Triadic concept analysis (TCA) is an extension of formal concept analysis (dyadic case) which takes into account modi (e.g. time instances, conditions, etc.) in addition to objects and attributes. Thus instead of 2-dimensional binary tables TCA concerns with 3-dimensional binary tables. In our previous work we generalized TCA to work with grades instead of binary data; in the present paper we study TCA in even more general way. In order to cover up an analogy of isotone conceptforming operators (known from dyadic case in fuzzy setting) we developed an unifying framework in which both kinds of concept-forming operators are particular cases of more general operators. We describe the unifying framework, properties of the general concept-forming operators, show their relationship to those we used in our previous work.
The triadic approach to concept analysis (TCA) was introduced by Wille and
Liehman in [
(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. [
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 [
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. 2
Triadic Formal Concept Analysis This section introduces the notions needed in
our paper. For further information we refer to [
A triadic context is a quadruple hX, Y, Z, Ii 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, hx, y, zi ∈ 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 hX1, X2, X3, Ii.
Let K = hX1, X2, X3, Ii be a triadic context. For {i, j, k} = {1, 2, 3} (i.e. i, j, k ∈ {1, 2, 3}s.t.i 6= j 6= k) and Ck ⊆ Xk, we define a dyadic context
KiCjk = hXi, Xj , ICijk i by hxi, xj i ∈ ICijk
iff for each xk ∈ Ck : xi, xj , xk are related by I.
The concept-forming operators induced by KiCjk are defined as follows: C(ijCk) = {xj ∈ Xj | for each xi ∈ Ci : hxi, xj i ∈ ICijk },
i
Operators (ijCk) and (jiCk) form a Galois connection between Xi and Xj [
A triadic concept of hX1, X2, X3, Ii is a triplet hC1, C2, C3i of C1 ⊆ X1, C2 ⊆ X2, and C3 ⊆ X3, such that for every {i, j, k} = {1, 2, 3} we have Ci = C(ijCk); j C1, C2, and C3 are called the extent, intent, and modus of hC1, C2, C3i. The set of all triadic concepts of hX1, X2, X3, Ii is denoted by T (X1, X2, X3, I) and is called the concept trilattice of hX1, X2, X3, Ii; 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 [
Let LU 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 ∈ LU 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. 3
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 [
Let L = (L, ≤) be a bounded complete lattice and for i ∈ {1, 2, 3, 4}, Li = (Li, ≤i) be bounded lattice with Li = L and ≤i being either ≤ or ≤−1. That is, each Li is either (L, ≤) or (L, ≤−1). We denote the operations on Li by adding the subscript i, e. g. the operations in L2 are denoted by ∧2, ∨2, 02, and 12.
We consider a ternary operation ¤ : L1 × L2 × L3 → L4. We assume that ¤ commutes with suprema in all arguments. That is, for any a, aj ∈ L1, b, bj ∈ L2, c, cj ∈ L3 we have ¤(_
aj , b, c) = _ 1 j∈J ¤(a, _ 2 j∈J
bj , c) = _ ¤(a, b, _
cj ) = _ 3 j∈J 4 j∈J 4 j∈J 4 j∈J ¤(aj , b, c) ¤(a, bj , c) ¤(a, b, cj ) (1) Furthermore, for i, j, k ∈ 1, 2, 3 we define the operations ¤i : Lj × Lk × L4 as
For convenience we denote ¤(a, b, c) also by ¤{b, a, c} or ¤{c, b, a} etc., and
¤i(ai, aj, a4) also by ¤i{aj, ai, a4} or ¤i{ai, aj, a4}
Example 1. Complete residuated lattice [
Let L = hL, ∧, ∨, ⊗, →, 0, 1i be a complete residuated lattice and ≤ be its order. (1) Let Li = Lj = (L, ≤) for each i, j ∈ {1, 2, 3, 4} and let ¤(a1, a2, a3) = a1 ⊗ a2 ⊗ a3. Then operations ¤i are defined as follows: ¤1(a2, a3, a4) = (a2 ⊗ a3) → a4 ¤2(a3, a1, a4) = (a3 ⊗ a1) → a4 ¤3(a1, a2, a4) = (a1 ⊗ a2) → a4 ¤1(a2, a3, a4) = (a4 ⊗ a2) → a3 ¤2(a3, a1, a4) = (a4 ⊗ a1) → a3 ¤3(a1, a2, a4) = a1 ⊗ a2 ⊗ a4 (2) (3) (4) (5) (6) (7) (8) (2) Let L1 = L2 = hL, ≤i and L3 = L4 = hL, ≤−1i and let ¤(a1, a2, a3) = (a1 ⊗ a2) → a3. Then operations ¤i are defined as follows
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. (ab) ¤i are monotone in first two arguments and antitone in third argument. (c) ¤(a1, a2, ¤3(a1, a2, a4)) ≤ a4, analogous formulas hold for ¤1 and ¤2. (d) ¤3(a1, a2, ¤(a1, a2, a3)) ≥ a3, analogous formulas hold for ¤1 and ¤2. Proof. (a) follows directly from (1) (b) follows directly from (2) (c)
¤(a1, a2, ¤3(a1, a2, a4)) = = ¤(a1, a2, _{a3 | ¤(a1, a2, a3) ≤ a4}) =
3 = _{¤(a1, a2, a3) | ¤(a1, a2, a3) ≤ a4} ≤ a4
4
¤3(a1, a2, ¤(a1, a2, a3)) = = _{x3 | ¤(a1, a2, x3) ≤ ¤(a1, a2, a3)} ≥ a3
3
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 hX, Y, Z, Ii 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 → L4. 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 hX1, X2, X3, Ii.
L-context K = hX1, X2, X3, Ii induces three concept-forming operators. For {i, j, k} = {1, 2, 3} and the sets Ai ∈ LXi and Ak ∈ LXk , the concept-forming operator is a map: Li × Lk × L4 → Lj which assigns to Ai and Ak a fuzzy set Aj ∈ LXj defined by
Aj(xj) = ^ ¤j{Ai(xi), Ak(xk), I{xi, xj, xk}}. (9) In this case, the concept-forming operator is denoted by (ijAk), i.e. fuzzy set Aj is denoted by Aj = A(ijAk).
i
Example 2. (1) Let Li and ¤ be as in Example 1(1). Then the concept-forming
operators are as follows
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 [
(2) Let Li and ¤ be as in Example 1(2). Then the concept-forming operators are defined as follows:
A(12A3)(x2) = Vx1∈X1 (I(x1, x2, x3) ⊗ A1(x1)) → A3(x3)
1 x3∈X3 A(23A1)(x3) = Vx1∈X1 (I(x1, x2, x3) ⊗ A2(x2)) → A1(x1)
2 x2∈X2 A(31A2)(x1) = Wx2∈X2 (I(x1, x2, x3) ⊗ A1(x1) ⊗ A3(x3)) 3 x3∈X3 (11) (12)
Note that operators (12)–(13) are selected as a triadic counterpart to (dyadic)
isotone galois connections [
A triadic fuzzy concept of hX1, X2, X3, Ii is a triplet hC1, C2, C3i consisting of fuzzy sets C1 ∈ L1X1 , C2 ∈ L2X2 , and C3 ∈ L3X3 , such that for every {i, j, k} = {1, 2, 3} we have Ci = C(ijCk), Cj = C(jkCi), and Ck = C(ikCj). The C1, C2, j k i and C3 are called the extent, intent, and modus of hC1, C2, C3i. The set of all triadic concepts of K = hX1, X2, X3, Ii is denoted by T (X1, X2, X3, 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. (c) Ai ⊆ (Ai(ijAk))(jiAk) Lemma 2. (a) A(ijCk) = C(kjAi)
i k (b) if Ck ⊆ Dk and Ai ⊆ Bi then B(ijDk) ⊆ A(ijCk)
i i Proof. (a)
A(ijCk)(xj) = ^ i ¤j(Ai(xi), Ck(xk), I{xi, xk, xj}) = C(kjAi)(xj) k B(ijDk)(xj) = ^ i ¤j(Bi(xi), Dk(xk), I{xi, xk, xj}) ≤ ¤j(Ai(xi), Ck(xk), I{xi, xk, xj}) = A(ijCk) i (b) (c) = ^ i xj∈Xj
xk∈Xk ≥ ^ i xj∈Xj xk∈Xk (Ai(ijAk))(jiAk)(xi) = j x′′i∈Xi
xk∈Xk ¤i(^
¤j(Ai(x′i), Ak(x′k), I{x′i, x′k, xj}), Ak(xk), I{xi, xj, xk}) ≥ ¤i(¤j(Ai(xi), Ak(xk), I{xi, xk, xj}), Ak(xk), I{xi, xj, xk}) = = ^ _{ai | ¤(¤j(Ai(xi), Ak(xk), I{xi, xk, xj}), xk, ai) ≤ I{xi, xk, xj}} i xj∈Xj i
xk∈Xk the proof.
Lemma 1(c) yields that one of the possible values of ai is Ai(xi). Therefore, the previous formula is greater than Vi xj∈Xj Ai(xi) = Ai(xi) which concludes xk∈Xk Theorem 1. Let {i, j, k} = {1, 2, 3}. Then for all triadic fuzzy concepts hA1, A2, A3i and hB1, B2, B3i from T (K), if hA1, A2, A3i ¹i hB1, B2, B3i and hA1, A2, A3i ¹j hB1, B2, B3i then hB1, B2, B3i ¹k hA1, A2, A3i.
Proof. We have Ak = Ai(ikAj) and Bk = Bi(ikBj). Since Ai ⊆ Bi and Aj ⊆ Bj, Lemma 2 yields Bk ⊆ Ak.
The following theorem describes a way how to compute a triadic concept. Starting with two fuzzy sets Ci ∈ LXi and Ck ∈ LXk we obtain a triadic concept i k hA1, A2, A3i by three projections using the concept-forming operators. Firstly we project Ci and Ck onto Aj, then we project Aj and Ck onto Ai, and finally we project Ai and Aj onto Ak Theorem 2. For Ci ∈ LiXi , Ck ∈ LXk with {i, j, k} = {1, 2, 3}, let Aj = k C(ijCk), Ai = A(jiCk), and Ak = A(ikAj). Then hA1, A2, A3i is a triadic fuzzy i j i concept bik(Ci, Ck).
Moreover, hA1, A2, A3i has the smallest k-th component among all triadic fuzzy concepts hB1, B2, B3i with the greatest j-th component satisfying Ci ⊆ Bi and X = Ck ⊆ Bk. In particular, bik(Ai, Ak) = hA1, A2, A3i for each triadic fuzzy concept hA1, A2, A3i.
Proof. By lemma 2(c) we have Ci ⊆ Ai and since Ak = A(ikAj) = (A(jjiCk))(ikAj) = i (Ck(kiAj))(ikAj) we have also Ck ⊆ Ak.
First, we prove that hA1, A2, A3i is a triadic fuzzy concept. Ak = A(ikAj) is i satisfied by definition. Consider Aj. We have Aj = C(ijCk) ⊇ A(ijAk) (Lemma 2(b)) i i and Aj ⊆ (A(jjkAi))(kjAi) = A(kjAi) = A(ijAk). Therefore, Aj = AijAk . The proof k i i for Ai is similar.
Let hB1, B2, B3i be a triadic fuzzy concept with Xi ⊆ Bi and Xk ⊆ Bk. Then Bj = B(ijBk) ⊆ X(ijXk), so the maximal j-th component is Aj. Let Bj = Aj.
i i Then Ai = A(jijXk) ⊇ Bj(jiBk) = Bi and thus Ak = Ai(ikAj) ⊆ Bi(ikBj) = Bk.
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 A1 ∈ L1X1 , A2 ∈ L2X2 , and A3 ∈ LX3 we denote by A1 × A2 × A3 the ternary 3 L4-relation between X1, X2, and X3 defined by (A1 × A2 × A3)(x1, x2, x3) = ¤(A1(x1), A2(x2), A3(x3)).
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) If hA1, A2, A3i ∈ T (K) then A1 × A2 × A3 ⊆ I. (b) If A1 × A2 × A3 ⊆ I then there is hB1, B2, B3i ∈ T (K) such that Ai ⊆ Bi for i = 1, 2, 3. (c) Each hA1, A2, A3i ∈ T (K) is maximal w.r.t. to set inclusion, i.e. there is no hB1, B2, B3i ∈ T (K) other than hA1, A2, A3i for which Ai ⊆ Bi.
Proof. (a) ¤(Ai(xi), Aj(xj), ^
(Ai(xi), Aj(xj), I{xi, xj, xk}) ≤ k xxki∈∈XXik ≤ ¤(Ai(xi), Aj(xj), ¤k(Ai(xi), Aj(xj), I{xi, xj, xk})) ≤ ≤ I{xi, xj, xk} (b) Let {i, j, k} = {1, 2, 3} and bik(Ai, Ak) = hB1, B2, B3i. Due Theorem 2 we have Ai ⊆ Bi and Ak ⊆ Bk. Moreover,
Bj(xj) = A(ijAk)(xj) =
i Theorem 3 (crisp representation). Let K = hX1, X2, X3, Ii be a fuzzy triadic context and Kcrisp = hX1×L1, X2×L2, X3×L3, Icrispi with Icrisp defined by ((x1, a), (x2, b), (x3, c)) ∈ Icrisp iff ¤(a, b, c) ≤4 I(x1, x2, x3) be a triadic context. Then T (K) is isomorphic to T (Kcrisp).
Proof. Define maps ⌊...⌋i : LXi → Xi × L and ⌈...⌉i : Xi × L → LXi for i ∈ {1, 2, 3} as follows: ⌊Ai⌋i = {(xi, ai) | ai ≤i Ai(xi)} ⌈A′i⌉i = Wi{ai | (xi, ai) ∈ A′i} (14) (15)
In what follows we skip subscripts and write just ⌊Ai⌋ and ⌈A′i⌉ instead of ⌊Ai⌋i and ⌈A′i⌉i.
Let ϕ be a mapping ϕ : T (K) → T (Kcrisp) defined by ϕ(hA1, A2, A3i) = h⌊A1⌋, ⌊A2⌋, ⌊A3⌋i.
We show, that ϕ(hA1, A2, A3i) ∈ T (Kcrisp). We have (xi, b) ∈ (⌊Aj⌋(ji⌊Ak⌋) iff for each ((xj, a), (xk, c)) ∈ ⌊Aj⌋ × ⌊Ak⌋ it holds that ((xi, b), (xj, a), (xk, c)) ∈ Icrisp iff for each xj ∈ Xj, xk ∈ Xk, and for each a ≤j Aj(xj), b ≤k Ak(xk) it holds ¤(a, b, c) ≤4 I{xi, xj, xk} iff for each xj ∈ Xj, xk ∈ Xk we have ¤(Aj(xj), Ak(xk), b) ≤4 I{xi, xj, xk} iff b ≤i Ai(xi), therefore (⌊Aj⌋(ji⌊Ak⌋) × ⌊Ak⌋)i = ⌊Ai⌋.
Let ψ be a mapping ψ : T (Kcrisp) → T (K) defined by ψ(hA1, A2, A3i) = h⌈A1⌉, ⌈A2⌉, ⌈A3⌉i. We show, that ψ(hA1, A2, A3)i ∈ T (K).
We have (⌈Aj⌉(ji⌈Ak⌉)(xi) = b iff b is the maximal degree with the property that for each xj ∈ Xj, xk ∈ Xk it holds ¤(⌈Aj⌉(xj), ⌈Ak⌉(xj), b) ≤4 I(xi, xj, xk) iff b is the maximal degree with the property that for each a ≤i b and each xj ∈ Xj, xk ∈ Xk we have ¤(⌈Aj⌉(xj), ⌈Ak⌉(xj), a) ≤4 I(xi, xj, xk) iff b is the maximal degree with the property that for each a ≤i b and each ((xj, c), (xk, d)) ∈ Aj × Ak we have that ((xi, a), (xj, c), (xk, d)) ∈ Icrisp iff b is the maximal degree with the property that for each a ≤ b we have (xi, a) ∈ A(jiAk) = Ai. Therefore j ⌈Aj⌉(ji⌈Ak⌉) = ⌈Ai⌉.
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. 5
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 [
Consider the following relations hA1, A2, A3i .i hB1, B2, B3i iff Ai ⊆ Bi, hA1, A2, A3i hi hB1, B2, B3i iff Ai = Bi.
It is easy to check that .i and hi are a quasiorder and an equivalence on T (K).
Denote by T (K)/ hi the corresponding factor set with equivalence classes denoted by [hA1, A2, A3i]i. Letting
[hA1, A2, A3i]i ¹i [hB1, B2, B3i]i iff hA1, A2, A3i .i hB1, B2, B3i, ¹i is an order on T (K)/ h .
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 (Vi, Vk), Vi, Vk ⊆ V , if x .i v for all x ∈ Vi and x .k v for all x ∈ Vk. An ik-bound v is called an ik-limit of (Vi, Vk) if u .j v for all ik-bounds of (V1, V2) u. In an triordered set (V, .1, .2, .3) there is at most one ik-limit of (V1, V2) v with a property u .k v for all ik-limits of (V1, V2) u. Then we call v an ik-join of (Vi, Vk) and denote it ∇ik(Vi, Vk). The triordered set (V, .1, .2, .3) in which the ik-join exists for all i 6= k and all pairs of subsets of V is a complete trilattice.
For a complete trilattice V = (V, .1, .2, .3), an order filter Fi on ordered set V / ∼i is defined as a subset Fi of V with the property: x ∈ Fi and x .i y implies y ∈ Fi for all x, y ∈ V . We denote the set of all order filters on V / ∼i by Fi(V). A principal filter generated by x ∈ V is the filter [X)i = {y ∈ V | x .i y}. We call a subset X ∈ Fi(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 κi : Xi × Li → T (K) be a mapping defined by κi(xi, b) = {hA1, A2, A3i ∈ T (K)|Ai(xi) ≥i b} for i ∈ {1, 2, 3}, xi ∈ Xi and b ∈ L. Since the principal filter generated by hA1, A2, A3i is [hA1, A2, A3i)i = ∩xi∈Xi κi(xi, Ai(xi)), the set κi(Xi × Li) is i-dense. Moreover, κi happens to satisfy κi(xi, a) ⊆ κi(xi, b) iff b ≤i a.
Theorem 4 (basic theorem). Let K = (X1, X2, X3, 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(Xi, Xj ) = bik ³[{Ai|hA1, A2, A3i ∈ Xi}, [{Ak|hA1, A2, A3i ∈ Xk}´ .
A complete trilattice V = (V, .1, .2, .3) is isomorphic to T (K) if and only if there are mappings κ˜i : Xi × Li → Fi(V), i = 1, 2, 3, such that (a) κ˜i(Xi × Li) is i-dense with respect to V, (b) κ˜i(xi, a) ⊆ κ˜i(xi, b) iff b ≤i a,
3 (c) A1 × A2, ×A3 ⊆ I ⇔ Ti=1 Txi∈Xi κ˜i(xi, Ai(xi)) 6= ∅ for all Ai ∈ LXi . Proof. The first assertion follows from Theorem 2.
From Theorem 3 we know that T (K) is isomorphic to T (Kcrisp). 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 (Kcrisp)).
Consider the map κ˜iw : (Xi ×Li) → Fi(V) defined by κ˜iw((xi, a)) = κ˜i(xi, a). Obviously, κ˜iw is i-dense iff κ˜i is i-dense. Furthermore, we have A1 × A2 × A3 ⊆ I ⇔ ⌊A1⌋ × ⌊A2⌋ × ⌊A3⌋ ⊆ Icrisp, and since if a ≤ b then κ˜iw((xi, b)) ⊆ κ˜iw((xi, a)), we obtain ∩i=1 ∩(xi,a)∈Ai κ˜iw((xi, a)) 6= ∅ ⇔
3 ⇔ ∩i=1 ∩xi∈Xi κ˜iw(xi, ∨{c | (xi, c) ∈ Ai}) 6= ∅ ⇔ 3 3 ⇔ ∩i=1 ∩xi∈Xi κ˜i(xi, Ai(xi)) 6= ∅ This concludes the proof.
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
Li to be different.