We present a new approach for handling fuzzy triadic data in the setting of Formal Concept Analysis. The starting point is a fuzzyvalued triadic context (K1; K2; K3; Y ), where K1; K2 and K3 are sets and Y is a ternary fuzzy relation between these sets. First, we generalise the methods of Triadic Concept Analysis to our setting and show how they t other approaches to Fuzzy Triadic Concept Analysis. Afterwards, we develop the fuzzy-valued triadic implications as counterparts of the various triadic implications studied in the literature. These are of major importance for the integrity of Fuzzy and Fuzzy-Valued Triadic Concept Analysis.
So far, the fuzzy approaches to Triadic Concept Analysis considered all three components of a triadic concept as fuzzy sets. In [1] the methods from Triadic Concept Analysis were generalised to the fuzzy setting. A more general approach was presented in [2], where di erent residuated lattices were considered for each fuzzy set. A somehow di erent strategy was considered in [3] using alpha-cuts.
Our approach di ers from the other ones in considering just two components as fuzzy and one as crisp in a triadic concept. This is motivated by the fact that in some situations it is not appropriate to regard all sets as fuzzy. For example, it is not natural to say that half of a person is old, however we may say a person is half old. First, we translate methods of Triadic Concept Analysis to our setting. Compared to other works, we generalise all triadic derivation operators and show how they change for the fuzzy approaches considered by other authors. Besides these results, the main achievement of this paper is the generalisation of the various triadic implications presented in [4].
Due to the large amount of results in this paper, we concentrate on giving an intuition of the methods and omit proofs whenever they do not in uence the understanding. The missing proofs, further results concerning fuzzy-valued triadic concepts and trilattices can be found in [5]. There, we also study the fuzzy-valued triadic approach to Factor Analysis.
The paper is structured as follows: In Section 2 we give brief introductions to Triadic and Formal Fuzzy Concept Analysis. In Section 3 we develop our fuzzy-valued setting, de ning context, concept, derivation operators and show how they correspond other to approaches to Fuzzy Triadic Concept Analysis. We also comment on the reasons why our setting is a proper generalisation. In Section 4 we present the fuzzy-valued triadic implications. The developed methods are accompanied by illustrative examples. The last section contains concluding remarks and further topics of research. 2
We assume basic familiarities with Formal Concept Analysis and refer the reader to [6]. In the following we give brief introductions to Triadic Concept Analysis [7, 8] and Formal Fuzzy Concept Analysis [9, 10]. 2.1
As introduced in [7], the underlying structure of Triadic Concept Analysis is a triadic context de ned as a quadruple (K1; K2; K3; Y ) where K1; K2 and K3 are sets and Y is a ternary relation, i.e., Y K1 K2 K3. The elements of K1; K2 and K3 are called (formal) objects, attributes and conditions, respectively, and (g; m; b) 2 Y is read: object g has attribute m under condition b. A triadic concept (shortly triconcept) of a triadic context (K1; K2; K3; Y ) is de ned as a triple (A1; A2; A3) with Ai Ki, i 2 f1; 2; 3g that is maximal with respect to component-wise set inclusion. For a triconcept (A1; A2; A3), the components A1; A2 and A3 are called the extent, the intent, and the modus of (A1; A2; A3), respectively.
Small triadic contexts can be represented through three-dimensional cross tables (see Example 1). Pictorially, a triconcept is a rectangular box full of crosses in the three-dimensional cross table representation of (K1; K2; K3; Y ), where this \box" is maximal under proper permutation of rows, columns and layers of the cross table.
For fi; j; kg = f1; 2; 3g with j < k and for X Ki and Z Kj Kk, the ( )(i)-derivation operators are de ned by
X 7! X(i) := f(kj; kk) 2 Kj
Kk j (ki; kj; kk) 2 Y for all ki 2 Xg;
Z 7! Z(i) := fki 2 Ki j (ki; kj; kk) 2 Y for all (kj; kk) 2 Zg:
(
Xi 7! XiXk := fkj 2 Kj j (ki; kj; kk) 2 Y for all (ki; kk) 2 Xi
Xj 7! XjXk := fki 2 Ki j (ki; kj; kk) 2 Y for all (kj; kk) 2 Xj
Xkg;
Xkg:
(
Bi and (A1; A2; A3) i (B1; B2; B3) :() Ai = Bi (i = 1; 2; 3): The triconcepts ordered in this way form complete trilattices, the triadic counterparts of concept lattices, as proved in the Basic Theorem of Triadic Concept Analysis [8]. However, unlike the dyadic case, the extents, intents and modi, respectively, do not form a closure system in general.
Example 1. The triadic context displayed below consists of the object set K1 =
f1; 2; 3g, the attribute set K2 = fa; b; cg and the condition set K3 = fA; Bg. The
context has 12 triconcepts which are displayed in the same gure on the right.
For example, the rst concept means that object 1 has attributes a and b under
all conditions from K3. However, as two components of a triconcept are necessary
to determine the third one, fa; bg is also an intent of another triconcept, namely
of the fth one.
A complete residuated lattice L := (L; ^; _; ; !; 0; 1) is an algebra such
that: (
the following properties: (
a fuzzy relation between the sets G and M and L is the support set of some residuated lattice. Elements from G and M are called objects and attributes, respectively. The fuzzy relation I assigns to each g 2 G and each m 2 M a truth degree I(g; m) 2 L to which the object g has the attribute m. For fuzzy sets A 2 LG and B 2 LM the derivation operators are de ned by for g 2 G and m 2 M . Then, Ap(m) is the truth degree of the statement \m is shared by all objects from A" and Bp(g) is the truth degree of \g has all attributes from B". For now, we take for the identity. It plays an important role in the computation of the stem base, as we will see later.
A fuzzy concept is a tuple (A; B) 2 LG LM such that Ap = B and
Bp = A. Then, A is called the (fuzzy) extent and B the (fuzzy) intent of
(A; B). Fuzzy concepts represent maximal rectangles with truth values di erent
from zero in the fuzzy context. The fuzzy concepts ordered by the fuzzy set
inclusion form fuzzy concept lattices [9, 10]. Taking in (
Fuzzy implications were studied in a series of papers by R. Belohlavek and V. Vychodil, as for example in [12, 13]. For fuzzy sets A; B 2 LX the subsethood degree of A being a subset of B is given by tv(A B) = Vx2X (A(x) ! B(x)). Let A and B be fuzzy attribute sets, then the truth value of the implication A ! B is given by tv(A ! B) := tv(8g 2 G((8m 2 A; (g; m) 2 I) ! (8n 2 B; (g; n) 2 I))) = ^ ( ^ (A(m) ! I(g; m)) !
^ (B(n) ! I(g; n))) g2G m2M = tv(B
App):
n2M
Example 3. Let us go back to our fuzzy context from Example 2. Consider the
Godel logic and the derivation operators from (
Due to the large number of implications in a fuzzy and even in a crisp formal
context, one is intrested in the stem base of the implications. The stem base is
a set of implications which is non-redundant and complete. The existence and
construction of the stem base for the discrete case was studied in [14], see also [6].
The problem for the fuzzy case was studied in [13]. There, the authors showed
that using in (
Now, we are ready to develop our fuzzy-valued triadicsetting. We will de ne fuzzy-valued triadic contexts, concepts and derivation operators.
For a triadic context K = (K1; K2; K3; Y ) a dyadic-cut (shortly d-cut) is de ned as ci := (Kj ; Kk; Y jk), where fi; j; kg = f1; 2; 3g and 2 Ki. A d-cut is actually a special case of KiXjk = (Ki; Kj ; YXijk ) for Xk Kk and jXkj = 1. Each d-cut is itself a dyadic context.
De nition 1. A fuzzy-valued triadic context ( f-valued triadic context) is a quadruple K := (K1; K2; K3; Y ), where Y is a ternary fuzzy relation between the sets Ki with i 2 f1; 2; 3g, i.e., Y : K1 K2 K3 ! L and L is the support set of some residuated lattice. The elements of K1; K2 and K3 are called objects, attributes and conditions, respectively. To every triple (k1; k2; k3) 2 K1 K2 K3, Y assigns a truth value tvk3 (k1; k2) to which object k1 has attribute k2 under condition k3.
The f-valued triadic context can be represented as a three-dimensional table, the entries of which are fuzzy values (see Example 4). In K one can interchange the roles played by the sets K1; K2 and K3 requiring, for example, that Y assigns to every triple (k2; k3; k1) a truth value tvk1 (k2; k3) to which attribute k2 exists under condition k3 having object k1.
De nition 2. A fuzzy-valued triadic concept (shortly f-valued triconcept) of an f-valued triadic context (K1; K2; K3; Y ) is a triple (A1; A2; A3) with A1 LK1 , A2 LK2 and A3 K3 that is maximal with respect to componentwise set inclusion. The components A1; A2 and A3 are called (f-valued) extent, (f-valued) intent, and the modus of (A1; A2; A3), respectively. We denote by T(K) the set of all f-valued triconcepts.
This de nition immediately implies that the d-cut (K1; K2; Yk132) is a fuzzy context for every k3 2 K3.
Example 4. We consider an f-valued triadic context with values from the 3element chain f0; 0:5; 1g. The object set K1 = f1; 2; 3; 4; 5g contains 5 groups of students, the attribute set K2 = ff; s; vg contains 3 feelings, namely, fevered (f), serious (s), vigilant (v) and the condition set K3 = fE; P; F g contains the events: Doing an exam (E), giving a presentation (P) and meeting friends (F). Using the Lukasiewicz logic, we obtain 30 f-valued triconcepts and with the Godel logic 34. For example, (f1; 1; 0:5; 0; 0g; f1; 0:5; 1g; fEg) is an f-valued triconcept meaning that while doing an exam the rst two student groups and half of the third one are fevered, vigilant and moderately serious. Another example is (f1; 1; 1; 1; 1g; f0:5; 0; 0g; fE; P g) meaning that all students are moderately fevered while giving a presentation. Yet another example is (f1; 0; 0; 0:5; 1g; f1; 0; 0:5g; fE; P g) signifying that the rst, the last and half of the 4-th group of students are fevered and moderately vigilant while doing an exam and giving a presentation. Lemma 1. Every f-valued triadic context is isomorphic to a triadic context. Proof. According to [9], every fuzzy context is isomorphic to a formal context, namely to its double-scaled context. Every condition d-cut is a fuzzy context. By double-scaling each condition d-cut we obtain the corresponding double-scaled triadic context Ke for an f-valued triadic context K.
Formally, suppose that Ke := (K1+; K2+; K3; Ye ) is the double-scaled triadic context of K := (K1; K2; K3; Y ), the construction of which is given below. We have to show that the considered isomorphism is given by
' : T(K) ! T(Ke ) with '(A1; A2; A3) := (A1+; A2+; A3) and that the inverse map is given by
: T(Ke ) ! T(K) with (A1; A2; A3) := (A1}; A2}; A3): Therefore, we have to prove the following statements: For all f-valued triconcepts (A1; A2; A3); (B1; B2; B3) 2 T(K) and for all (X1; X2; X3) 2 T(Ke ) : '(A1; A2; A3) 2 T(Ke ); (X1; X2; X3) 2 T(K); '(A1; A2; A3) = (A1; A2; A3); ' (X1; X2; X3) = (X1; X2; X3); (A1; A2; A3) .i (B1; B2; B3) , '(A1; A2; A3) .i '(B1; B2; B3); for all i 2 f1; 2; 3g. These statements can be proven by basic properties of fuzzy sets and triadic derivation operators. Due to limitation of space, we skip the proof. tu
We present the construction of Ke := (K1+; K2+; K3; Ye ), the double-scaled triadic context, for a given f-valued triadic context K = (K1; K2; K3; Y ). Let Xi LKi with i 2 f1; 2g and let L be the support set of some residuated lattice. We de ne
Xi+ := f(ki; ) j ki 2 Ki; 2 L; Xi} := _f j (ki; ) 2 Xig
LKi :
Xi(ki)g
Ki := Ki
L;
K1+
K2+
K3 and ((k1; ); (k2; ); k3) 2 Ye :() tvk3 (k1; k2) ()
Y (k1; k2; k3):
According to the above lemma, the f-valued triadic contexts ful ll all the properties the triadic contexts have. The f-valued triconcepts ordered by the (fuzzy) set inclusion form a complete fuzzy trilattice. Due to limitation of space we omit the proofs.
For our f-valued setting we want to obtain the corresponding ( )Ak and ( )(i) derivation operators. However, these can be de ned in various ways. We distinguish between more cases for the ( )(i)-derivation operators. In case of the ( )(i)-derivation operators with Z = Xj X3 LKj K3 and Xi LKi for fi; jg = f1; 2g the situation is easy. They are de ned as
Z 7! Z(i) :=
^ fXj(kj) ! Y (i)(ki; (kj; k3)) j 8k3 2 K3g; kj2Kj Xi 7! Xi(i) := (Tl3 ; fk3 2 K3 j Tk3
Tl3 g) for l3 2 K3;
where Tl3 := Vki2Ki (Xi(ki) ! Y (i)(ki; (kj; l3)) with the derivation operators
from the fuzzy dyadic context K(i) := (Ki; Kj K3; Y (i)) and Y (i)(ki; (kj; k3)) :=
Y (ki; kj; k3). The ( )(
Z 7! Z(
^ = k2
tvk3 (k1; k2); 8(k1; k2) 2 Zg
(Z(k1; k2) ! Y (
The situation for X3(
For other approaches of fuzzy triadic data the derivation operators given in
(
Proof. Suppose X1
LK1 ; X2
LK2 and X3
K3. We have X1(
^ (X1(k1) ! Y (
^ (X1(k1) ! Yl132(k1; k2)):
Since Kl132 is a dyadic fuzzy context, (X1; Tl3 ) =: (X1; A2) is a fuzzy preconcept
in Kl132, i.e., X1p A2 and Ap2 X1 with the derivation operators of Kl132 given
by Equation (
A3)(
^ fA2(k2) ! Y 1(k1; (k2; k3)) j 8k3 2 A3g k22K2 k22K2 =
^ (A2(k2) ! YA132(k1; k2)); which is A2 derivated in K1A23 , i.e., the st component of the triconcept, namely A1. Since (A1; A2) is a fuzzy concept, it is a maximal rectangle and A3 is the largest set containing this maximal rectangle.
We still have to check the other pair of derivation operators. Let X3 K3,
then the maximality of X3(
As a direct consequence of this proposition, we have the following statement: Proposition 2. For an f-valued triconcept (A1; A2; A3) it holds that Ai = (Aj Ak)(i) for fi; j; kg = f1; 2; 3g with j < k. tu
For the ( )Ak -derivation operators we also distinguish between two cases, namely when Ak is a crisp set and when it is fuzzy. When Ak is crisp, i.e., Ak := A3 we proceed as follows: For Xi LKi with i 2 f1; 2g and A3 K3 we de ne
X1 7! X1A3 :=
X2 7! X2A3 :=
k12K1
k22K2
^ (X1(k1) ! YA132(k1; k2));
^ (X2(k2) ! YA132(k1; k2))
tu
(
YA132 : K1
K2
A3 ! L; YA132(k1; k2) := ^ftvk3 (k1; k2) j 8(k1; k2; k3) 2 K1 K2
A3g:
These derivation operators are the fuzzy counterparts of the ( )Ak -derivation
operators, because Ak is crisp. In the discrete case we have (ki; kj) 2 YAi;kj if
and only if for all kk 2 Ak it holds that (ki; kj; kk) 2 Y . Therefore, in the fuzzy
setting for YAij3 (ki; kj), we take the minimum of the values tvk3 (ki; kj). Since K1A23
is a fuzzy context, the ( )A3 -derivation operators form fuzzy Galois connections.
In (
For the ( )Aj -derivation operators with fi; jg = f1; 2g the situation is different, because Aj is a fuzzy set. In the following we discuss more possibilities to obtain these derivation operators. In such cases we are interested in the relation between Ki and K3 for the values of Aj. This means that we are interested in just a part of the double-scaled context Ke , namely in Ke Aj := Vaj2Aj (Ki+; K3; aj; Ye ).
and afterwards transform them into fuzzy concepts. However, this is a laborious task and was presented just for a better understanding of the problem.
Another approach for the ( )Aj -derivation operators is the following: Xi 7! XiAj := fk3 2 K3 j ki X3 7! X3Aj := _fki 2 LKi j ki kj tvk3 (ki; kj); 8(ki; kj) 2 Xi
Ajg; kj tvk3 (ki; kj); 8(k3; kj) 2 X3
Ajg: In this case we do not need to double-scale the context. We compute the fuzzy concept induced by Xi and Aj and check under which conditions it exists. This way we obtain XiAj , i.e., the third component of the f-valued triconcept that is induced by Xi and Aj. To obtain X3Aj we consider each ki 2 LKi and check whether the maximal rectangle ki Aj exists under the xed conditions of X3. Afterwards, we take the maximum of these ki's due to the maximality property of f-valued triconcepts. This approach is laborious, especially the computation of X3Aj due to the large number of ki's we have to check.
We will consider a more straight-forward approach by computing the fuzzy context induced by Aj. A similar approach was presented in [1]. For Xi 2 LKi , Aj 2 LKj with fi; jg = f1; 2g and A3 K3 we have ki2Ki kj2Kj Xi 7! XiAj :=
^ (Xi(ki) ! YAi3j (ki; k3)) ;
X3 7! X3Aj :=
^ (Xj(kj) ! YAi3j (ki; k3));
(
Considering in (
Proposition 3. For fi; j; kg = f1; 2; 3g there are (fuzzy) sets Xi 2 LKi (Xi 2 Ki, if i = 3) and Xk 2 LKk (Xk 2 Kk, if k = 3) such that Aj := XiXk , Ai := AjXk and Ak := (Ai Aj )(k) (if i < j) or Ak := (Aj Ai)(k) (if i > j). Then, (A1; A2; A3) is an f-valued triconcept denoted by bik(Xi; Xk) having the smallest k-th component under all f-valued triconcepts (B1; B2; B3) with the largest jth component satisfying Xi Bi and Xk Bk. Particularly, bik(Ai; Ak) = (A1; A2; A3) for each f-valued triconcept (A1; A2; A3) of K.
Proof. Without loss of generality we can assume (i; j; k) = (1; 2; 3). Obviously,
X1 A1 and X3 A3. We start by proving that (A1; A2; A3) is indeed
an f-valued triconcept. From Proposition 1 we have A3 = (A1 A2). Then,
A2 A(1A1 A2)(
In this section we will study f-valued implications, as generalisations of those elaborated for the discrete case in [4]. There, the authors presented various triadic implications, which are stronger than the ones developed in [15]. For a given discrete triadic context K = (K1; K2; K3; Y ) and for R; S K2 and C K3 the expression R !C S was called conditional attribute implication. For R; S K3 and C K2 the expression R !C S was called attributional condition implication. Implications of the form R ! S with R; S K2 K3 were called attribute condition implications. Our main aim in the upcoming subsections is to generalise such implications to our setting. 4.1
In this subsection we study implications of the form: If we are moderately vigilant during an exam, then we are also fevered and If we are serious during an exam, then we feel the same during our presentation.
De nition 3. For R; S LK2 ; C K3 and globalization we call the expression R !C S f-valued conditional attribute implication and its truth value is given by R !C S := tv(8g 2 K1((8m 2 R; (g; m) = ^ ( ^ (R(m) ! YC12(g; m)) !
C 2 Y )
! (8n 2 S; (g; n) ^ (S(n) ! YC12(g; n)))
C 2 Y ))) = tv(S g2K1 m2K2
RCC ): n2K2 Note that these implications are ordinary fuzzy implications since we are working in the fuzzy context K1C2.
Example 5. For the context given in Figure 2 we have, for example, the f-valued
conditional attribute implication s(0:5) !E f (
For an f-valued triadic context K we denote by
Imp(K2) := fR ! S j R; S 2 LK2 g
the set of all fuzzy implications on K2. We construct the dyadic context
Cimp(K) := (Imp(K2); K3; I)
c
where Imp(K2) is a fuzzy set, K3 is a crisp set and I(R ! S; c) := R ! S.
In order to keep the condition set crisp, we use in Cimp(K) a slightly di erent
version of the dyadic fuzzy derivation operators de ned in (
The number of fuzzy implications can be very large, since we have all implications A ! B with A; B LK2 . In the crisp case an implication either holds or not, whereas in the fuzzy case an implication holds with a given truth value, i.e., with tv(A ! B). We have tv(a ! b; c) = Vftv(a ! b); tv(a ! c)g and tv(a; b ! c) = Vftv(a ! c); tv(b ! c)g for all a; b; c 2 LK2 . Hence, for the structure of Cimp(K) it is enough to compute implications of the form a ! b and a( ) ! a( ) for all a; b 2 LK2 with b 6= a and ; 2 L with . As discussed before, the other implications are in mum reducible elements in the lattice.
In accordance with the idea presented in [4] we label the concept lattice of Cimp(K) as follows: The attribute labelling is done in the usual way. For the object labelling the situation is more cumbersome. Each set of implications from Imp(K2) is an extent of Cimp(K) and an implicational theory, as discussed above. The object labels shall be distributed such that every extent is generated as an implicational theory by the labels attached to it and to its subconcepts. Therefore, the bottom element of the lattice will contain the stem base of all f-valued conditional attribute implications.
2 3 8 Friends
Exam
Presentation
4
6
5
7
E; F ! P
On the left part in Figure 3 the lattice of Cimp(K) is displayed. For better
legibility we used just the attribute labels (the conditions) and one object label
(conditional attribute implication). The implication v(
An implication C ! D between the intents of Cimp(K) means that if R !C S holds, then R !D S must hold as well. For our example the stem base of Cimp(K) is P; F ! E. We could perform a condition attribute exploration as proposed in [4] for the discrete case, however this would go beyond the scope of this paper.
In a triadic context we may arbitrarily interchange the roles of objects, attributes and conditions. Therefore, a triadic context has a sixfold symmetry. By interchanging attributes with conditions in De nition 3, we obtain the attributional condition implications de ned as follows:
K3 and M
LK2 the expression R !M S := tv(8g 2 K1((8a 2 R; g
M a 2 Y ) ! (8b 2 S; g
M b 2 Y ))) = ^ ( ^ (R(a) ! YM13(g; a)) !
^ (S(b) ! YM13(g; a)) ); g2K1 a2K3 b2K3 is called f-valued attributional condition implication, where alization. is the glob
We use the globalization hedge operator because this time R !M S is a crisp
implication. For example, for the f-valued triadic context from Table 2 we have
the attributional condition implication P v!(
In analogy to the conditional attribute implications, we can also build the context Cimp(Ke ) := (Imp(K3); K2 L; I) for the attributional condition implications. This time we have Imp(K3) := fR ! S j R; S 2 K3g, i.e., all implications on K3 and I(R ! S; m) := R !m S. The extents of Cimp(Ke ) consist of all implications that hold in (K1; K3; Ym13) with m 2 K2. The concept lattice is displayed on the right in Figure 3. For example the implication E; F ! P means that if the students during an exam and while meeting friends are (partially) fevered and (partially) serious, then they have the same feelings during their presentation.
The connection between the two classes of implications is an open question even for the discrete case and it remains open for the f-valued triadic case as well. 4.2
As presented for the discrete case, the two classes of implications studied so far are not powerful enough to express all possible kinds of implications in a triadic context. Therefore, we will generalise the so-called attribute condition implications to our setting. These express implications of the form If we are serious during our presentation, then we are moderately fevered during the exam. De nition 5. For R; S LK2 K3 the expression R ! S is an f-valued attribute condition implication and its truth value is given by ^ (
^ g2K1 (m;b)2K2 K3 (R(m; b) ! Y (g; m; b)) !
(S(n; c) ! Y (g; n; c))); ^ (n;c)2K2 K3 where is the globalization, if we want to compute the unique stem base, otherwise the identity.
Their stem base is given by the stem base of the attribute implications from
(K1; K2 K3; Y (
Obviously, such implications can be easily obtained by the f-valued condiC tional attribute and attributional condition implications, i.e., if we have R ! S for R; S LK2 ; C K3, then we can compute R fcg ! S fcg for all c 2 C. Going the other way around, namely transforming the f-valued attribute condition implications into f-valued conditional attribute and attributional condition implications, is of course also possible.
One could also be interested in f-valued object attribute or object condition implications. For our example this would mean If the rst group of students is fevered, then the second one is serious. 5
First, we presented a new framework for treating triadic fuzzy data. For this setting we generalised the notions of the ( )Ak and ( )(i) derivation operators, triconcepts and trilattices. We also showed how our notions can be translated into di erent approaches to Fuzzy Triadic Concept Analysis studied by other authors. One of our main results is the generalisation of the ( )(i) derivation operator for the f-valued triadic and fuzzy triadic setting, since it is absent in other works dealing with fuzzy triadic data. Second, we generalised triadic implications to our f-valued setting. These are of major importance for the development of Fuzzy and Fuzzy-Valued Triadic Concept Analysis.
Future research will focus on the connection between the di erent classes of f-valued triadic implications. As mentioned at the beginning, [5] is an extended version of this paper including the factorization problem. In the future we want to apply the f-valued triadic factorization to real world data.