=Paper=
{{Paper
|id=None
|storemode=property
|title=General Approach to Triadic Concept Analysis
|pdfUrl=https://ceur-ws.org/Vol-672/paper11.pdf
|volume=Vol-672
|dblpUrl=https://dblp.org/rec/conf/cla/KonecnyO10
}}
==General Approach to Triadic Concept Analysis==
https://ceur-ws.org/Vol-672/paper11.pdf
General approach to triadic concept analysis
Jan Konecny and Petr Osicka
Dept. Computer Science
Palacky University, Olomouc
17. listopadu 12, CZ-77146 Olomouc
Czech Republic
email: {osicka,konecny}@inf.upol.cz
Abstract. 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 concept-
forming operators (known from dyadic case in fuzzy setting) we devel-
oped an unifying framework in which both kinds of concept-forming op-
erators 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.
1 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
General approach to triadic concept analysis 117
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 re-
lational 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 ba-
sic properties. In Section 4 we turn our attention to triadic concept-forming
operators, triadic concepts. Section 5 brings an analogy to basic theorem of con-
cept (tri)-lattices. Our conclusions and future research ideas are summarized in
Section 6.
2 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 hX, Y, Z, Ii where X, Y , and Z are non-
empty 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
Kij ij
Ck = hXi , Xj , ICk i
by
hxi , xj i ∈ ICijk iff for each xk ∈ Ck : xi , xj , xk are related by I.
The concept-forming operators induced by Kij
Ck are defined as follows:
(ijCk )
Ci = {xj ∈ Xj | for each xi ∈ Ci : hxi , xj i ∈ ICijk },
118 Jan Konecny and Petr Osicka
Operators (ijCk ) and (jiCk ) form a Galois connection between Xi and Xj [9].
A triadic concept of hX1 , X2 , X3 , Ii is a triplet hC1 , C2 , C3 i of C1 ⊆ X1 , C2 ⊆
(ijC )
X2 , and C3 ⊆ X3 , such that for every {i, j, k} = {1, 2, 3} we have Ci = Cj k ;
C1 , C2 , and C3 are called the extent, intent, and modus of hC1 , C2 , C3 i. 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 [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 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 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}, 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) = ¤(aj , b, c)
1 j∈J 4 j∈J
_ _
¤(a, bj , c) = ¤(a, bj , c) (1)
2 j∈J 4 j∈J
_ _
¤(a, b, cj ) = ¤(a, b, cj )
3 j∈J 4 j∈J
General approach to triadic concept analysis 119
Furthermore, for i, j, k ∈ 1, 2, 3 we define the operations ¤i : Lj × Lk × L4 as
_
¤i (aj , ak , a4 ) = {ai | ¤(ai , aj , ak ) ≤ a4 } (2)
i
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 [11] is an algebra L = hL, ∧, ∨, ⊗, →
, 0, 1i such that hL, ∧, ∨, 0, 1i is a complete lattice with 0 and 1 being the least
and greatest element of L, respectively; hL, ⊗, 1i 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 = 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 (3)
2
¤ (a3 , a1 , a4 ) = (a3 ⊗ a1 ) → a4 (4)
¤3 (a1 , a2 , a4 ) = (a1 ⊗ a2 ) → a4 (5)
(2) Let L1 = L2 = hL, ≤i and L3 = L4 = hL, ≤−1 i and let ¤(a1 , a2 , a3 ) =
(a1 ⊗ a2 ) → a3 . Then operations ¤i are defined as follows
¤1 (a2 , a3 , a4 ) = (a4 ⊗ a2 ) → a3 (6)
¤2 (a3 , a1 , a4 ) = (a4 ⊗ a1 ) → a3 (7)
¤3 (a1 , a2 , a4 ) = a1 ⊗ a2 ⊗ a4 (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.
(b) ¤i are monotone in first two arguments and antitone in third argument.
(a)
(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
120 Jan Konecny and Petr Osicka
(d)
¤3 (a1 , a2 , ¤(a1 , a2 , a3 )) =
_
= {x3 | ¤(a1 , a2 , x3 ) ≤ ¤(a1 , a2 , a3 )} ≥ a3
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 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)
j xi ∈Xi
xk ∈Xk
In this case, the concept-forming operator is denoted by (ijAk ) , i.e. fuzzy set Aj
(ijA )
is denoted by Aj = Ai k .
Example 2. (1) Let Li and ¤ be as in Example 1(1). Then the concept-forming
operators are as follows
(ijAk )
^
Ai (xj ) = (Ai (xi ) ⊗ Ak (xk )) → I(x1 , x2 , x3 ) (10)
xi ∈Xi
xk ∈Xk
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 Li and ¤ be as in Example 1(2). Then the concept-forming operators
are defined as follows:
(12A ) V
A1 3 (x2 ) = x1 ∈X1 (I(x1 , x2 , x3 ) ⊗ A1 (x1 )) → A3 (x3 ) (11)
x3 ∈X3
(23A1 ) V
A2 (x3 ) = x1 ∈X1 (I(x1 , x2 , x3 ) ⊗ A2 (x2 )) → A1 (x1 ) (12)
x2 ∈X2
(31A ) W
A3 2 (x1 ) = x2 ∈X2 (I(x1 , x2 , x3 ) ⊗ A1 (x1 ) ⊗ A3 (x3 )) (13)
x3 ∈X3
General approach to triadic concept analysis 121
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 hX1 , X2 , X3 , Ii is a triplet hC1 , C2 , C3 i consisting
of fuzzy sets C1 ∈ LX X2 X3
1 , C2 ∈ L2 , and C3 ∈ L3 , such that for every {i, j, k} =
1
(ijC ) (jkC ) (ikC )
{1, 2, 3} we have Ci = Cj k , Cj = Ck i , and Ck = Ci j . The C1 , C2 ,
and C3 are called the extent, intent, and modus of hC1 , C2 , C3 i. 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 com-
plete lattices which consists of common support set, but they may be ordered
dually.
The following lemma describes basic properties of concept-forming operators.
(ijC ) (kjA )
Lemma 2. (a) Ai k = Ck i
(ijD ) (ijC )
(b) if Ck ⊆ Dk and Ai ⊆ Bi then Bi k ⊆ Ai k
(ijA )
(c) Ai ⊆ (Ai k )(jiAk )
Proof. (a)
(ijCk ) (kjAi )
^
Ai (xj ) = ¤j (Ai (xi ), Ck (xk ), I{xi , xk , xj }) = Ck (xj )
j xi ∈Xi
xk ∈Xk
(b)
(ijDk )
^
Bi (xj ) = ¤j (Bi (xi ), Dk (xk ), I{xi , xk , xj }) ≤
j xi ∈Xi
xk ∈Xk
(ijCk )
^
≤ ¤j (Ai (xi ), Ck (xk ), I{xi , xk , xj }) = Ai
j xi ∈Xi
xk ∈Xk
(c)
(ijAk ) (jiAk )
(Ai ) (xi ) =
^ ^
= ¤(i
¤j (Ai (x′i ), Ak (x′k ), I{x′i , x′k , xj }), Ak (xk ), I{xi , xj , xk }) ≥
i xj ∈Xj
′
j xi ∈Xi
xk ∈Xk x′k ∈Xk
^
≥ ¤i (¤j (Ai (xi ), Ak (xk ), I{xi , xk , xj }), Ak (xk ), I{xi , xj , xk }) =
i xj ∈Xj
xk ∈Xk
^ _
= {ai | ¤(¤j (Ai (xi ), Ak (xk ), I{xi , xk , xj }), xk , ai ) ≤ I{xi , xk , xj }}
i xj ∈Xj i
xk ∈Xk
122 Jan Konecny and Petr Osicka
Lemma 1(c) yields that one of theVpossible values of ai is Ai (xi ). Therefore,
the previous formula is greater than i xj ∈Xj Ai (xi ) = Ai (xi ) which concludes
xk ∈Xk
the proof.
Theorem 1. Let {i, j, k} = {1, 2, 3}. Then for all triadic fuzzy concepts hA1 , A2 , A3 i
and hB1 , B2 , B3 i from T (K), if hA1 , A2 , A3 i ¹i hB1 , B2 , B3 i and hA1 , A2 , A3 i ¹j
hB1 , B2 , B3 i then hB1 , B2 , B3 i ¹k hA1 , A2 , A3 i.
(ikAj ) (ikBj )
Proof. We have Ak = Ai and Bk = Bi . Since Ai ⊆ Bi and Aj ⊆ Bj ,
Lemma 2 yields Bk ⊆ Ak .
The following theorem describes a way how to compute a triadic concept.
Xk
Starting with two fuzzy sets Ci ∈ LX i and Ck ∈ Lk we obtain a triadic concept
i
hA1 , A2 , A3 i 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
Xk
Theorem 2. For Ci ∈ LX i , Ck ∈ Lk
i
with {i, j, k} = {1, 2, 3}, let Aj =
(ijCk ) (jiCk ) (ikAj )
Ci , Ai = Aj , and Ak = Ai . Then hA1 , A2 , A3 i is a triadic fuzzy
concept bik (Ci , Ck ).
Moreover, hA1 , A2 , A3 i has the smallest k-th component among all triadic
fuzzy concepts hB1 , B2 , B3 i with the greatest j-th component satisfying Ci ⊆ Bi
and X = Ck ⊆ Bk . In particular, bik (Ai , Ak ) = hA1 , A2 , A3 i for each triadic
fuzzy concept hA1 , A2 , A3 i.
(ikAj ) (jiCk ) (ikAj )
Proof. By lemma 2(c) we have Ci ⊆ Ai and since Ak = Ai = (Aj ) =
(kiA )
(Ck j )(ikAj ) we have also Ck ⊆ Ak .
(ikA )
First, we prove that hA1 , A2 , A3 i is a triadic fuzzy concept. Ak = Ai j is
(ijC ) (ijA )
satisfied by definition. Consider Aj . We have Aj = Ci k ⊇ Ai k (Lemma 2(b))
(jkAi ) (kjAi ) (kjAi ) (ijAk )
and Aj ⊆ (Aj ) = Ak = Ai . Therefore, Aj = AijA
i
k
. The proof
for Ai is similar.
Let hB1 , B2 , B3 i be a triadic fuzzy concept with Xi ⊆ Bi and Xk ⊆ Bk . Then
(ijB ) (ijX )
Bj = Bi k ⊆ Xi k , so the maximal j-th component is Aj . Let Bj = Aj .
(ijX ) (jiB ) (ikA ) (ikB )
Then Ai = Aj k ⊇ Bj k = Bi and thus Ak = Ai j ⊆ Bi j = 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 ∈ LX X2
1 , A2 ∈ L2 , and A3 ∈ L3
1 X3
we denote by A1 × A2 × A3 the ternary
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 , A3 i ∈ T (K) then A1 × A2 × A3 ⊆ I.
General approach to triadic concept analysis 123
(b) If A1 × A2 × A3 ⊆ I then there is hB1 , B2 , B3 i ∈ T (K) such that Ai ⊆ Bi
for i = 1, 2, 3.
(c) Each hA1 , A2 , A3 i ∈ T (K) is maximal w.r.t. to set inclusion, i.e. there is no
hB1 , B2 , B3 i ∈ T (K) other than hA1 , A2 , A3 i for which Ai ⊆ Bi .
Proof. (a)
^
¤(Ai (xi ), Aj (xj ), (Ai (xi ), Aj (xj ), I{xi , xj , xk }) ≤
k xi ∈Xi
xk ∈Xk
≤ ¤(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 , B3 i. Due Theorem 2 we
have Ai ⊆ Bi and Ak ⊆ Bk . Moreover,
(ijA )
Bj (xj ) = Ai k (xj ) =
^
= ¤j (Ai (xi ), Ak (xk ), I{xi , xk , xj }) ≥
j xi ∈Xi
xk ∈Xk
^
≥ ¤j (Ai (xi ), Ak (xk ), ¤(Ai (xi ), Ak (xk ), Aj (xj ))) =
j xi ∈Xi
xk ∈Xk
= Aj (xj )
(c) Let hA1 , A2 , A3 i and hB1 , B2 , B3 i be triadic concepts with Ai ⊆ Bi and
k ∈ {1, 2, 3} be an index such that Aj ⊂ Bj . From Theorem 1 follows that
(ijA )
there is an index j ∈ {1, 2, 3} such that Ak = Bk . Then having Aj = Ai k
(ijB )
Bj = Bi k Lemma 2(c) yields Bj ⊆ Aj which is a contradiction.
Theorem 3 (crisp representation). Let K = hX1 , X2 , X3 , Ii be a fuzzy tri-
adic context and Kcrisp = hX1 ×L1 , X2 ×L2 , X3 ×L3 , Icrisp i 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 )} (14)
⌈A′i ⌉i = i {ai | (xi , ai ) ∈ A′i }
W
(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 , A3 i) =
h⌊A1 ⌋, ⌊A2 ⌋, ⌊A3 ⌋i.
We show, that ϕ(hA1 , A2 , A3 i) ∈ 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)) ∈
124 Jan Konecny and Petr Osicka
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 , A3 i) =
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 max-
imal 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
(jiA )
with the property that for each a ≤ b we have (xi , a) ∈ Aj k = Ai . Therefore
⌈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 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
hA1 , A2 , A3 i .i hB1 , B2 , B3 i iff Ai ⊆ Bi ,
hA1 , A2 , A3 i hi hB1 , B2 , B3 i 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 , A3 i]i . Letting
[hA1 , A2 , A3 i]i ¹i [hB1 , B2 , B3 i]i iff hA1 , A2 , A3 i .i hB1 , B2 , B3 i,
¹i is an order on T (K)/ hi .
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
General approach to triadic concept analysis 125
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 , A3 i ∈ T (K)|Ai (xi ) ≥i b} for i ∈
{1, 2, 3}, xi ∈ Xi and b ∈ L. Since the principal filter generated by hA1 , A2 , A3 i is
[hA1 , A2 , A3 i)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 con-
text. Then T (K) is a complete trilattice of K for which the ik-joins are defined
as follows:
³[ [ ´
∇ik (Xi , Xj ) = bik {Ai |hA1 , A2 , A3 i ∈ Xi }, {Ak |hA1 , A2 , A3 i ∈ Xk } .
A complete trilattice V = (V, .1 , .2 , .3 ) is isomorphic to T (K) if and only
if there are mappings κ̃i : Xi × Li → Fi (V), i = 1, 2, 3, such that
(a) κ̃i (Xi × Li ) is i-dense with respect to V,
(b) κ̃i (xi , a) ⊆ κ̃i (xi , b) iff b ≤i a,
T3 T
(c) A1 × A2 , ×A3 ⊆ I ⇔ i=1 xi ∈Xi κ̃i (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 equiv-
alent with the conditions from Wille’s original basic theorem (for T (Kcrisp )).
Consider the map κ̃i w : (Xi ×Li ) → Fi (V) defined by κ̃i w ((xi , a)) = κ̃i (xi , a).
Obviously, κ̃i w is i-dense iff κ̃i is i-dense. Furthermore, we have A1 × A2 ×
A3 ⊆ I ⇔ ⌊A1 ⌋ × ⌊A2 ⌋ × ⌊A3 ⌋ ⊆ Icrisp , and since if a ≤ b then κ̃i w ((xi , b)) ⊆
κ̃i w ((xi , a)), we obtain
∩3i=1 ∩(xi ,a)∈Ai κ̃i w ((xi , a)) 6= ∅ ⇔
⇔ ∩3i=1 ∩xi ∈Xi κ̃i w (xi , ∨{c | (xi , c) ∈ Ai }) 6= ∅ ⇔
⇔ ∩3i=1 ∩xi ∈Xi κ̃i (xi , Ai (xi )) 6= ∅
This concludes the proof.
126 Jan Konecny and Petr Osicka
6 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 oper-
ators, are just particular cases of a more general approach. We provided defi-
nitions 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 uni-
fying framework for dyadic case.
– Generalization of the unifying framework assuming supports of the lattices
Li to be different.
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