=Paper=
{{Paper
|id=None
|storemode=property
|title=Fuzzy-Valued Triadic Implications
|pdfUrl=https://ceur-ws.org/Vol-959/paper11.pdf
|volume=Vol-959
|dblpUrl=https://dblp.org/rec/conf/cla/Glodeanu11
}}
==Fuzzy-Valued Triadic Implications==
https://ceur-ws.org/Vol-959/paper11.pdf
Fuzzy-Valued Triadic Implications
Cynthia Vera Glodeanu
Technische Universität Dresden,
01062 Dresden, Germany
Cynthia_Vera.Glodeanu@mailbox.tu-dresden.de
Abstract. We present a new approach for handling fuzzy triadic data
in the setting of Formal Concept Analysis. The starting point is a fuzzy-
valued 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 fit 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.
Keywords: Formal Concept Analysis, fuzzy data, three-way data
1 Introduction
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 different residuated lattices were considered for each
fuzzy set. A somehow different strategy was considered in [3] using alpha-cuts.
Our approach differs 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 influence
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
2 Fuzzy-Valued Triadic Implications
fuzzy-valued setting, defining 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 Preliminaries
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 Triadic Concept Analysis
As introduced in [7], the underlying structure of Triadic Concept Analysis is a
triadic context defined 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) ∈ 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 defined as a triple (A1 , A2 , A3 ) with Ai ⊆ Ki , i ∈ {1, 2, 3} 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 {i, j, k} = {1, 2, 3} with j < k and for X ⊆ Ki and Z ⊆ Kj × Kk , the
(−)(i) -derivation operators are defined by
X 7→ X (i) := {(kj , kk ) ∈ Kj × Kk | (ki , kj , kk ) ∈ Y for all ki ∈ X}, (1)
(i)
Z 7→ Z := {ki ∈ Ki | (ki , kj , kk ) ∈ Y for all (kj , kk ) ∈ Z}. (2)
These derivation operators correspond to the derivation operators of the dyadic
contexts defined by K(i) := (Ki , Kj × Kk , Y (i) ) for {i, j, k} = {1, 2, 3}, where
k1 Y (1) (k2 , k3 ) :⇐⇒ k2 Y (2) (k1 , k3 ) :⇐⇒ k3 Y (3) (k1 , k2 ) :⇐⇒ (k1 , k2 , k3 ) ∈ Y .
Due to the structure of triadic contexts further derivation operators can be
defined. For {i, j, k} = {1, 2, 3} and for Xi ⊆ Ki , Xj ⊆ Kj and Xk ⊆ Kk the
(−)Xk -derivation operators are defined by
Xi 7→ XiXk := {kj ∈ Kj | (ki , kj , kk ) ∈ Y for all (ki , kk ) ∈ Xi × Xk }, (3)
Xj 7→ XjXk := {ki ∈ Ki | (ki , kj , kk ) ∈ Y for all (kj , kk ) ∈ Xj × Xk }. (4)
Fuzzy-Valued Triadic Implications 3
These derivation operators correspond to the derivation operators of the dyadic
contexts defined by Kij ij ij
Xk := (Ki , Kj , YXk ) where (ki , kj ) ∈ YXk if and only if
(ki , kj , kk ) ∈ Y for all kk ∈ Xk . The structure on the set of all triconcepts T(K)
is the set inclusion in each component of the triconcept. For each i ∈ {1, 2, 3}
there is a quasiorder .i and its corresponding equivalence relation ∼i defined by
(A1 , A2 , A3 ) .i (B1 , B2 , B3 ) :⇐⇒ Ai ⊆ 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 coun-
terparts 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 =
{1, 2, 3}, the attribute set K2 = {a, b, c} and the condition set K3 = {A, B}. The
context has 12 triconcepts which are displayed in the same figure on the right.
For example, the first concept means that object 1 has attributes a and b under
No. Extent Intent Modus No. Extent Intent Modus
A B 1 {1} {a, b} {K3 } 7 {3} {K2 } {B}
a b c a b c 2 {K1 } {b} {A} 8 {K1 } {a} {B}
1 ×× ×× 3 {2, 3} {b, c} {A} 9 {2, 3} {c} {K3 }
2 ×× × × 4 {∅} {K2 } {K3 } 10 {3} {b, c} {K3 }
3 ×× ××× 5 {1, 3} {a, b} {B} 11 {K1 } {K2 } {∅}
6 {2, 3} {a, c} {B} 12 {K1 } {∅} {K3 }
Fig. 1. Triadic context and the associated triconcepts
all conditions from K3 . However, as two components of a triconcept are necessary
to determine the third one, {a, b} is also an intent of another triconcept, namely
of the fifth one.
2.2 Formal Fuzzy Concept Analysis
A complete residuated lattice L := (L, ∧, ∨, ⊗, →, 0, 1) is an algebra such
that: (1) (L, ∧, ∨, 0, 1) is a complete lattice, (2) (L, ⊗, 1) is a commutative
monoid, (3) 0 is the least and 1 the greatest element, (4) the adjointness property
holds for all a, b, c ∈ L, i.e., a ⊗ b ≤ c ⇔ a ≤ b → c. Then, ⊗ is called mul-
tiplication, → residuum and (⊗, →) adjoint couple. Each of the following
adjoint couples make L a complete residuated lattice:
4 Fuzzy-Valued Triadic Implications
Lukasiewicz: a ⊗ b := max(0, a + b − 1) with a → b := min(1, 1 − a + b)
1, a ≤ b
Gödel: a ⊗ b := min(a, b) with a → b :=
b, a b
1, a ≤ b
Product: a ⊗ b := ab with a → b :=
b/a, a b
The hedge operator is defined as a unary function ∗ : L → L which satisfies
the following properties: (1) 1∗ = 1, (2) a∗ ≤ a, (3) (a → b)∗ ≤ a∗ → b∗ , and
(4) a∗∗ = a∗ . Typical examples are the identity, i.e., for all a ∈ L it holds that
a∗ = a, and the globalization, i.e., a∗ = 0 for all a ∈ L \ {1} and a∗ = 1 if and
only if a = 1.
A triple (G, M, I) is called a formal fuzzy context if I : G × M → L is
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 ∈ G and each m ∈ M a truth
degree I(g, m) ∈ L to which the object g has the attribute m. For fuzzy sets
A ∈ LG and B ∈ LM the derivation operators are defined by
^ ^
Ap (m) := (A(g)∗ → I(g, m)), B p (g) := (B(m) → I(g, m)), (5)
g∈G m∈M
for g ∈ G and m ∈ M . Then, Ap (m) is the truth degree of the statement “m
is shared by all objects from A” and B p (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) ∈ LG × LM such that Ap = B and
p
B = A. Then, A is called the (fuzzy) extent and B the (fuzzy) intent of
(A, B). Fuzzy concepts represent maximal rectangles with truth values different
from zero in the fuzzy context. The fuzzy concepts ordered by the fuzzy set
inclusion form fuzzy concept lattices [9, 10]. Taking in (5) for ∗ hedges different
from the identity, we obtain the so-called fuzzy concept lattices with hedges [11].
Example 2. The fuzzy context displayed below has the object set G = {x, y, z},
the attribute set M = {a, b, c, d} and the set of truth values is the 3-element
chain L = {0, 0.5, 1}. Using the Gödel logic and the derivation operators defined
in Equation 5 with the hedge ∗ being the iden-
a b c d
tity we obtain 10 fuzzy concepts. For example
x 1 1 0.5 0
({1, 0.5, 0}, {1, 1, 0, 0}) is a fuzzy concept. The extent
y 1 0.5 0 1
contains the truth values of each object belonging to
z 1 0 0 0.5
the extent, i.e., in this case x belongs fully to the set, y
belongs to it with a truth value 0.5 and z does not belong to the extent. Similar
affirmations can be done for the intent. Using the Lukasiewicz logic in the same
setting we obtain 13 fuzzy concepts. On this set of truth values the only possible
hedge operators are the identity and globalization. As one of the major roles
of the hedge operators is to control the size of the fuzzy concept lattice, the
Fuzzy-Valued Triadic Implications 5
number of fuzzy concepts will be smaller, when using in (5) a hedge different
from the identity. In our example, using the globalization operator as the hedge,
we obtain 6 fuzzy concepts both with the Gödel and Lukasiewicz logic. As we
will see immediately, the hedges play also an important role for the attribute
implications, especially for the stem base.
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 ∈VLX the subsethood
degree of A being a subset of B is given by tv(A ⊆ B) = x∈X (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(∀g ∈ G((∀m ∈ A, (g, m) ∈ I) → (∀n ∈ B, (g, n) ∈ I)))
^ ^ ^
= ( (A(m) → I(g, m)) → (B(n) → I(g, n)))
g∈G m∈M n∈M
= tv(B ⊆ App ).
Example 3. Let us go back to our fuzzy context from Example 2. Consider the
Gödel logic and the derivation operators from (5) with ∗ being the identity. Then,
b(1)pp = {1, 0.5, 0}p = {1, 1, 0, 0}. Now, tv(b(1) → a(1)) = tv({a(1)} ⊆ b(1)pp ) = 1
and tv(b(1) → {a(1), c(0.5)}) = 0 because c(0.5) ∈ / b(1)pp . On the other hand,
considering in (5) the globalization as the hedge, we obtain b(1)pp = {1, 0.5, 0}p =
{1, 1, 0.5, 0} and therefore tv(b(1) → {a(1), c(0.5)}) = 1. Yet another example is
tv(b(0.5) → b(1)) = tv({b(1)} ⊆ b(0.5)pp ) = tv({b(1)} ⊆ {1, 0.5, 0, 0}) = 0.5.
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 (5) the globalization, the stem base of a fuzzy context is uniquely
determined. Using hedges different from the globalization, a fuzzy context may
have more than one stem base.
3 Fuzzy-Valued Triadic Concept Analysis
Now, we are ready to develop our fuzzy-valued triadicsetting. We will define
fuzzy-valued triadic contexts, concepts and derivation operators.
For a triadic context K = (K1 , K2 , K3 , Y ) a dyadic-cut (shortly d-cut) is
defined as ciα := (Kj , Kk , Yαjk ), where {i, j, k} = {1, 2, 3} and α ∈ Ki . A d-cut is
actually a special case of Kij ij
Xk = (Ki , Kj , YXk ) for Xk ⊆ Kk and |Xk | = 1. Each
d-cut is itself a dyadic context.
Definition 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 ∈ {1, 2, 3}, i.e., Y : K1 ×K2 ×K3 → L and L is the support set
6 Fuzzy-Valued Triadic Implications
of some residuated lattice. The elements of K1 , K2 and K3 are called objects,
attributes and conditions, respectively. To every triple (k1 , k2 , k3 ) ∈ 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 .
Definition 2. A fuzzy-valued triadic concept (shortly f-valued tricon-
cept) 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 component-
wise 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 definition immediately implies that the d-cut (K1 , K2 , Yk12
3
) is a fuzzy
context for every k3 ∈ K3 .
Example 4. We consider an f-valued triadic context with values from the 3-
element chain {0, 0.5, 1}. The object set K1 = {1, 2, 3, 4, 5} contains 5 groups
of students, the attribute set K2 = {f, s, v} contains 3 feelings, namely, fevered
(f), serious (s), vigilant (v) and the condition set K3 = {E, P, F } 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 Gödel
E P F
f s v f s v f s v
1 1 1 1 1 0.5 0.5 0 0.5 1
2 1 0.5 1 0.5 0 0 0 0 0.5
3 0.5 0.5 0.5 0.5 0.5 0 0 0 0.5
4 0.5 0 0.5 0.5 0.5 0.5 0 0.5 0.5
5 1 1 1 1 0.5 0.5 0 0.5 1
Fig. 2. F-valued triadic context
logic 34. For example, ({1, 1, 0.5, 0, 0}, {1, 0.5, 1}, {E}) is an f-valued triconcept
meaning that while doing an exam the first two student groups and half of
the third one are fevered, vigilant and moderately serious. Another example
is ({1, 1, 1, 1, 1}, {0.5, 0, 0}, {E, P }) meaning that all students are moderately
fevered while giving a presentation. Yet another example is ({1, 0, 0, 0.5, 1}, {1, 0,
0.5}, {E, P }) signifying that the first, the last and half of the 4-th group of
students are fevered and moderately vigilant while doing an exam and giving a
presentation.
Fuzzy-Valued Triadic Implications 7
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 K
e for an f-valued triadic context K.
Formally, suppose that K e := (K + , K + , K3 , Ye ) is the double-scaled triadic
1 2
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
e with ϕ(A1 , A2 , A3 ) := (A+ , A+ , A3 )
ϕ : T(K) → T(K) 1 2
and that the inverse map is given by
e → T(K) with ψ(A1 , A2 , A3 ) := (A♦ , A♦ , A3 ).
ψ : T(K) 1 2
Therefore, we have to prove the following statements: For all f-valued triconcepts
(A1 , A2 , A3 ), (B1 , B2 , B3 ) ∈ T(K) and for all (X1 , X2 , X3 ) ∈ T(K)
e :
ϕ(A1 , A2 , A3 ) ∈ T(K),
e
ψ(X1 , X2 , X3 ) ∈ T(K),
ψϕ(A1 , A2 , A3 ) = (A1 , A2 , A3 ),
ϕψ(X1 , X2 , X3 ) = (X1 , X2 , X3 ),
(A1 , A2 , A3 ) .i (B1 , B2 , B3 ) ⇔ ϕ(A1 , A2 , A3 ) .i ϕ(B1 , B2 , B3 ),
for all i ∈ {1, 2, 3}. These statements can be proven by basic properties of fuzzy
sets and triadic derivation operators. Due to limitation of space, we skip the
proof. t
u
We present the construction of K e := (K + , K + , K3 , Ye ), the double-scaled
1 2
triadic context, for a given f-valued triadic context K = (K1 , K2 , K3 , Y ). Let
Xi ⊆ LKi with i ∈ {1, 2} and let L be the support set of some residuated lattice.
We define
Xi+ := {(ki , µ) | ki ∈ Ki , µ ∈ L, µ ≤ Xi (ki )} ⊆ Ki∗ := Ki × L,
_
Xi♦ := {µ | (ki , µ) ∈ Xi } ⊆ LKi .
Then, Ye ⊆ K1+ × K2+ × K3 and
((k1 , µ), (k2 , λ), k3 ) ∈ Ye :⇐⇒ µ ⊗ λ ≤ tvk3 (k1 , k2 ) ⇐⇒ µ ⊗ λ ≤ Y (k1 , k2 , k3 ).
According to the above lemma, the f-valued triadic contexts fulfill 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 defined in various ways. We
8 Fuzzy-Valued Triadic Implications
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
{i, j} = {1, 2} the situation is easy. They are defined as
^
Z 7→ Z (i) := {Xj (kj ) → Y (i) (ki , (kj , k3 )) | ∀k3 ∈ K3 },
kj ∈Kj
(i)
Xi 7→ Xi := (Tl3 , {k3 ∈ K3 | Tk3 ⊆ Tl3 }) for l3 ∈ K3 ,
where Tl3 := ki ∈Ki (Xi (ki ) → Y (i) (ki , (kj , l3 )) with the derivation operators
V
from the fuzzy dyadic context K(i) := (Ki , Kj ×K3 , Y (i) ) and Y (i) (ki , (kj , k3 )) :=
Y (ki , kj , k3 ). The (−)(3) -derivation operator for Z := X1 × X2 ⊆ LK1 × LK2
and X3 ⊆ K3 is defined by
Z 7→ Z (3) : = {k3 ∈ K3 | k1 ⊗ k2 ≤ tvk3 (k1 , k2 ), ∀(k1 , k2 ) ∈ Z} (6)
^
= (Z(k1 , k2 ) → Y (3) ((k1 , k2 ), k3 ))∗ , (7)
(k1 ,k2 )∈K1 ×K2
where Z(k1 , k2 ) := X1 (k1 ) ⊗ X2 (k2 ), ∗ is the globalization in order to assure that
Z (3) is crisp and we have the dyadic fuzzy context K(3) := (K1 × K2 , K3 , Y (3) )
with Y (3) ((k1 , k2 ), k3 ) := Y (k1 , k2 , k3 ). We search for the conditions which con-
tain the maximal rectangle generated by Z.
(3)
The situation for X3 is quite tricky. Applying the derivation operators in
(3)
K for X3 , we get a truth value l ∈ L such that l = k1 ⊗ k2 instead of a tuple
(k1 , k2 ). To obtain such a tuple, we first have to compute the double-scaled con-
text K.e Afterwards, we use the crisp (−)(3) -derivation operator in K e to find the
components of the triconcept. Finally, we transform these into fuzzy sets as de-
scribed in the construction of K. e This way, we obtain the tuples ((k1 , µ), (k2 , ν))
consisting of objects and attributes with their truth values instead of the truth
value k1 ⊗ k2 .
For other approaches of fuzzy triadic data the derivation operators given in
(7) and the above construction suffice for any (−)(i) derivation operator.
Proposition 1. The (−)(i) -derivation operators with i ∈ {1, 2, 3} yield f-valued
triconcepts.
(1)
Proof. Suppose X1 ⊆ LK1 , X2 ⊆ LK2 and X3 ⊆ K3 . We have X1 = (Tl3 , {k3 |
Tk3 ⊆ Tl3 }), where
^
Tl3 = (X1 (k1 ) → Y (1) (k1 , (k2 , l3 )))
k1 ∈K1
^
= (X1 (k1 ) → Yl12
3
(k1 , k2 )).
k1 ∈K1
Since K12l3 is a dyadic fuzzy context, (X1 , Tl3 ) =: (X1 , A2 ) is a fuzzy preconcept
in K12 p p
l3 i.e., X1 ⊆ A2 and A2 ⊆ X1 with the derivation operators of Kl3 given
, 12
Fuzzy-Valued Triadic Implications 9
by Equation (5). In particular we have X1p ⊆ A2 . For any k3 ∈ K3 if Tk3 ⊆ Tl3 ,
then (X1 , A2 ) is a fuzzy preconcept also in K12 l3 ∪k3 . Proceeding alike, we obtain
the largest set A3 ⊆ K3 containing l3 such that TA3 ⊆ Tl3 . Then, (X1 , A2 ) is
a fuzzy preconcept in K12 A3 . So far, we obtained the last two components of the
f-valued triconcept and apply on them the (−)(1) -derivation operator to obtain
the first one. Now, we have
^
(A2 × A3 )(1) = {A2 (k2 ) → Y 1 (k1 , (k2 , k3 )) | ∀k3 ∈ A3 }
k2 ∈K2
^
= (A2 (k2 ) → YA123 (k1 , k2 )),
k2 ∈K2
which is A2 derivated in K12 A3 , i.e., the fist 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 ,
(3)
then the maximality of X3 = (A1 , A2 ) is automatically satisfied, as we obtain
(3)
X3 from the double scaled context. The maximality of (A1 × A2 )(3) follows
analogously to the first case. tu
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 {i, j, k} = {1, 2, 3} with j < k. t
u
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 ∈ {1, 2} and A3 ⊆ K3 we
define
^
X1 7→ X1A3 := (X1 (k1 )• → YA123 (k1 , k2 )), (8)
k1 ∈K1
^
X2 7→ X2A3 := (X2 (k2 ) → YA123 (k1 , k2 )) (9)
k2 ∈K2
for the dyadic fuzzy context K12 12
A3 := (K1 , K2 , YA3 ). where
YA123 : K1 × K2 × A3 → L,
^
YA123 (k1 , k2 ) := {tvk3 (k1 , k2 ) | ∀(k1 , k2 , k3 ) ∈ K1 × K2 × A3 }.
These derivation operators are the fuzzy counterparts of the (−)Ak -derivation
operators, because Ak is crisp. In the discrete case we have (ki , kj ) ∈ YAi,jk if
and only if for all kk ∈ Ak it holds that (ki , kj , kk ) ∈ Y . Therefore, in the fuzzy
setting for YAij3 (ki , kj ), we take the minimum of the values tvk3 (ki , kj ). Since K12
A3
is a fuzzy context, the (−)A3 -derivation operators form fuzzy Galois connections.
10 Fuzzy-Valued Triadic Implications
In (8) we will need the hedge • for the computation of the unique stem base,
however in general we take the identity for this hedge.
For the (−)Aj -derivation operators with {i, j} = {1, 2} the situation is dif-
ferent, 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
e A := V +
a part of the double-scaled context K,e namely in K
j aj ∈Aj (Ki , K3 , aj , Y ).
e
So, we could use discrete derivation operators to compute the concepts of K eA
j
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:
A
Xi 7→ Xi j := {k3 ∈ K3 | ki ⊗ kj ≤ tvk3 (ki , kj ), ∀(ki , kj ) ∈ Xi × Aj },
A
_
X3 7→ X3 j := {ki ∈ LKi | ki ⊗ kj ≤ tvk3 (ki , kj ), ∀(k3 , kj ) ∈ X3 × Aj }.
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
A
way we obtain Xi j , i.e., the third component of the f-valued triconcept that is
A
induced by Xi and Aj . To obtain X3 j we consider each ki ∈ LKi and check
whether the maximal rectangle ki ⊗ Aj exists under the fixed 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
A
of X3 j 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 ∈ LKi ,
Aj ∈ LKj with {i, j} = {1, 2} and A3 ⊆ K3 we have
A
^
Xi 7→ Xi j := (Xi (ki )• → YAi3j (ki , k3 ))∗ , (10)
ki ∈Ki
A
^
X3 7→ X3 j := (Xj (kj ) → YAi3j (ki , k3 )), (11)
kj ∈Kj
where • and ∗ are hedge operators. The • operator is optional, as it is needed
just for the computation of the stem base. It is the identity if i = 1. The ∗
A
hedge is always a compulsory globalization in order to assure that Xi j yields a
crisp set. Then, (10) and (11) are the V derivation operators of the fuzzy context
(Ki , K3 , YAi3j ) where YAi3j (ki , k3 ) := kj ∈Kj (Aj (kj ) → Y (ki , kj , kk )).
Considering in (10) and (11) all values for the indices, i.e., instead of (−)Aj
we take (−)Ak for {i, j, k} = {1, 2, 3}, and ignoring ∗ , these derivation operators
suffice for other approaches to Fuzzy Triadic Concept Analysis. This happens
due to the fact that such derivation operators yield triconcepts in which all three
components are fuzzy sets.
Proposition 3. For {i, j, k} = {1, 2, 3} there are (fuzzy) sets Xi ∈ LKi (Xi ∈
Ki , if i = 3) and Xk ∈ LKk (Xk ∈ Kk , if k = 3) such that Aj := XiXk , Ai :=
Fuzzy-Valued Triadic Implications 11
AXj
k
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 j-
th 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,
(A ×A )(3)
A2 ⊆ A1 1 2 = AA 1
3
⊆ X1X3 = A2 . Hence, A2 = AA 1
3
= (A1 × A3 )(2) ,
(1)
similarly A1 = (A2 × A3 ) and together with Proposition 2 they yield an
f-valued triconcept. The rest of the proof is analogous to the crisp case. Let
(B1 , B2 , B3 ) ∈ T(K) with X1 ⊆ B1 and X3 ⊆ B3 . Then, B2 ⊆ A2 , because B2 =
(B1 ×B3 )(2) = B1B3 ⊆ X1X3 = A2 . If B2 = A2 , by similar consideration as before,
we obtain B1 ⊆ A1 . Therefore, we have A3 = (A1 × A2 )(3) ⊆ (B1 × B2 )(3) = B3 ,
finishing the first part of the proof. Now, if (A1 , A2 , A3 ) is an f-valued tricon-
cept, then AA 1 = (A1 × A3 )
3 (2)
= A2 and AA2 = (A2 × A3 )
3 (1)
= A1 . Therefore,
bik (A1 , A3 ) = (A1 , A2 , A3 ) follows by the first part of the proposition. t
u
4 F-valued Implications
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
C ⊆ K3 the expression R → S was called conditional attribute implication. For
C
R, S ⊆ K3 and C ⊆ K2 the expression R → 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 F-valued Conditional Attribute vs. Attributional Condition
Implications
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.
Definition 3. For R, S ⊆ LK2 , C ⊆ K3 and globalization • we call the expres-
C
sion R → S f-valued conditional attribute implication and its truth value
is given by
C
R → S := tv(∀g ∈ K1 ((∀m ∈ R, (g, m) × C ∈ Y )• → (∀n ∈ S, (g, n) × C ∈ Y )))
^ ^ ^
= ( (R(m) → YC12 (g, m)) → (S(n) → YC12 (g, n)))
g∈K1 m∈K2 n∈K2
CC
= tv(S ⊆ R ).
12 Fuzzy-Valued Triadic Implications
Note that these implications are ordinary fuzzy implications since we are working
in the fuzzy context K12
C.
Example 5. For the context given in Figure 2 we have, for example, the f-valued
E P
conditional attribute implication s(0.5) → f (1) = s(0.5) → f (1) = 0.5 and yet
F
another is s(0.5) → f (1) = 0. The first implication means that whenever the
students are partially serious during an exam then they are also fevered. The
same holds for this implication during a presentation given by the students. The
implication does not hold when they are meeting their friends. In such situations
the students can be serious but have a relaxed attitude.
For an f-valued triadic context K we denote by
Imp(K2 ) := {R → S | R, S ∈ LK2 }
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 different
version of the dyadic fuzzy derivation operators defined in (5), namely
^ ^
Ap (m) := (A(g)∗ → I(g, m)), B p (g) := ( (B(m) → I(g, m)))•
g∈Imp(K2 ) m∈K3
for A ∈ Imp(K2 ), B ∈ K3 and ∗ is the globalization. Then, (A, B) ∈ B(Cimp (K))
contains in its extent all the implications that hold under all conditions of B. As
in the crisp case, each extent is an implicational theory and hence, every extent
has a stem base. In the concept lattice of Cimp (K) the implicational theories are
hierarchically ordered by the conditions under which they hold. The extent A is
the set of all implications that hold in (K1 , K2 , Yc12 ) with c ∈ C.
The number of fuzzy implications can be very large, since we have all impli-
cations A → B with A, B ⊆ LK2 . In the crisp case an implication either holds
or not, whereas in the fuzzy case an implication holds
V with a given truth value,
i.e., with tv(A → B). We have tv(a → b, c) = {tv(a → b), tv(a → c)} and
tv(a, b → c) = {tv(a → c), tv(b → c)} for all a, b, c ∈ LK2 . Hence, for the
V
structure of Cimp (K) it is enough to compute implications of the form a → b and
a(µ) → a(ν) for all a, b ∈ LK2 with b 6= a and µ, ν ∈ L with µ ν. As discussed
before, the other implications are infimum 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.
Fuzzy-Valued Triadic Implications 13
18
v(0.5) f (0.5)
2 17 s(1) 15
16
Friends Exam Presentation v(1) 13 f (1)
4 3 5 14 12
s(0.5)
6 7 11 10
v(1) → f (1) E, F → P
8 9
Fig. 3. Conditional attribute vs. attributional condition implications
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(1) → f (1) from the lattice
means that whenever the students are vigilant in degree (truth value) 1 during
an exam and presentation they are also fevered in degree 1 in these situations.
C
An implication C → D between the intents of Cimp (K) means that if R → S
D
holds, then R → 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, at-
tributes and conditions. Therefore, a triadic context has a sixfold symmetry. By
interchanging attributes with conditions in Definition 3, we obtain the attribu-
tional condition implications defined as follows:
Definition 4. For R, S ⊆ K3 and M ⊆ LK2 the expression
M
R → S := tv(∀g ∈ K1 ((∀a ∈ R, g × M × a ∈ Y ∗ ) → (∀b ∈ S, g × M × b ∈ Y ∗ )))
^ ^ ^
= ( (R(a) → YM 13
(g, a))∗ → (S(b) → YM13
(g, a))∗ ),
g∈K1 a∈K3 b∈K3
is called f-valued attributional condition implication, where ∗ is the glob-
alization.
M
We use the globalization hedge operator because this time R → S is a crisp
implication. For example, for the f-valued triadic context from Table 2 we have
v(1)
the attributional condition implication P → E, F = 1, meaning that students
who are vigilant during a presentation are also vigilant during an exam and while
f (1)
meeting friends. On the other hand, P → E, F = 0 means that a student being
fevered during a presentation does not imply that he/she is fevered during an
exam and while spending time with friends.
14 Fuzzy-Valued Triadic Implications
In analogy to the conditional attribute implications, we can also build the
e := (Imp(K3 ), K2 ×L, I) for the attributional condition implica-
context Cimp (K)
tions. This time we have Imp(K3 ) := {R → S | R, S ∈ K3 }, i.e., all implications
m
on K3 and I(R → S, m) := R → S. The extents of Cimp (K) e consist of all impli-
cations that hold in (K1 , K3 , Ym13 ) with m ∈ 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 F-valued Attribute×Condition Implications
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.
Definition 5. For R, S ⊆ LK2 × K3 the expression R → S is an f-valued
attribute×condition implication and its truth value is given by
^ ^ ^
( (R(m, b) → Y (g, m, b))• → (S(n, c) → Y (g, n, c))),
g∈K1 (m,b)∈K2 ×K3 (n,c)∈K2 ×K3
where • is the globalization, if we want to compute the unique stem base, other-
wise the identity.
These are the attribute implications of the fuzzy context (K1 , K2 ×K3 , Y (1) ).
Their stem base is given by the stem base of the attribute implications from
(K1 , K2 × K3 , Y (1) ).
Obviously, such implications can be easily obtained by the f-valued condi-
C
tional attribute and attributional condition implications, i.e., if we have R →
S for R, S ⊆ LK2 , C ⊆ K3 , then we can compute R × {c} → S × {c} for
all c ∈ C. Going the other way around, namely transforming the f-valued
attribute×condition implications into f-valued conditional attribute and attri-
butional condition implications, is of course also possible.
One could also be interested in f-valued object×attribute or object×condi-
tion implications. For our example this would mean If the first group of students
is fevered, then the second one is serious.
5 Conclusion and Further Research
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,
Fuzzy-Valued Triadic Implications 15
triconcepts and trilattices. We also showed how our notions can be translated into
different 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 different 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.
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