Introduction

The triadic approach to concept analysis (TCA) was introduced by Wille and Liehman in [12]. TCA is an extension of Formal Concept Analysis; it is based on a formalization of the triadic relation connecting objects, attributes, and conditions (we recall the basic notions of TCA in Section 2). In our previous work [3], we generalized TCA for graded data (fuzzy setting). The present paper generalizes TCA even more.

(Antitone) Galois connections and concept lattices of data represented by a fuzzy relation (graded data) were studied in a series of papers, see e.g. [1,14]. An alternative approach, based on antitone Galois connections, was studied in [10,13]. The concept lattices based on the antitone and the isotone Galois connections have distinct, natural meaning. It is well known that in the ordinary (two-valued) setting, the antitone and isotone cases are mutually reducible due to the law of double negation and that such a reducibility fails in a fuzzy setting because the law of double negation is not available in fuzzy logic. Nevertheless, a framework which enables a unifying approach to both the antitone and isotone cases was recently proposed in [5,7], see also [10]. In this paper, inspired by this approach, we provide a unifying framework that enables us to treat the isotone and antitone cases within TCA. We provide mathematical foundations of the unifying framework, show its properties, and describe the way it covers the two types of concept-forming operators. We also show that an analogy of the basic theorem holds true.

The main inspiration for the presented work, in addition to developing a general framework which covers distinct particular cases, is the recent work in relational factor analysis [4,5,3,6], in which concept lattices, both the antitone and the isotone are of crucial importance because they are optimal factors for relational matrix decompositions. In particular, triadic concept lattices were shown to play the role of the space of optimal factors for factor analysis of three-way binary data in [6]. To develop a general mathematical framework that can be used for a factor analysis of ordinal (graded) data is the main goal of this paper. Proper generalization can simplify definitions (as we demonstrate in Examples and ) and open door to new extenstions -we intent to use it to generalize factor analysis of three-way graded data.

The paper is organized as follows: Section 2 recalls basic notions from (crisp) triadic concept analysis. Section 3 describes the unifying framework and its basic properties. In Section 4 we turn our attention to triadic concept-forming operators, triadic concepts. Section 5 brings an analogy to basic theorem of concept (tri)-lattices. Our conclusions and future research ideas are summarized in Section 6.

Preliminaries

Triadic Formal Concept Analysis This section introduces the notions needed in our paper. For further information we refer to [12,16] (triadic FCA).

A triadic context is a quadruple X, Y, Z, I where X, Y , and Z are nonempty sets, and I is a ternary relation between X, Y , and Z, i.e. I ⊆ X × Y × Z. X, Y , and Z are interpreted as the sets of objects, attributes, and conditions, respectively; I is interpreted as the incidence relation ("to have-under relation"). That is, x, y, z ∈ I is interpreted as: object x has attribute y under condition z. In this case, we say that x, y, z (or x, z, y, or the result of listing x, y, z in any other sequence) are related by I. For convenience, a triadic context is denoted by X 1 , X 2 , X 3 , I .

Let K = X 1 , X 2 , X 3 , I be a triadic context. For {i, j, k} = {1, 2, 3} (i.e. i, j, k ∈ {1, 2, 3}s.t.i = j = k) and C k ⊆ X k , we define a dyadic context

K ij C k = X i , X j , I ij C k by x i , x j ∈ I ij C k iff for each x k ∈ C k : x i , x j , x k are related by I.

The concept-forming operators induced by K ij C k are defined as follows:

C (ijC k ) i = {x j ∈ X j | for each x i ∈ C i : x i , x j ∈ I ij C k },

Operators (ijC k ) and (jiC k ) form a Galois connection between X i and X j [9].

A triadic concept of X 1 , X 2 , X 3 , I is a triplet C 1 , C 2 , C 3 of C 1 ⊆ X 1 , C 2 ⊆ X 2 ,

and C 3 ⊆ X 3 , such that for every {i, j, k} = {1, 2, 3} we have

C i = C (ijC k ) j ; C 1 , C 2 ,

and C 3 are called the extent, intent, and modus of C 1 , C 2 , C 3 . The set of all triadic concepts of X 1 , X 2 , X 3 , I is denoted by T (X 1 , X 2 , X 3 , I) and is called the concept trilattice of X 1 , X 2 , X 3 , I ; we refer to Section 5 where the notion of a trilattice is defined.

Fuzzy sets As a structure of truth-degrees we use a complete lattice. Given a complete lattice L, we define the usual notions [1,11]: an L-set (fuzzy set, graded set) A in a universe U is a mapping A : U → L, A(u) being interpreted as "the degree to which u belongs to A".

Let L U denote the collection of all L-sets in U . The operations with L-sets are defined componentwise. For instance, the intersection of

L-sets A, B ∈ L U is an L-set A ∩ B in U such that (A ∩ B)(u) = A(u) ∧ B(u) for each u ∈ U , etc. We write A ⊆ B iff A(u) ≤ B(u)

for each u ∈ U . Note that 2-sets and operations with 2-sets can be identified with ordinary sets and operations with ordinary sets, respectively. Binary L-relations (binary fuzzy relations) between X and Y can be thought of as L-sets in the universe X × Y ; similarly for ternary L-relations.

Unifying framework

In this section we describe the structure of truth degrees we use. In our previous work we used residuated lattices as a scale of truth degrees. Our current approach differs in that we allow the fuzzy sets which constitutes triadic concepts, and the input table to have complete lattices with common support set and dual order as their scales of truth degrees. Moreover, we define operations on the structure of truth degrees that are counterparts of operations from residuated lattices. This approach is inspired by [5,7,10].

Let L = (L, ≤) be a bounded complete lattice and for i ∈ {1, 2, 3, 4}, L i = (L i , ≤ i ) be bounded lattice with L i = L and ≤ i being either ≤ or ≤ −1 . That is, each L i is either (L, ≤) or (L, ≤ −1 ). We denote the operations on L i by adding the subscript i, e. g. the operations in L 2 are denoted by ∧ 2 , ∨ 2 , 0 2 , and 1 2 .

We consider a ternary operation :

L 1 × L 2 × L 3 → L 4 .

We assume that commutes with suprema in all arguments. That is, for any a,

a j ∈ L 1 , b, b j ∈ L 2 , c, c j ∈ L 3 we have ( 1 j∈J a j , b, c) = 4 j∈J (a j , b, c) (a, 2 j∈J b j , c) = 4 j∈J (a, b j , c)(1)

(a, b,

3 j∈J c j ) = 4 j∈J (a, b, c j )

Furthermore, for i, j, k ∈ 1, 2, 3 we define the operations i :

L j × L k × L 4 as i (a j , a k , a 4 ) = i {a i | (a i , a j , a k ) ≤ a 4 }(2)

For convenience we denote (a, b, c) also by {b, a, c} or {c, b, a} etc., and i (a i , a j , a 4 ) also by i {a j , a i , a 4 } or i {a i , a j , a 4 } Example 1. Complete residuated lattice [11] is an algebra L = L, ∧, ∨, ⊗, → , 0, 1 such that L, ∧, ∨, 0, 1 is a complete lattice with 0 and 1 being the least and greatest element of L, respectively; L, ⊗, 1 is a commutative monoid (i.e. ⊗ is commutative, associative, and a ⊗ 1 = a for each a ∈ L); ⊗ and → satisfy so-called adjointness property:

a ⊗ b ≤ c iff a ≤ b → c for each a, b, c ∈ L.

Let L = L, ∧, ∨, ⊗, →, 0, 1 be a complete residuated lattice and ≤ be its order. (1) Let L i = L j = (L, ≤) for each i, j ∈ {1, 2, 3, 4} and let (a 1 , a 2 , a 3 ) = a 1 ⊗ a 2 ⊗ a 3 . Then operations i are defined as follows:

1 (a 2 , a 3 , a 4 ) = (a 2 ⊗ a 3 ) → a 4 (3) 2 (a 3 , a 1 , a 4 ) = (a 3 ⊗ a 1 ) → a 4 (4) 3 (a 1 , a 2 , a 4 ) = (a 1 ⊗ a 2 ) → a 4(5)

(2

) Let L 1 = L 2 = L, ≤ and L 3 = L 4 = L, ≤ −1 and let (a 1 , a 2 , a 3 ) = (a 1 ⊗ a 2 ) → a 3 .

Then operations i are defined as follows

1 (a 2 , a 3 , a 4 ) = (a 4 ⊗ a 2 ) → a 3 (6) 2 (a 3 , a 1 , a 4 ) = (a 4 ⊗ a 1 ) → a 3 (7) 3 (a 1 , a 2 , a 4 ) = a 1 ⊗ a 2 ⊗ a 4(8)

We show usability of both sets of operators in Example 2.

The following lemma describes basic properties of the previously defined operations that we will need in rest of the paper.

Lemma 1.

x is monotone in all arguments. (a) (b) i are monotone in first two arguments and antitone in third argument. (c) (a 1 , a 2 , 3 (a 1 , a 2 , a 4 )) ≤ a 4 , analogous formulas hold for 1 and 2 . (d) 3 (a 1 , a 2 , (a 1 , a 2 , a 3 )) ≥ a 3 , analogous formulas hold for 1 and 2 .

Proof. (a) follows directly from (1)

(b) follows directly from (2) (c) (a 1 , a 2 , 3 (a 1 , a 2 , a 4 )) = = (a 1 , a 2 , 3 {a 3 | (a 1 , a 2 , a 3 ) ≤ a 4 }) = = 4 { (a 1 , a 2 , a 3 ) | (a 1 , a 2 , a 3 ) ≤ a 4 } ≤ a 4 (d) 3 (a 1 , a 2 , (a 1 , a 2 , a 3 )) = = 3 {x 3 | (a 1 , a 2 , x 3 ) ≤ (a 1 , a 2 , a 3 )} ≥ a 3 4 Triadic context, concept-

forming operators, and concepts

In this section we develop the basic notions of the general approach to triadic concept analysis. We define the notions of L-context, concept-forming operators and triadic concepts in our setting and investigate their properties. Triadic L-context is a quadruple X, Y, Z, I where X, Y , Z are non-empty sets interpreted as sets of objects, attributes, and conditions, respectively. I is a ternary L-relation between X, Y and Z, i.e.: I : X × Y × Z → L 4 . For every x ∈ X, y ∈ Y , and z ∈ Z, the degree I(x, y, z) in which are x,y, and z related is interpreted as the degree to which object x has attribute y under condition z. For convenience, we denote I(x, y, z) also by I{x, y, z} or I{x, z, y} or I{z, x, y}, and the triadic L-context by X 1 , X 2 , X 3 , I .

L-context K = X 1 , X 2 , X 3 , I induces three concept-forming operators. For {i, j, k} = {1, 2, 3} and the sets

A i ∈ L Xi and A k ∈ L X k , the concept-forming operator is a map: L i × L k × L 4 → L j which assigns to A i and A k a fuzzy set A j ∈ L Xj defined by A j (x j ) = j xi∈Xi x k ∈X k j {A i (x i ), A k (x k ), I{x i , x j , x k }}.(9)

In this case, the concept-forming operator is denoted by (ijA k ) , i.e. fuzzy set A j is denoted by

A j = A (ijA k ) i . Example 2. (1)

Let L i and be as in Example 1 (1). Then the concept-forming operators are as follows

A (ijA k ) i (x j ) = xi∈Xi x k ∈X k (A i (x i ) ⊗ A k (x k )) → I(x 1 , x 2 , x 3 )(10)

for {i, j, k} ∈ {1, 2, 3}. Note that these operators are fuzzy generalizations of those described in Section 2. These concept-forming operators also appear in [8].

(2) Let L i and be as in Example 1 (2). Then the concept-forming operators are defined as follows:

A (12A3) 1 (x 2 ) = x1∈X1 x3∈X3 (I(x 1 , x 2 , x 3 ) ⊗ A 1 (x 1 )) → A 3 (x 3 )(11)A (23A1) 2 (x 3 ) = x1∈X1 x2∈X2 (I(x 1 , x 2 , x 3 ) ⊗ A 2 (x 2 )) → A 1 (x 1 )(12)A (31A2) 3 (x 1 ) = x2∈X2 x3∈X3 (I(x 1 , x 2 , x 3 ) ⊗ A 1 (x 1 ) ⊗ A 3 (x 3 ))(13)

Note that operators ( 12)-( 13) are selected as a triadic counterpart to (dyadic) isotone galois connections [10]. Formulas ( 12)-( 13) are rather complicated in comparisson with the general definition (9).

A triadic fuzzy concept of X 1 , X 2 , X 3 , I is a triplet C 1 , C 2 , C 3 consisting of fuzzy sets C 1 ∈ L X1 1 , C 2 ∈ L X2

2 , and C 3 ∈ L X3 3 , such that for every {i, j, k} = {1, 2, 3} we have

C i = C (ijC k ) j , C j = C (jkCi) k

, and

C k = C (ikCj ) i . The C 1 , C 2 ,

and C 3 are called the extent, intent, and modus of C 1 , C 2 , C 3 . The set of all triadic concepts of K = X 1 , X 2 , X 3 , I is denoted by T (X 1 , X 2 , X 3 , I) and is called the concept trilattice of K.

We view the triadic concepts as triplets of fuzzy sets of objects, attributes, and modi. That is, a concept applies to objects to degrees; similarly for attributes and conditions. In our setting, the scales of truth degrees in which objects belong to extent, attributes belong to intent, and conditions belongs to modus are complete lattices which consists of common support set, but they may be ordered dually.

The following lemma describes basic properties of concept-forming operators.

Lemma 2. (a) A

(ijC k ) i = C (kjAi) k (b) if C k ⊆ D k and A i ⊆ B i then B (ijD k ) i ⊆ A (ijC k ) i (c) A i ⊆ (A (ijA k ) i ) (jiA k )

Proof. (a)

A (ijC k ) i (x j ) = j xi∈Xi x k ∈X k j (A i (x i ), C k (x k ), I{x i , x k , x j }) = C (kjAi) k (x j ) (b) B (ijD k ) i (x j ) = j xi∈Xi x k ∈X k j (B i (x i ), D k (x k ), I{x i , x k , x j }) ≤ ≤ j xi∈Xi x k ∈X k j (A i (x i ), C k (x k ), I{x i , x k , x j }) = A (ijC k ) i (c) (A (ijA k ) i ) (jiA k ) (x i ) = = i xj ∈Xj x k ∈X k i ( j x ′ i ∈Xi x ′ k ∈X k j (A i (x ′ i ), A k (x ′ k ), I{x ′ i , x ′ k , x j }), A k (x k ), I{x i , x j , x k }) ≥ ≥ i xj ∈Xj x k ∈X k i ( j (A i (x i ), A k (x k ), I{x i , x k , x j }), A k (x k ), I{x i , x j , x k }) = = i xj ∈Xj x k ∈X k i {a i | ( j (A i (x i ), A k (x k ), I{x i , x k , x j }), x k , a i ) ≤ I{x i , x k , x j }}

Lemma 1(c) yields that one of the possible values of a i is A i (x i ). Therefore, the previous formula is greater than i xj ∈Xj

x k ∈X k A i (x i ) = A i (x i ) which concludes the proof.

Theorem 1. Let {i, j, k} = {1, 2, 3}. Then for all triadic fuzzy concepts A 1 , A 2 , A 3 and

B 1 , B 2 , B 3 from T (K), if A 1 , A 2 , A 3 i B 1 , B 2 , B 3 and A 1 , A 2 , A 3 j B 1 , B 2 , B 3 then B 1 , B 2 , B 3 k A 1 , A 2 , A 3 . Proof. We have A k = A (ikAj ) i and B k = B (ikBj ) i . Since A i ⊆ B i and A j ⊆ B j , Lemma 2 yields B k ⊆ A k .

The following theorem describes a way how to compute a triadic concept. Starting with two fuzzy sets C i ∈ L Xi i and C k ∈ L X k k we obtain a triadic concept A 1 , A 2 , A 3 by three projections using the concept-forming operators. Firstly we project C i and C k onto A j , then we project A j and C k onto A i , and finally we project A i and

A j onto A k Theorem 2. For C i ∈ L Xi i , C k ∈ L X k k with {i, j, k} = {1, 2, 3}, let A j = C (ijC k ) i , A i = A (jiC k ) j

, and

A k = A (ikAj ) i . Then A 1 , A 2 , A 3 is a triadic fuzzy concept b ik (C i , C k ).

Moreover, A 1 , A 2 , A 3 has the smallest k-th component among all triadic fuzzy concepts B 1 , B 2 , B 3 with the greatest j-th component satisfying

C i ⊆ B i and X = C k ⊆ B k . In particular, b ik (A i , A k ) = A 1 , A 2 , A 3 for each triadic fuzzy concept A 1 , A 2 , A 3 . Proof. By lemma 2(c) we have C i ⊆ A i and since A k = A (ikAj ) i = (A (jiC k ) j ) (ikAj ) = (C (kiAj ) k ) (ikAj ) we have also C k ⊆ A k . First, we prove that A 1 , A 2 , A 3 is a triadic fuzzy concept. A k = A (ikAj ) i is satisfied by definition. Consider A j . We have A j = C (ijC k ) i ⊇ A (ijA k ) i (Lemma 2(b)) and A j ⊆ (A (jkAi) j ) (kjAi) = A (kjAi) k = A (ijA k ) i . Therefore, A j = A ijA k i . The proof for A i is similar. Let B 1 , B 2 , B 3 be a triadic fuzzy concept with X i ⊆ B i and X k ⊆ B k . Then B j = B (ijB k ) i ⊆ X (ijX k ) i , so the maximal j-th component is A j . Let B j = A j . Then A i = A (ijX k ) j ⊇ B (jiB k ) j = B i and thus A k = A (ikAj ) i ⊆ B (ikBj ) i = B k .

The last assertion is easily observable from the definition of triadic fuzzy concept.

In the rest of the paper we need the following notation. For fuzzy sets

A 1 ∈ L X1 1 , A 2 ∈ L X2 2 , and A 3 ∈ L X3 3 we denote by A 1 × A 2 × A 3 the ternary L 4 -relation between X 1 , X 2 , and X 3 defined by (A 1 × A 2 × A 3 )(x 1 , x 2 , x 3 ) = (A 1 (x 1 ), A 2 (x 2 ), A 3 (x 3 )).

The following lemma describes a "geometric view" on triadic fuzzy concepts, i.e. that triadic fuzzy concepts can be viewed as maximal clusters contained in the input data.

Lemma 3. (a) IfA 1 , A 2 , A 3 ∈ T (K) then A 1 × A 2 × A 3 ⊆ I. (b) If A 1 × A 2 × A 3 ⊆ I then there is B 1 , B 2 , B 3 ∈ T (K) such that A i ⊆ B i for i = 1, 2, 3. (c) Each A 1 , A 2 , A 3 ∈ T (K) is maximal w.r.t. to set inclusion, i.e. there is no B 1 , B 2 , B 3 ∈ T (K) other than A 1 , A 2 , A 3 for which A i ⊆ B i .

Proof. (a)

(A i (x i ), A j (x j ), k xi∈Xi x k ∈X k (A i (x i ), A j (x j ), I{x i , x j , x k }) ≤ ≤ (A i (x i ), A j (x j ), k (A i (x i ), A j (x j ), I{x i , x j , x k })) ≤ ≤ I{x i , x j , x k } (b) Let {i, j, k} = {1, 2, 3} and b ik (A i , A k ) = B 1 , B 2 , B 3 . Due Theorem 2 we have A i ⊆ B i and A k ⊆ B k . Moreover, B j (x j ) = A (ijA k ) i (x j ) = = j xi∈Xi x k ∈X k j (A i (x i ), A k (x k ), I{x i , x k , x j }) ≥ ≥ j xi∈Xi x k ∈X k j (A i (x i ), A k (x k ), (A i (x i ), A k (x k ), A j (x j ))) = = A j (x j ) (c) Let A 1 , A 2 , A 3 and B 1 , B 2 , B 3 be triadic concepts with A i ⊆ B i and k ∈ {1,2

, 3} be an index such that A j ⊂ B j . From Theorem 1 follows that there is an index j ∈ {1, 2, 3} such that

A k = B k . Then having A j = A (ijA k ) i B j = B (ijB k ) i

Lemma 2(c) yields B j ⊆ A j which is a contradiction.

Theorem 3 (crisp representation). Let K = X 1 , X 2 , X 3 , I be a fuzzy triadic context and

K crisp = X 1 ×L 1 , X 2 ×L 2 , X 3 ×L 3 , I crisp with I crisp defined by ((x 1 , a), (x 2 , b), (x 3 , c)) ∈ I crisp iff (a, b, c) ≤ 4 I(x 1 , x 2 , x 3 ) be a triadic context. Then T (K) is isomorphic to T (K crisp ).

Proof. Define maps ⌊...⌋ i : L Xi → X i × L and ⌈...⌉ i : X i × L → L Xi for i ∈ {1, 2, 3} as follows:

⌊A i ⌋ i = {(x i , a i ) | a i ≤ i A i (x i )} (14) ⌈A ′ i ⌉ i = i {a i | (x i , a i ) ∈ A ′ i }(15)

In what follows we skip subscripts and write just ⌊A i ⌋ and

⌈A ′ i ⌉ instead of ⌊A i ⌋ i and ⌈A ′ i ⌉ i . Let ϕ be a mapping ϕ : T (K) → T (K crisp ) defined by ϕ( A 1 , A 2 , A 3 ) = ⌊A 1 ⌋, ⌊A 2 ⌋, ⌊A 3 ⌋ . We show, that ϕ( A 1 , A 2 , A 3 ) ∈ T (K crisp ). We have (x i , b) ∈ (⌊A j ⌋ (ji⌊A k ⌋) iff for each ((x j , a), (x k , c)) ∈ ⌊A j ⌋ × ⌊A k ⌋ it holds that ((x i , b), (x j , a), (x k , c)) ∈ I crisp iff for each x j ∈ X j , x k ∈ X k , and for each a ≤ j A j (x j ), b ≤ k A k (x k ) it holds (a, b, c) ≤ 4 I{x i , x j , x k } iff for each x j ∈ X j , x k ∈ X k we have (A j (x j ), A k (x k ), b) ≤ 4 I{x i , x j , x k } iff b ≤ i A i (x i ), therefore (⌊A j ⌋ (ji⌊A k ⌋) × ⌊A k ⌋) i = ⌊A i ⌋.

Let ψ be a mapping ψ : T (K crisp ) → T (K) defined by ψ( A 1 , A 2 , A 3 ) = ⌈A 1 ⌉, ⌈A 2 ⌉, ⌈A 3 ⌉ . We show, that ψ( A 1 , A 2 , A 3 ) ∈ T (K).

We have (⌈A j ⌉ (ji⌈A k ⌉) (x i ) = b iff b is the maximal degree with the property that for each x j ∈ X j , x k ∈ X k it holds (⌈A j ⌉(x j ), ⌈A k ⌉(x j ), b) ≤ 4 I(x i , x j , x k ) iff b is the maximal degree with the property that for each a ≤ i b and each x j ∈ X j , x k ∈ X k we have (⌈A j ⌉(x j ), ⌈A k ⌉(x j ), a) ≤ 4 I(x i , x j , x k ) iff b is the maximal degree with the property that for each a ≤ i b and each ((x j , c), (x k , d)) ∈ A j × A k we have that ((x i , a), (x j , c), (x k , d)) ∈ I crisp iff b is the maximal degree with the property that for each a ≤ b we have (x i , a) ∈ A

(jiA k ) j = A i . Therefore ⌈A j ⌉ (ji⌈A k ⌉) = ⌈A i ⌉.

Since ⌈⌊A⌋⌉ = A for each fuzzy set A, the mappings ϕ and ψ are mutually inverse and ϕ is a bijection. Moreover, ⌊A⌋ ⊆ ⌊B⌋ iff A ⊆ i B for all fuzzy sets A and B and thus ϕ preserves 1 , 2 , 3 .

Basic theorem

In this section, we define important structural relations on the set of triadic concepts. These relations are based on the subsethood relations on the sets of objects, attributes, and modi, and are fundamental for an understanding of the structure of the set of all triadic concepts. In the final part of this section, we prove a theorem which is a generalization of the basic theorem of triadic concept analysis [16].

Consider the following relations

A 1 , A 2 , A 3 i B 1 , B 2 , B 3 iff A i ⊆ B i , A 1 , A 2 , A 3 i B 1 , B 2 , B 3 iff A i = B i .

It is easy to check that i and i are a quasiorder and an equivalence on T (K). Denote by T (K)/ i the corresponding factor set with equivalence classes denoted by

[ A 1 , A 2 , A 3 ] i . Letting [ A 1 , A 2 , A 3 ] i i [ B 1 , B 2 , B 3 ] i iff A 1 , A 2 , A 3 i B 1 , B 2 , B 3 , i is an order on T (K)/ i .

Let V be a non-empty set, and for i ∈ {1, 2, 3} let i be quasiorder relations on V . Then we call (V, 1 , 2 , 3 ) a triordered set if and only if it holds that v i w and v j w implies w k v for {i, j, k} = {1, 2, 3} and each v, w ∈ V and

∼ i ∩ ∼ j ∩ ∼ k (∼ i = i ∩ i ) is an identity relation. Clearly, ∼ i = i ∩ i is an equivalence, and ∼ i ∩ ∼ j is an identity relation on V . Moreover, ∼ i turns i into an ordering on V / ∼ i and so (V / ∼ i , i ) is an ordered set. An element v ∈ V is an ik-bound of (V i , V k ), V i , V k ⊆ V , if x i v for all x ∈ V i and x k v for all x ∈ V k . An ik-bound v is called an ik-limit of (V i , V k ) if u j v for all ik-bounds of (V 1 , V 2 ) u.

In an triordered set (V, 1 , 2 , 3 ) there is at most one ik-limit of (V 1 , V 2 ) v with a property u k v for all ik-limits of (V 1 , V 2 ) u. Then we call v an ik-join of (V i , V k ) and denote it ∇ ik (V i , V k ). The triordered set (V, 1 , 2 , 3 ) in which the ik-join exists for all i = k and all pairs of subsets of V is a complete trilattice.

For a complete trilattice V = (V, 1 , 2 , 3 ), an order filter F i on ordered set V / ∼ i is defined as a subset F i of V with the property: x ∈ F i and x i y implies y ∈ F i for all x, y ∈ V . We denote the set of all order filters on V / ∼ i by

F i (V). A principal filter generated by x ∈ V is the filter [X) i = {y ∈ V | x i y}.

We call a subset X ∈ F i (V) of filters i-dense with respect to V if each principal filter of (V, / ∼ i ) can be obtained as an intersection of some order filters from X .

It is easy to see that T (K) is a triordered set. Let κ i :

X i × L i → T (K) be a mapping defined by κ i (x i , b) = { A 1 , A 2 , A 3 ∈ T (K)|A i (x i ) ≥ i b} for i ∈ {1, 2, 3}, x i ∈ X i and b ∈ L. Since the principal filter generated by A 1 , A 2 , A 3 is [ A 1 , A 2 , A 3 ) i = ∩ xi∈Xi κ i (x i , A i (x i )), the set κ i (X i × L i ) is i-dense. Moreover, κ i happens to satisfy κ i (x i , a) ⊆ κ i (x i , b) iff b ≤ i a.

Theorem 4 (basic theorem). Let K = (X 1 , X 2 , X 3 , I) be a fuzzy triadic context. Then T (K) is a complete trilattice of K for which the ik-joins are defined as follows:

∇ ik (X i , X j ) = b ik {A i | A 1 , A 2 , A 3 ∈ X i }, {A k | A 1 , A 2 , A 3 ∈ X k } .

A complete trilattice V = (V, 1 , 2 , 3 ) is isomorphic to T (K) if and only if there are mappings κi : X i × L i → F i (V), i = 1, 2, 3, such that Proof. The first assertion follows from Theorem 2.

From Theorem 3 we know that T (K) is isomorphic to T (K crisp ). To prove our assertion it suffices to show that conditions (a),(b), and (c) (for T (K)) are equivalent with the conditions from Wille's original basic theorem (for T (K crisp )).

Consider the map κi w : (X i ×L i ) → F i (V) defined by κi w ((x i , a)) = κi (x i , a).

Obviously, κi

w is i-dense iff κi is i-dense. Furthermore, we have A 1 × A 2 × A 3 ⊆ I ⇔ ⌊A 1 ⌋ × ⌊A 2 ⌋ × ⌊A 3 ⌋ ⊆ I crisp , and since if a ≤ b then κi w ((x i , b)) ⊆ κi w ((x i , a)), we obtain

∩ 3 i=1 ∩ (xi,a)∈Ai κi w ((x i , a)) = ∅ ⇔ ⇔ ∩ 3 i=1 ∩ xi∈Xi κi w (x i , ∨{c | (x i , c) ∈ A i }) = ∅ ⇔ ⇔ ∩ 3 i=1 ∩ xi∈Xi κi (x i , A i (x i )) = ∅

This concludes the proof.

Conclusion

We presented how foundations of triadic concept analysis can be developed in a very general way. We showed that the previously studied cases of fuzzy TCA, namely the TCA with isotone and TCA with antitone concept-forming operators, are just particular cases of a more general approach. We provided definitions of basic notions, described properties of concept-forming operators and triadic concepts, and proved the analogy of basic theorem of TCA using crisp representation of triadic concepts.

Our future research topics on general approach to TCA include:

-Investigation of attribute implications in the unifying framework we have developed. At the current moment we study attribute implications in a unifying framework for dyadic case. -Generalization of the unifying framework assuming supports of the lattices L i to be different.