Fuzzy Concept Lattice is Made by Proto-Fuzzy Concepts

An L-fuzzy context is a triple consisting of a set of objects, a set of attributes and an L-fuzzy binary relation between them. An l-cut is a classical context over the same sets with relation as a set of all object attribute pairs, which fuzzy relation assigns truth degree geater or equal than l. Proto-fuzzy concept is a triple made of a set of objects and a set of attributes, which form a concept in some cut of L-fuzzy context and a supremum of all degrees in which cuts this concept exists. Aim of the paper is to show the connection of the structure of proto-fuzzy concepts and fuzzy concept lattice constructed in the way of [1][5]. This connection can help to generate of all fuzzy concepts. Keywords— formal concept analysis, fuzzy concept lattice, fuzzy Galois connection 1 Preliminaries Basic notions of Formal Concept Analysis(FCA) are formal context and formal concept. Definition 1 A formal context 〈B,A,R〉 consists of a set of objects B, a set of attributes A and a relation R between B and A. Definition 2 Define the mappings ↑: 2 → 2 and ↓: 2 → 2. The first assigns to the set X ⊆ B the set of all attributes common to all objects of the set X ↑ (X) = {a ∈ A : (∀o ∈ X)(o, a) ∈ R} and the second assigns to the set Y ⊆ A the set of all objects common to all attributes of the set Y ↓ (Y ) = {o ∈ B : (∀a ∈ Y )(o, a) ∈ R}. Definition 3 A formal concept of the context 〈B,A,R〉 is a pair 〈X,Y 〉 such that X ⊆ B, Y ⊆ A, ↑ (X) = Y and ↓ (Y ) = X . Ganter and Wille in [3] showed that the pair of mappings ( ↑, ↓ ) is a Galois connection and the composite mappings ↑↓: 2 → 2 and ↓↑: 2 → 2 are closure operators. Authors prooved an important theorem in FCA well known as The Basic Theorem On Concept Lattices. Theorem 1 (The Basic Theorem on Concept Lattices)The Concept Lattice (lattice of concepts with ordering 〈X1, Y1〉 ≤ 〈X2, Y2〉 iff X1 ⊆ X2 iff Y1 ⊇ Y2) is a complete lattice in which infimum and supremum are given by ∧ i∈I 〈Xi, Yi〉 = 〈⋂ i∈I Xi, ↑↓ (⋃ i∈I Yi )〉 ∨ i∈I 〈Xi, Yi〉 = 〈 ↓↑ (⋃ i∈I Xi ) , ⋂ i∈I Yi 〉 . A complete lattice V is isomorphic to the concept lattice of some context 〈B,A,R〉 if and only if there are mappings β : B → V and α : A → V , such that β(B) is supremumdense in V and α(A) is infimum-dense in V and (o, a) ∈ R is equivalent to β(o) ≤ α(a) for all o ∈ B and a ∈ A. In particular V is isomorphic to the concept lattice of context 〈V, V,≤〉. Bělohlávek and Krajči in [1, 2, 5, 6] showed that above mentioned basic notions may be generalized by applying the fuzzy logic. Everybody knows that reality provides situations where many of attributes are rather fuzzy than crisp. Answer of question “Does the object has the attribute?” is rather somewhere in the middle of false (0) and true (1). Definition 4 An L-fuzzy formal context is a triple 〈B,A, r〉 consists of a set of objects B, a set of attributes A and an Lfuzzy binary relation r, i.e. the L-fuzzy subset of B × A or mapping from B × A to L, where L is a complete residuated lattice. The class of all L-fuzzy sets in X will be denoted by L. If L is complete then the relation ⊆ (defined by f ⊆ g iff f(x) ≤ g(x) for all x ∈ X) makes L into a complete lattice. Definition 5 A complete residuated lattice is an algebra L = 〈L,∧,∨,⊗,→, 0, 1〉 where (a) 〈L,∧,∨, 0, 1〉 is a complete lattice with the least element 0 and the greatest element 1, (b) 〈L,⊗, 1〉 is a commutative monoid, (c) ⊗ and → satisfy adjointness, i.e. a⊗ b ≤ c ⇐⇒ a ≤ b → c for each a, b, c ∈ L (≤ is the lattice ordering). Definition 6 (Bělohlávek) A triple 〈B,A, r〉 is an L-fuzzy context where r : B × A → L and L is a complete residuated lattice. Define mappings ↑: L → L and ↓: L → L such that for every f ∈ L and g ∈ L ↑ (f)(a) = ∧ o∈B ( f(o) → r(o, a)) ↓ (g)(o) = ∧ a∈A ( g(a) → r(o, a)). ISBN: 978-989-95079-6-8 IFSA-EUSFLAT 2009