Formal concept analysis (FCA) introduced by Ganter and Wille [9] has become an extremely useful theoretical and practical tool for formally describing structural and hierarchical properties of data with "object-attribute" character. Regarding applications, we can find papers ranging from ontology merging [20], to applications to the Semantic Web by using the notion of concept similarity [7], and from processing of medical records in the clinical domain [11] to the development of recommender systems [6].

Soon after the introduction of "classical" formal concept analysis, several approaches towards its generalization were introduced and, nowadays, there are recent works which extend the theory by using ideas from fuzzy set theory, or fuzzy logic reasoning, or from rough set theory, or some integrated approaches such as fuzzy and rough, or rough and domain theory [1, 15-18, 21, 22].

In this paper, we are concerned with extensions of Bělohlávek's approach. In [2,4] he provided an L-fuzzy extension of the main notions of FCA, such as context and concept, by extending its underlying interpretation on classical logic to the more general framework of L-fuzzy logic [10].

In this work, we aim at formally describing some structural properties of intercontextual relationships [8] of L-fuzzy formal contexts. The categorical treatment of morphisms as fundamental structural properties has been advocated by [14] as a means for the modelling of data translation, communication, and distributed computing, among other applications. Research on (extensions of) the theory of Chu spaces studies morphisms among contexts in order to obtain categories with certain specific properties. Previous work in this line has been developed by the authors in [12] by using category theory following the results in [19].

The main result here is the extension of the relationship between bonds and extents of direct products of contexts to the realm of L-fuzzy FCA.

In order to make this contribution as self-contained as possible, we proceed now with the preliminary definitions of complete residuated lattice, L-fuzzy context, L-fuzzy concept.

2.1 L-fuzzy concept lattice Definition 1. An algebra L, ∧, ∨, ⊗, →, 0, 1 is said to be a complete residuated lattice if 1. L, ∧, ∨, 0, 1 is a complete bounded lattice with the least element 0 and the greatest element 1, 2. L, ⊗, 1 is a commutative monoid, 3. ⊗ and → are adjoint, i.e. a⊗b ≤ c if and only if a ≤ b → c, for all a, b, c ∈ L, where ≤ is the ordering in the lattice generated from ∧ and ∨.

Now, the natural extension of the notion of context is given below.

Definition 2. Let L be a complete residuated lattice, an L-fuzzy context is a triple B, A, r consisting of a set of objects B, a set of attributes A and an Lfuzzy binary relation r, i.e. a mapping r : B × A → L, which can be alternatively understood as an L-fuzzy subset of B × A.

We now introduce the L-fuzzy extension provided by Bělohlávek [2,3], where we will use the notation Y X to refer to the set of mappings from X to Y . Definition 3. Given an L-fuzzy context B, A, r , a pair of mappings ↑ : L B → L A and ↓ : L A → L B can be defined for every f ∈ L B and g ∈ L A as follows:

Lemma 1. Let L be a complete residuated lattice, let r ∈ L B×A be an L-fuzzy relation between B and A. Then the pair of operators ↑ and ↓ form a Galois connection between L B ; ⊆ and L A ; ⊆ , that is, ↑ : L B → L A and ↓ : L A → L B are antitonic and, furthermore, for all f ∈ L B and g ∈ L A we have f ⊆ ↓↑ f and g ⊆ ↑↓ g.

Lemma 2. Under the conditions of Lemma 1, the following equalities hold for arbitrary f ∈ L B and g ∈ L A , ↑ f =↑↓↑ f and ↓ g =↓↑↓ g, that is, both ↓↑ f and ↑↓ g are closed in C.

The first component f is said to be the extent of the concept, whereas the second component g is the intent of the concept.

The set of all L-fuzzy concepts associated to a fuzzy context (B, A, r) will be denoted as L-F CL(B, A, r).

An ordering between L-fuzzy concepts is defined as follows:

The corresponding notions of negation, disjunction and complement on an Lcontext are introduced below. Definition 6. Let us consider a unary operator negation and a binary disjunction operator on the underlying structure of truth values L as follows:

Some of the properties of negation appear in the following lemma.

Lemma 3 (Bělohlávek [5]). For any a, b, c ∈ L the following holds.

From Property 6 above and the definition of disjunction, we can see that disjunction needs not be, in general, commutative. However, this property will be very important for the definition and properties of direct product of two L-contexts. Notice that commutativity will hold if the law of double negation (¬¬a = a) holds. The following result states some properties of residuated lattices satisfying double negation.

Proposition 2 (Bělohlávek [5]). If a residuated lattice satisfies the law of double negation then it also satisfies the following conditions:

It is convenient here to recall that adding conditions of our underlying residuated lattice may change the class of structures we are working with. In particular, a residuated lattice satisfying the double negation law and divisibility (that is, x ≤ y implies the existence of z such that x = y ⊗ z), we are working with an MV-algebra. If divisibility is replaced by the fact that the product ⊗ coincides with the infimum of the lattice, then we are have just a Boolean algebra.

We finish this section with a specific notion of complement of a given L-fuzzy formal context. ⊓ ⊔

The definition of L-bonds is based on a suitable extension of the theory of multifunctions (also called, many-valued functions, or correspondences) whose notation and terminology is introduced below.

The L-multifunction ϕ : X → L Y can be extended to a mapping ϕ * :

The set L-Mfn(X, Y ) of all the L-multifunctions from X to Y can be endowed with a poset structure by defining the ordering ϕ 1 ≤ ϕ 2 as ϕ 1 (x)(y) ≤ ϕ 2 (x)(y) for all x ∈ X and y ∈ Y .

The usual definition of curry and uncurry operations can be adapted to the framework of L-multifunctions as follows: Definition 9. Let us define for arbitrary L-multifunction ϕ ∈ L-Mfn(X, Y ) an L-fuzzy relation ϕ r ∈ L X×Y defined by ϕ r (x, y) = ϕ(x)(y) for all (x, y) ∈ X × Y . For arbitrary L-fuzzy relation r ∈ L X×Y lets define an L-multifunction from r mfn : X → L Y defined by r mfn (x)(y) = r(x, y). Proof. We will prove the two inequalities separately.

For the other inequality, consider the following chain

where (⋆) follows from the inequality (k → l) ⊗ (l → m) ≤ k → l which holds for all k, l, m ∈ L.

⊓ ⊔

Here we introduce the corresponding extension of the notion of direct product of two L-fuzzy contexts.

Definition 11. The direct product of two L-fuzzy contexts

The following result states properties of the just defined direct product of L-fuzzy contexts.

r 2 be two L-fuzzy contexts, where L satisfies the double negation law. Given two arbitrary L-multifunctions ϕ : B 1 → L A2 and ψ : A 2 → L B1 , for all o 1 , o 2 ∈ B 1 and a 1 , a 2 ∈ A 2 the following equalities hold

Theorem 1. Let C i = B i , A i , r i be L-fuzzy contexts for i ∈ {1, 2}, where L satisfies the double negation law. Let β ∈ L-Mfn(B 1 , A 2 ). Then:

Proof. 1. For the first item, let β be an extent of C 1 ∆C 2 , then we know that

Let us write ↑ ∆ (β r ) mfn = ψ, then

2. The proof for the second item is as follows:

where (⋆) follows, firstly, from the hypothesis, which states that it equals to β * (↓ ¬1 ↑ ¬1 (χ o1 ))(a 2 ) and, as β ∈ L-Bond(C 1 , C 2 ), by Lemma 5. ⊓ ⊔

We have introduced an adequate generalization of the study of L-bonds as morphisms among contexts, initiated in [14], by showing how the classical relationships between bonds and contexts can be lifted to a more general framework.

The contribution seems to pave the way towards determining possible categories on which to model knowledge transfer and information sharing. Other steps have been given in [12,13] where the category of L-Chu correspondences has been considered. However, much work still has to be done.

A thorough study of the properties of the extended categorical framework of Chu correspondences and L-Chu correspondences is needed, in order to identify their natural interpretation within the theory of knowledge representation.