor products of join-semilattices with zero ( [11,12]; see also [13] and references therein). In 1991, Cignoli et al. [8] proposed a representation theory for the category of bounded distributive lattices and zero-preserving join-homomorphisms.
Independently of [8], Cabrer and Celani introduced in [4] and [3] a categorical duality for algebras with fusion and implications. In [2], the author compares both dualities, looking at the latter as an extension of the former.
The aim of this paper is to develop the duality of Cabrer and Celani directly from [8]. This will be done on the basis of the following known results: (i) the tensor product of join-semilattices defines a monoidal structure on the category of join-semilattices with zero; Functor i 1 is the inclusion of the category of bounded distributive lattices and zero-preserving join-homomorphisms into the category of join-semilattices with zero; i 2 is the inclusion of the category of bounded distributive lattices into the category of bounded distributive lattices and zero-preserving joinhomomorphisms, and i 3 is the inclusion of the category of Priestley spaces into the category of spaces defined in [8]. Both i 2 and i 3 are surjective on objects and i 1 is full. Hence, a monoidal structure is inherited by J. Priestley duality is clearly monoidal, when we consider coproduct in the category of bounded distributive lattices and product in the category of Priestley spaces.
The main point to be observed is that the duality of Cignoli et al. is also monoidal, when we consider the monoidal structure inherited by J and the categorical product in P, as we shall show further on. Since J is monoidal, fusions and implications can be seen as objects of appropriate subcategories of the category of morphisms of J. Hence, they pass through the duality of Cignoli et al. to certain relations on Priestley spaces.
The paper is structured as follows. In Section 2, we recall some basic results and definitions about monoidal categories. In Section 3, the results from [8] relevant for the present work are recalled. In Section 4, we recall from [11,12,13] the results about tensor product of semilattices we shall need further on. In Section 5, appropriate subcategories of the category of morphisms of J are introduced, in order to represent in J fusions and implications. In Section 6, we study the monoidal structure of the duality of [8]. In Section 7, we see how fusions and implications pass through the duality. Section 8 is by far the longest and most technical, and is devoted to the comparison of our results with those of Cabrer and Celani. In the last section, we exemplify with the properties of associativity and commutativity of fusion, how diagramatic properties of the operations translate through the duality to certain conditions on the associated relations.

Monoidal categories
In this section, we shall recall from [15] the basic definitions and results on monoidal categories used in this article.
A monoidal category M is one with a bifunctor : M × M → M that is associative up to natural isomorphisms α : a (b c) → (a b) c, and equipped with an object e, which is a unit up to natural isomorphisms λ : e a → a and ρ : a e → a. These arrows must also make the following diagrams commute: I  I  I  I  I  I  I  I Note that the commutativity of the second diagram in (2.2) is equivalent to equality of the morphisms ρ and λ.
A monoidal category (M, , e) is said to be symmetric if there are natural isomorphisms σ : a b → b a making appropriate diagrams commute.
The category of all vector spaces over a fixed field F with the usual tensor product ⊗ F as and the one-dimensional vector space F as unit is a standard example of a (symmetric) monoidal category. Other standard examples of (symmetric) monoidal categories are given by any category C having finite (co)products, with the categorical (co)product as and the (initial) terminal object as e.
A (strong) monoidal functor between monoidal categories M = (M, , e) and M = (M , , e ) is a functor F : M → M , together with natural isomor- and an isomorphism θ : e → F (e) in M , making the following diagrams in M commute: .
The composition of monoidal functors is monoidal. A monoid in a monoidal category M is an object m of M together with two arrows μ : m m → m and ν : e → m such that the following diagrams commute: A relation R is said to be a Priestley relation provided the following conditions are satisfied: We write P for the category whose objects are Priestley spaces and whose morphisms are Priestley relations. If j : L → M is a morphism in J, then X(j) ⊆ X(M ) × X(L), given by is a morphism in P. Conversely, if R ⊆ X × Y is a morphism in P then the function D(R): D(Y ) → D(X) is a morphism in J. It was proved in [8] that there exists a dual categorical equivalence between J and P. We shall refer to this equivalence as CLP duality. A Priestley relation is said to be functional in the case that dom(R) = X and R(x) has a greatest element for each x ∈ X. If h : X → Y is a continuous and monotone function between Priestley spaces, . Hence, the category of Priestley spaces may be seen as a subcategory of P.
Since CLP duality extends Priestley's, we shall use the same notation for the former; i.e, we also write X : J P op : D for the functors that realize the adjunction, and ϕ and for the corresponding isomorphisms.

Tensor product
Write S 0 for the category of join-semilattices with zero. If A, B, and C are objects in S 0 , a function f : A × B → C is said to be a bimorphism if the functions g a : B → C and h b : A → C defined by g a (b) := f (a, b) and h b (a) := f (a, b) are morphisms in S 0 for all a ∈ A and b ∈ B. A joinsemilattice with zero T is a tensor product in S 0 of A and B if there is a bimorphism f : A × B → T such that for any join-semilattice with zero C and any bimorphism g : A × B → C there is a unique morphism h : T → C in S 0 satisfying g = hf . Note that T is generated by f (A × B). The tensor product of A and B in the category S 0 is often written as A ⊗ B and the image of (a, b) under the canonical bimorphism f : Let A and B be bounded distributive lattices. It is well known that in this case, A ⊗ B is a bounded distributive lattice ( [12,Lemma 3.15]) and it is isomorphic as a lattice to the coproduct in D of A and B, A * B. If F 1 ∈ X(A) and F 2 ∈ X(B), we write F 1,2 for the set Similarly to the case of join-semilattices studied in [11] (see Theorem 3.5), it can be seen that the following theorem holds. The following lemma is a part of the folklore of Priestley spaces.

(i) Let X be a Priestley space, and let A be an open upset of
Finally the following result will be central in what follows.
Proof. First note that j ∈ J. We define k : The universal property of the tensor product [13] allows us to claim that (S 0 , ⊗, 2) is monoidal. Here, 2 is the total order in two elements. Since J is a full subcategory of S 0 and A ⊗ B is in J whenever A and B are, J inherits the monoidal structure from S 0 . Hence, (J, ⊗, 2) is monoidal (in fact, symmetric monoidal).

Some categories of interest
Let us start this section by characterizing the monoids in J. Recall from [4] that a bounded distributive lattice with fusion (or DLF for short) is an algebra (L, ∧, ∨, •, 0, 1), where (L, ∧, ∨, 0, 1) is a bounded distributive lattice, • is a binary operation on L, and the following conditions hold: Lemma 5.1. Let L ∈ J and let • be a binary operation. The following are equivalent: : L → L are join-homomorphisms, by (a), (b) and (c) above, f is a bimorphism. Hence, by the universal property of the tensor product, there exists an unique morphismf : Then, straightforward computations show that • satisfies (a), (b) and (c) in the definition of a DLF.
Let f : L ⊗ L → L be a morphism in J. The proof of the above lemma shows that we can associate a DLF with f by defining on L a binary operation a • b := f (a ⊗ b). What properties must have • in order to get a monoid in J?
The commutativity of the first diagram in (2.5) reads as Applying both sides of this equality to a basic tensor Hence, the commutativity of this diagram just says that • is associative. For the second diagram in (2.5), note that a map ν : 2 → L in J is completely determined by ν(1) ∈ L. Then we can think of ν as a constant e = ν(1) ∈ L. Hence, the commutativity of the second diagram in (2.5) said that for every a ∈ L, e • a = a = a • e.
It follows that we have proved the following result.
Although Mon J is an interesting category, it is not general enough for the purpose of representing most algebras of interest in non-classical logic. In this work, we shall consider the following categories.
We write DB for the category whose objects are the morphisms of J of the form f : L ⊗ M → N , for L, M and N in J, and is such that of arrows in D that makes the following diagram commute, This is defined as the category whose objects are triples (A, f, B), with A ∈ A, B ∈ B, and f : F A → GB ∈ C, and whose morphisms from commutes. Composition is defined componentwise.
Note that DB could also be presented as the comma category (− ⊗ − ↓ i 2 ), where − ⊗ −: D × D → J is the semilattice tensor product and i 2 : D → J is the embedding of the category of distributive lattices in J.
We write DF for the full subcategory of DB whose objects are of the form It follows from Lemma 5.1 that category DF is categorically equivalent to the category of DLFs; so in the future, we shall identify both categories.
Remark 5.4. Let us observe that since every object L in J is a lattice, we implicitly have for every L, a monoid (L, inf, 1) in J given on basic tensors by We write DI for the full subcategory of DB whose objects are of the form Recall from [4] that a bounded distributive lattice with implication (or DLI for short) is an algebra (L, ∧, ∨, →, 0, 1) where (L, ∧, ∨, 0, 1) is a bounded distributive lattice, → is a binary operation on L, and the following conditions hold: It can be seen that the category DI is categorically equivalent to the category of DLIs; so in the future, we shall identify both categories.
Remark 5.5. To be more general, we could have taken categories whose objects are the operations of the form f : L ε1 This allows us to represent operations on L of the sort studied in [17].
For the sake of simplicity, in this article, we restrict ourselves to the categories DF and DI. To be consistent with this notation, we write SR for the composition of relations R : X → Y and S : Y → Z. Note that SR ⊆ X × Z. The facts that X : J P op : D acts on objects as Priestley duality does and that there is an isomorphism between A⊗B and A * B imply the commutativity of diagrams (2.3) and (2.4). Hence, CLP is a monoidal dual equivalence; i.e, the monoidal structures of J and P are preserved by D and X. In what follows, we shall take advantage of this fact to develop representation theories for the operations on lattices introduced in Section 5.

Dualizing operations
Since X is monoidal, diagram (7.2) commutes if and only if the following diagram does: The above observation suggests the following definition.
Definition 7.1. We write PB for the category whose object are relations R ⊆ X × (Y × Z) ∈ P, and whose morphisms from are the triples of Priestley spaces morphisms α := (α 1 , α 2 , α 3 ), with α 1 : X → X , α 2 : Y → Y , and α 3 : Z → Z , making the following diagram commute in P: Note that PB could also be presented as the comma category (i P ↓ − × −), where − × −: P × P → P is the product in P, seen as a functor, and i P : P → P is the identity functor.
The commutativity of diagram (7.4) can be also stated as in the following remark.

Remark 7.2.
For every x ∈ X , U ∈ D(Y ), and V ∈ D(Z), Indeed, straightforward computations show that condition (7.5) is equivalent to the commutativity of the following diagram in J: On the other hand, the commutativity of previous diagram is equivalent to the commutativity of diagram (7.4).
The following result is immediate from the definition of PB.

Theorem 7.3. There is a dual categorical equivalence between categories DB and PB.
We shall still use the letters X and D to denote the functors that realize the duality of Theorem 7.3.
We have already mentioned that since each object L of J is a lattice, also the dual lattice L op is an object of J. Definition 7.4. We write PF for the subcategory of PB whose object are relations R ⊆ X × (X × X) ∈ P, and whose morphisms are of the form α := (α, α, α). We write PI for the subcategory of PB whose object are relations R ⊆ X op × (X × X op ) ∈ P, and whose morphisms are of the form α := (α, α, α).
Note that the duality of Theorem 7.3 restricts to dualities between categories DF and PF and between categories DI and PI.
Also observe that, associated with any object R ⊆ X × (X × X) ∈ PF, we have a relational Priestley space (X, R), in the sense of [17], and similarly, to any R ⊆ X op × (X × X op ) ∈ PI. We call relational spaces with fusion (RSF for short) and relational spaces with implication (RSI for short) the just mentioned ordered topological spaces associated with the objects of PF and PI, respectively.

Relation with other representation theories
In this section, we shall give explicit descriptions of the ternary relations that appeared in the previous section, and compare the equivalences mentioned before with those developed in [2], [4], and [3].
Let L ∈ J; let R ⊆ X(L op ) × X(L ⊗ L op ) be a morphism in P, and let R be defined as in (7.
Observe that if R is a morphism in P, then R is a morphism in P.
Let R ⊆ X op × (X × X op ) ∈ P. We write R : D(X) ⊗ D(X op ) → D(X op ) for the morphism in J given by A direct computation proves that ζ R is a morphism in J. For U, V ∈ D(X), consider the binary relation in D(X): The following result can then be checked.

Proposition 8.1. If (X, R) is an RSI, then (D(X), ⇒ R ) is a DLI.
Another equivalence for the category of DLIs was introduced in [2] and [4]. We shall recall, for completeness, the basic definitions and relevant results of the mentioned articles.
Let (L, →) be a DLI. If F and G are filters of L, we define Let T L ⊆ X(L) × X(L) × X(L) be the ternary relation defined as Definition 8.2. If X is a Priestley space, T is a ternary relation in X, and U, V ∈ D(X), then we define the set Let X be a Priestley space, and let T be a ternary relation in X. A structure (X, T ) is a DLI-space provided that the following conditions hold.
We write DLI for the above-mentioned category.
There is a dual categorical equivalence between this category and the category of DLIs. Since PI is also dually equivalent to the category of DLIs, it follows that DLI and PI are equivalent. Let us make this equivalence explicit.
Let X be a Priestley space.
Let us now state some useful technical lemmata.
Similarly, we can get the following result.
Proof. Note that for every x, y, z ∈ X, we have that ( X (x), ( X (y), X (z))) ∈ (X(ζ R )) if and only if for every , and by hypothesis, we conclude that z ∈ B k , which is a contradiction. The first observation of this proof then implies the desired result.
We can now state one of the main results of this section.

Theorem 8.6. Let X be a Priestley space. Then R ⊆ X × (X × X) is in PI if and only if (X, R ) is a DLI-space.
Proof. For the necessity, take U, V ∈ D(X). An easy computation proves that Since R ∈ PI, we have, using Proposition 8.1, that U ⇒ R V ∈ D(X), and then U → R V ∈ D(X).
To show sufficiency, we have to see that R is a morphism in the category P.
It is enough to show that for every x ∈ X, R(x) is a closed downset. Take (y, z) ∈ R(x) and (a, b) ≤ (y, z). Hence, a ≤ y and z ≤ b. Suppose that (a, b) / ∈ R(x), so (x, a, b) / ∈ R . Thus, by hypothesis, we conclude that X (x) → R X (a) X (b), and as a consequence, there exists U ∈ D(X) such that U ∈ X (x) → R X (a) and Let us now see that R(x) is closed. Take (y, z) / ∈ R(x). Hence, (x, y, z) / ∈ R . Thus, we have, by hypothesis, that X (x) → R X (y) X (z). Then there exists U ∈ D(X) such that U ∈ X (x) → R X (y) and z / ∈ U . There exits V, W ∈ D(X) such that V ⊆ W → R U , x ∈ V , and y ∈ W .
Consider the family This is a family of closed subsets of X × X op .
To see F has the finite intersection property, let (g −1 (U i )×g −1 (V c i ))∩R(x) ∈ F for i = 1, . . . , n; then using that Using that X × X op is compact, we conclude that F ∈F F = ∅. Therefore, there exist y, z ∈ X such (x, y, z) ∈ R , g(y) ∈ U for every U ∈ D(X) such that y ∈ U , whereas g(z) ∈ V c for every V ∈ D(X) such that z ∈ V c .
Suppose now that y g(y). Thus, there exists U ∈ D(X) such that y ∈ U and g(y) / ∈ U ; which is a contradiction. Then y ≤ g(y). Similarly, we prove that g(z) ≤ z .
To show sufficiency, take U, Thus, there exist y , z ∈ Y such that (g(x), y , z ) ∈ S with (y , z ) ∈ U × V c . Since g is an i-morphism, there exist y, z ∈ X such that y ≤ g(y) and f (z) ≤ z . Hence, (g(y), g(z)) ∈ U × V c and as a consequence, (y, z) ∈ R(x) ∩ (g −1 (U ) × g −1 (V c )). Thus, We have already studied implications. Let us now study fusions. Let (L, •) be a DLF. If F and G are filters of L, we define Let R L ⊆ X(L) × X(L) × X(L) be the ternary relation defined as If X is a Priestley space, T is a ternary relation in X, and U, V ∈ D(X), then we define the set (II) For every x, y, z ∈ X, if X (x) • T X (y) ⊆ X (z), then (x, y, z) ∈ T .
For example, it follows that a fusion f on L is associative if and only if the following diagram commutes: i.e., when we have the equality f (1 L ⊗ f ) = f (f ⊗ 1 L ). Employing the adjunction, diagram (9.1) becomes: .
Hence, we get that a Priestley relation R ⊆ X × (X × X) represents an associative fusion if and only if the following diagram commutes: i.e., when the equality of relations (Δ X × R)R = (R × Δ X )R holds. Here, Δ X is the identity relation for X in P, which is given by (x, y) ∈ Δ X iff x ≤ y.
As another example, we can give a condition on a Priestley relation R so that D(R) becomes a commutative fusion.
Recall that a fusion f : L ⊗ L → L is commutative if and only if fτ = f , where τ : L ⊗ L → L ⊗ L is given by Note that the definition of τ together with Theorem 2.6 of [11] implies that it is a morphism of D. Thus, its associated Priestley relation is functional [8, As a consequence, fτ = f if and only if X(f ) = X(fτ ) = X(τ )X(f ), which is equivalent to the identity Hence, a Priestley relation R ⊆ X ×(X ×X) would represent a commutative fusion on D(X) if and only if R = τ * R. (9.2) Here, τ * ⊆ (X ×X)×(X ×X) is the relation in P such that ((x, y), (z, w)) ∈ τ * iff x ≤ w and y ≤ z.
The following lemma is useful in determining whether two given Priestley relations are equal. The proof of this lemma makes use of the Prime Filter Theorem. For the convenience of the reader, we recall this theorem (see [9, p. 9] for the dual of this theorem), before the statement of the lemma. Theorem 9.1 (Prime Filter Theorem). Let L be a distributive lattice, let F be a filter in L, and let I be an ideal in L that is disjoint from F . Then there is a prime filter containing F and not intersecting I. Proof. Easy computations show the necessity.
To show sufficiency, suppose that there is x ∈ L such that g(x) h(x). Then by the Prime Filter Theorem, there exists Q ∈ X(M ) such that g(x) ∈ Q and h(x) / ∈ Q. It is clear that g −1 (Q c ) is an ideal, and that if F ( x ) is the filter generated by {x}, then F ( x ) ∩ g −1 (Q c ) = ∅, so by the Prime Filter Theorem again there exists P ∈ X(L) such that x ∈ P and P ∩ g −1 (Q c ) = ∅. Thus, P ⊆ g −1 (Q). On the other hand, we have that X(g) ⊆ X(h), so P ⊆ h −1 (Q). Thus, using that x ∈ P , we conclude that h(x) ∈ Q, which is a contradiction.
The following results follow directly from Lemma 9.2 and from equation (9.2), respectively.

Conclusions
We have developed directly from [8] the duality of [2,4,3]. This was done on the basis of the following well-known results: (i) The tensor product of join-semilattices defines a monoidal structure on the category of join-semilattices with zero, see Section 4.
(ii) For any pair of bounded distributive lattices L and M , their tensor product as join-semilattices with zero is a lattice isomorphic to the coproduct of L and M in the category of bounded distributive lattices, see Section 4. (iii) The duality introduced in [8] extends Priestley duality, see Section 6.
More precisely, let DB be the comma category (− ⊗ − ↓ i 2 ), where we have that − ⊗ −: D × D → J is the semilattice tensor product and i 2 : D → J is the embedding of the category of distributive lattices in J. Let PB be the the comma category (i P ↓ − × −), where − × −: P × P → P is the product in P, seen as a functor, and i P : P → P is the identity functor. We have proved in Theorem 7.3 that there is a dual categorical equivalence between DB and PB. Write PF for the subcategory of PB whose objects are relations R ⊆ X × (X × X) ∈ P, and whose morphisms are of the form α := (α, α, α). Write PI for the subcategory of PB whose objects are relations R ⊆ X op × (X × X op ) ∈ P, and whose morphisms are of the form α := (α, α, α). The duality of Theorem 7.3 restricts to dualities between categories DF and PF and between categories DI and PI.
Associated with any object R ⊆ X × (X × X) ∈ PF we have a relational Priestley space (X, R) in the sense of [17], and similarly, with any R ⊆ X op × (X ×X op ) ∈ PI. This allows us to give, in Section 8, an alternative description of the representation theory studied in [2].
Since the duality CLP is monoidal, properties characterized by the commutativity of certain diagrams in J involving the monoidal bifunctor can be translated to commutative diagrams in P. We do this in Section 9.
Some questions still remain open. For example, it would be interesting to know if other Priestley type dualities are monoidal; and if so, try to adapt the formalism of this work to these dualities. It would also be interesting to know if the duality used in this article, and studied in [8], is natural in the sense of [9]; and if it is so, if our representation can be piggybacked from it. This last question was suggested to us by the anonymous referee, but we have no answer for it at the present.