=Paper=
{{Paper
|id=None
|storemode=property
|title=L-Bonds vs Extents of Direct Products of Two L-fuzzy Contexts
|pdfUrl=https://ceur-ws.org/Vol-672/paper7.pdf
|volume=Vol-672
|dblpUrl=https://dblp.org/rec/conf/cla/KridloKO10
}}
==L-Bonds vs Extents of Direct Products of Two L-fuzzy Contexts==
https://ceur-ws.org/Vol-672/paper7.pdf
L-Bonds vs extents of direct product of two
L-fuzzy contexts
Ondrej Krı́dlo1 , Stanislav Krajči1 , and Manuel Ojeda-Aciego2
1
Dept. of Computer Science, Univ. P.J. Šafárik, Košice, Slovakia⋆
2
Dept. Matemática Aplicada, Univ. Málaga, Spain⋆⋆
Abstract. We focus on the direct product of two L-fuzzy contexts,
which are defined with the help of a binary operation on a lattice of
truth-values L. This operation, essentially a disjunction, is defined as
k ⋉ l = ¬k → l, for k, l ∈ L where negation is interpreted as ¬l = l → 0.
We provide some results which extend previous work by Krötzsch, Hitzler
and Zhang.
1 Introduction
Formal concept analysis (FCA) introduced by Ganter and Wille [9] has be-
come an extremely useful theoretical and practical tool for formally describing
structural and hierarchical properties of data with “object-attribute” character.
Regarding applications, we can find papers ranging from ontology merging [20],
to applications to the Semantic Web by using the notion of concept similar-
ity [7], and from processing of medical records in the clinical domain [11] to the
development of recommender systems [6].
Soon after the introduction of “classical” formal concept analysis, several
approaches towards its generalization were introduced and, nowadays, there are
recent works which extend the theory by using ideas from fuzzy set theory, or
fuzzy logic reasoning, or from rough set theory, or some integrated approaches
such as fuzzy and rough, or rough and domain theory [1, 15–18, 21, 22].
In this paper, we are concerned with extensions of Bělohlávek’s approach.
In [2, 4] he provided an L-fuzzy extension of the main notions of FCA, such as
context and concept, by extending its underlying interpretation on classical logic
to the more general framework of L-fuzzy logic [10].
In this work, we aim at formally describing some structural properties of in-
tercontextual relationships [8] of L-fuzzy formal contexts. The categorical treat-
ment of morphisms as fundamental structural properties has been advocated
by [14] as a means for the modelling of data translation, communication, and
distributed computing, among other applications. Research on (extensions of)
the theory of Chu spaces studies morphisms among contexts in order to obtain
categories with certain specific properties. Previous work in this line has been
⋆
Partially supported by grant VEGA 1/0131/09
⋆⋆
Partially supported by Spanish Ministry of Science project TIN09-14562-C05-01 and
Junta de Andalucı́a projects P06-FQM-02049 and P09-FQM-5233.
L-Bonds vs extents of direct product of two L-fuzzy contexts 71
developed by the authors in [12] by using category theory following the results
in [19].
The main result here is the extension of the relationship between bonds and
extents of direct products of contexts to the realm of L-fuzzy FCA.
2 Preliminary definitions
In order to make this contribution as self-contained as possible, we proceed now
with the preliminary definitions of complete residuated lattice, L-fuzzy context,
L-fuzzy concept.
2.1 L-fuzzy concept lattice
Definition 1. An algebra hL, ∧, ∨, ⊗, →, 0, 1i is said to be a complete resid-
uated lattice if
1. hL, ∧, ∨, 0, 1i is a complete bounded lattice with the least element 0 and the
greatest element 1,
2. hL, ⊗, 1i is a commutative monoid,
3. ⊗ and → are adjoint, i.e. a⊗b ≤ c if and only if a ≤ b → c, for all a, b, c ∈ L,
where ≤ is the ordering in the lattice generated from ∧ and ∨.
Now, the natural extension of the notion of context is given below.
Definition 2. Let L be a complete residuated lattice, an L-fuzzy context is a
triple hB, A, ri consisting of a set of objects B, a set of attributes A and an L-
fuzzy binary relation r, i.e. a mapping r : B × A → L, which can be alternatively
understood as an L-fuzzy subset of B × A.
We now introduce the L-fuzzy extension provided by Bělohlávek [2,3], where
we will use the notation Y X to refer to the set of mappings from X to Y .
Definition 3. Given an L-fuzzy context hB, A, ri, a pair of mappings ↑ : LB →
LA and ↓ : LA → LB can be defined for every f ∈ LB and g ∈ LA as follows:
^ ^¡ ¢
↑ f (a) = (f (o) → r(o, a)) ↓ g(o) = g(a) → r(o, a) (1)
o∈B a∈A
Lemma 1. Let L be a complete residuated lattice, let r ∈ LB×A be an L-fuzzy
relation between B and A. Then the pair of operators ↑ and ↓ form a Galois
connection between hLB ; ⊆i and hLA ; ⊆i, that is, ↑ : LB → LA and ↓ : LA → LB
are antitonic and, furthermore, for all f ∈ LB and g ∈ LA we have f ⊆ ↓↑ f
and g ⊆ ↑↓ g.
Definition 4. Consider an L-fuzzy context C = hB, A, ri. An L-fuzzy set of
objects f ∈ LB (resp. an L-fuzzy set of attributes g ∈ LA ) is said to be closed
in C iff f =↓↑ f (resp. g =↑↓ g).
72 Ondrej Krı́dlo, Stanislav Krajči, and Manuel Ojeda-Aciego
Lemma 2. Under the conditions of Lemma 1, the following equalities hold for
arbitrary f ∈ LB and g ∈ LA , ↑ f =↑↓↑ f and ↓ g =↓↑↓ g, that is, both ↓↑ f
and ↑↓ g are closed in C.
Definition 5. An L-fuzzy concept is a pair hf, gi such that ↑ f = g, ↓ g = f .
The first component f is said to be the extent of the concept, whereas the second
component g is the intent of the concept.
The set of all L-fuzzy concepts associated to a fuzzy context (B, A, r) will be
denoted as L-F CL(B, A, r).
An ordering between L-fuzzy concepts is defined as follows: hf1 , g1 i ≤ hf2 , g2 i
if and only if f1 ⊆ f2 if and only if g1 ⊇ g2 .
Proposition 1. The poset (L-F CL(B, A, r), ≤) is a complete lattice where
^ D^ ¡^ E _ D ¡^ ^ E
hfj , gj i = fj , ↑ fj ) and hfj , gj i = ↓ gj ), gj
j∈J j∈J j∈J j∈J j∈J j∈J
3 Other operations on an L-context
The corresponding notions of negation, disjunction and complement on an L-
context are introduced below.
Definition 6. Let us consider a unary operator negation and a binary disjunc-
tion operator on the underlying structure of truth values L as follows:
1. Negation ¬ : L → L is defined by ¬(l) = ¬l = l → 0
2. Disjunction ⋉ : L × L → L is defined by l1 ⋉ l2 = ¬l1 → l2
Some of the properties of negation appear in the following lemma.
Lemma 3 (Bělohlávek [5]). For any a, b, c ∈ L the following holds.
1. a ≤ ¬b ⇐⇒ a ⊗ b = 0
2. a ⊗ ¬a = 0
3. a ≤ ¬¬a
4. ¬0 = 1
5. ¬a = ¬¬¬a
6. a → b ≤ ¬b → ¬a
7. a ≤ b =⇒ ¬b ≤ ¬a
8. ¬(a ∨ b) = ¬a ∧ ¬b
From Property 6 above and the definition of disjunction, we can see that
disjunction needs not be, in general, commutative. However, this property will
be very important for the definition and properties of direct product of two
L-contexts. Notice that commutativity will hold if the law of double negation
(¬¬a = a) holds. The following result states some properties of residuated lat-
tices satisfying double negation.
L-Bonds vs extents of direct product of two L-fuzzy contexts 73
Proposition 2 (Bělohlávek [5]). If a residuated lattice satisfies the law of
double negation then it also satisfies the following conditions:
1. l →
Vk = ¬(k ⊗ W ¬l)
2. ¬( i∈I li ) = i∈I ¬li
3. l → k = ¬k → ¬l
It is convenient here to recall that adding conditions of our underlying residu-
ated lattice may change the class of structures we are working with. In particular,
a residuated lattice satisfying the double negation law and divisibility (that is,
x ≤ y implies the existence of z such that x = y ⊗ z), we are working with an
MV-algebra. If divisibility is replaced by the fact that the product ⊗ coincides
with the infimum of the lattice, then we are have just a Boolean algebra.
We finish this section with a specific notion of complement of a given L-fuzzy
formal context.
Definition 7. The complement of an L-fuzzy formal context is a formal context
with the binary relation ¬r defined by ¬r(o, a) = r(o, a) → 0 for all o ∈ B and
a ∈ A. The uparrow and downarrow mappings on the complement are denoted
by ↑¬ and ↓¬ .
Lemma 4. Let C = hB, A, ri be an L-fuzzy formal context. For all objects o, b ∈
B the inequality ↓↑ (χo )(b) ≤↓¬ ↑¬ (χb )(o) holds. If, moreover, the law of double
negation holds we have the equality ↓↑ (χo )(b) =↓¬ ↑¬ (χb )(o).
Proof. In the following, we use the common notation χe to denote the charac-
teristic function of an element e in a set:
^
↓↑ (χo )(b) = (↑ (χo )(a) → r(b, a))
a∈A
^ ^
= ( (χo (c) → r(c, a)) → r(b, a))
a∈A c∈B
^ ^
= (( (χo (c) → r(c, a)) ∧ (χo (o) → r(o, a))) → r(b, a))
a∈A c∈B,c6=o
^ ^
= (( (0 → r(c, a)) ∧ (1 → r(o, a))) → r(b, a))
a∈A c∈B,c6=o
^
= ((1 ∧ (1 → r(o, a))) → r(b, a))
a∈A
^
= ((1 → r(o, a)) → r(b, a))
a∈A
^
= (r(o, a) → r(b, a))
a∈A
∗
^
= (¬r(b, a) → ¬r(o, a)) = · · · =↓¬ ↑¬ (χb )(o)
a∈A
Equality (∗) follows from the law of double negation, otherwise we can only
obtain the inequality ↓↑ (χo )(b) ≤↓¬ ↑¬ (χb )(o). ⊓
⊔
74 Ondrej Krı́dlo, Stanislav Krajči, and Manuel Ojeda-Aciego
4 L-Multifunctions and L-fuzzy relations
The definition of L-bonds is based on a suitable extension of the theory of
multifunctions (also called, many-valued functions, or correspondences) whose
notation and terminology is introduced below.
Definition 8. An L-multifunction from X to Y is a mapping ϕ : X → LY .
The transposed of an L-multifunction ϕ : X → LY is an L-multifunction
t
ϕ : Y → X L defined by t ϕ(y)(x) = ϕ(x)(y).
The L-multifunction ϕ : X → LY can be extended to a mapping ϕ∗ : LX →
LY by ϕ∗ (f )(y) = x∈X (f (x) ⊗ ϕ(x)(y)), for f ∈ LX .
W
The set L-Mfn(X, Y ) of all the L-multifunctions from X to Y can be en-
dowed with a poset structure by defining the ordering ϕ1 ≤ ϕ2 as ϕ1 (x)(y) ≤
ϕ2 (x)(y) for all x ∈ X and y ∈ Y .
The usual definition of curry and uncurry operations can be adapted to the
framework of L-multifunctions as follows:
Definition 9. Let us define for arbitrary L-multifunction ϕ ∈ L-Mfn(X, Y ) an
L-fuzzy relation ϕr ∈ LX×Y defined by ϕr (x, y) = ϕ(x)(y) for all (x, y) ∈ X × Y .
For arbitrary L-fuzzy relation r ∈ LX×Y lets define an L-multifunction from
rmfn : X → LY defined by rmfn (x)(y) = r(x, y).
Finally, the notion of L-bond is given in the following definition:
Definition 10. An L-bond between two formal contexts C1 = hB1 , A1 , r1 i and
C2 = hB2 , A2 , r2 i is a multifunction b : B1 → LA2 satisfying the condition that
for all o1 ∈ B1 and a2 ∈ A2 both b(o1 ) and t b(a2 ) are closed L-fuzzy sets of,
respectively, attributes in C2 and objects in C1 . The set of all bonds from C1 to
C2 is denoted as L-Bonds(C1 , C2 ).
Lemma 5. Let hBi , Ai , ri i be two L-fuzzy formal contexts for i ∈ {1, 2}, where L
satisfies the double negation law. For all L-bonds β ∈ L-Bonds(C1 , C2 ) and for
all objects o1 ∈ B1 the equation β(o1 ) = β ∗ (↓¬1 ↑¬1 (χo1 )) holds.
Proof. We will prove the two inequalities separately.
_
β(o1 )(a2 ) = (β(b1 )(a2 ) ⊗ χo1 (b1 ))
b1 ∈B1
_
≤ (β(b1 )(a2 )⊗ ↓¬1 ↑¬1 (χo1 )(b1 )) = β ∗ (↓¬1 ↑¬1 (χo1 ))(a2 )
b1 ∈B1
L-Bonds vs extents of direct product of two L-fuzzy contexts 75
For the other inequality, consider the following chain
_
β ∗ (↓¬1 ↑¬1 (χo1 ))(a2 ) = (β(b1 )(a2 )⊗ ↓¬1 ↑¬1 (χo1 )(b1 ))
b1 ∈B1
∗
_
= (β(b1 )(a2 )⊗ ↓1 ↑1 (χb1 )(o1 ))
b1 ∈B1
_ ^
= (t β(a2 )(b1 ) ⊗ (↑1 (χb1 )(a1 ) → r1 (o1 , a1 )))
b1 ∈B1 a1 ∈A1
t
β(a2 ) is a closed L-set in B1 , then t β(a2 )(b1 ) =↓1 (g)(b1 ) for some g ∈ LA1
_ ^
= (↓1 (g)(b1 ) ⊗ ((1 → r1 (b1 , a1 )) → r1 (o1 , a1 )))
b1 ∈B1 a1 ∈A1
_ ^ ^
= ( g(a1 ) → r1 (b1 , a1 )) ⊗ (r1 (b1 , a1 ) → r1 (o1 , a1 )))
b1 ∈B1 a1 ∈A1 a1 ∈A1
⋆
_ ^
= ((g(a1 ) → r1 (b1 , a1 )) ⊗ (r1 (b1 , a1 ) → r1 (o1 , a1 )))
b1 ∈B1 a1 ∈A1
_ ^
≤ (g(a1 ) → r1 (o1 , a1 )) =
b1 ∈B1 a1 ∈A1
_ _ _
t
= ↓1 (g)(o1 ) = β(a2 )(o1 ) = β(o1 )(a2 )
b1 ∈B1 b1 ∈B1 b1 ∈B1
= β(o1 )(a2 )
where (⋆) follows from the inequality (k → l) ⊗ (l → m) ≤ k → l which holds
for all k, l, m ∈ L. ⊓
⊔
5 Direct product of two L-fuzzy contexts
Here we introduce the corresponding extension of the notion of direct product
of two L-fuzzy contexts.
Definition 11. The direct product of two L-fuzzy contexts C1 = hB1 , A1 , r1 i
and C2 = hB2 , A2 , r2 i is an L-fuzzy context C1 ∆C2 = hB1 × A2 , A1 × B2 , ∆i,
such that ∆((o1 , a2 ), (a1 , o2 )) = ¬r1 (o1 , a1 ) → r2 (o2 , a2 ).
The following result states properties of the just defined direct product of
L-fuzzy contexts.
Lemma 6. Let C1 = hB1 , A1 , r1 i and C2 = hB2 , A2 , r2 i be two L-fuzzy contexts,
where L satisfies the double negation law. Given two arbitrary L-multifunctions
ϕ : B1 → LA2 and ψ : A2 → LB1 , for all o1 , o2 ∈ B1 and a1 , a2 ∈ A2 the following
equalities hold
↑∆ (ϕr )(o2 , a1 ) = ↓2 (ϕ∗ (↓¬1 (χa1 )))(o2 ) = ↑1 (t ϕ∗ (↑¬2 (χo2 )))(a1 )
↓∆ (ψ r )(o1 , a2 ) = ↑2 (ψ ∗ (↑¬1 (χo1 )))(a2 ) = ↓1 (t ψ ∗ (↓¬2 (χa2 )))(o1 )
76 Ondrej Krı́dlo, Stanislav Krajči, and Manuel Ojeda-Aciego
Proof. Consider the following chain of equalities:
↑∆ (ϕr )(o2 , a1 )
^
= (ϕr (o1 , a2 ) → ∆((o1 , a2 ), (o2 , a1 )))
(o1 ,a2 )∈B1 ×A2
^
= (ϕr (o1 , a2 ) → (¬r1 (o1 , a1 ) → r2 (o2 , a2 )))
(o1 ,a2 )∈B1 ×A2
^
= ((ϕr (o1 , a2 ) ⊗ ¬r1 (o1 , a1 )) → r2 (o2 , a2 ))
(o1 ,a2 )∈B1 ×A2
^
= ((ϕr (o1 , a2 ) ⊗ (1 → ¬r1 (o1 , a1 ))) → r2 (o2 , a2 ))
(o1 ,a2 )∈B1 ×A2
^ ^
= ((ϕr (o1 , a2 ) ⊗ (χa1 (t1 ) → ¬r1 (o1 , t1 ))) → r2 (o2 , a2 ))
(o1 ,a2 )∈B1 ×A2 t1 ∈A1
^
= ((ϕr (o1 , a2 )⊗ ↓¬1 (χa1 )(o1 )) → r2 (o2 , a2 ))
(o1 ,a2 )∈B1 ×A2
^ ^
= ((ϕr (o1 , a2 )⊗ ↓¬1 (χa1 )(o1 )) → r2 (o2 , a2 ))
o1 ∈B1 a2 ∈A2
^ ^
= (↓¬1 (χa1 )(o1 ) ⊗ (ϕ(o1 )(a2 )) → r2 (o2 , a2 ))
o1 ∈B1 a2 ∈A2
^
= ↓2 (↓¬1 (χa1 )(o1 ) ⊗ ϕ(o1 ))(o2 )
o1 ∈B1
_
= ↓2 ( (↓¬1 (χa1 )(o1 ) ⊗ ϕ(o1 )))(o2 )
o1 ∈B1
= ↓2 (ϕ(↓¬1 (χa1 )))(o2 )
Similarly we have
^
↑∆ (ϕr )(o2 , a1 ) = (ϕr (o1 , a2 ) → (¬r1 (o1 , a1 ) → r2 (o2 , a2 )))
(o1 ,a2 )∈B1 ×A2
^
= (t ϕ(a2 )(o1 ) → (¬r2 (o2 , a2 ) → r1 (o1 , a1 )))
(o1 ,a2 )∈B1 ×A2
..
.
= ↑1 (t ϕ∗ (↑¬2 (χo2 )))(a1 )
⊓
⊔
6 L-bonds vs direct products of L-fuzzy contexts
The main contribution of the paper is presented in this section, in which a
relationship between L-bonds and extents of direct products of L-fuzzy contexts
is drawn by the following theorem.
L-Bonds vs extents of direct product of two L-fuzzy contexts 77
Theorem 1. Let Ci = hBi , Ai , ri i be L-fuzzy contexts for i ∈ {1, 2}, where L
satisfies the double negation law. Let β ∈ L-Mfn(B1 , A2 ). Then:
1. If β r is an extent of C1 ∆C2 , then β ∈ L-Bond(C1 , C2 ).
2. If β ∈ L-Bond(C1 , C2 ) and
^
β ∗ (↓¬1 ↑¬1 (χo1 ))(a2 ) = (↑¬1 (χo1 )(a1 ) → ↑2 ↓2 (β ∗ (↓¬1 (χa1 )))(a2 ))
a1 ∈A1
then r β is an extent of C1 ∆C2 .
Proof. 1. For the first item, let β be an extent of C1 ∆C2 , then we know that
β(o1 )(a2 ) = ↓∆ ↑∆ (β r )(o1 , a2 )
Let us write ↑∆ (β r )mfn = ψ, then
β(o1 )(a2 ) = ↓∆ (ψ)(o1 , a2 ) = ↑2 (ψ ∗ (↑¬1 (χo1 )))(a2 )
As a result, β(o1 ) is a closed L-set from LA2 .
Similarly, we have that t β(a2 )(o1 ) =↓1 (t ψ ∗ (↓¬2 (χa2 )))(o1 ). Hence t β(a2 ) is
a closed L-set of objects from LB1 .
2. The proof for the second item is as follows:
^
(↑¬1 (χo1 )(a1 ) → ↑2 ↓2 (β ∗ (↓( χa1 )))(a2 )) =
a1 ∈A1
^ ^
= (↑¬1 (χo1 )(a1 ) → (↓2 (β ∗ (↓¬1 (χa1 )))(o2 ) → r2 (o2 , a2 )))
a1 ∈A1 o2 ∈B2
^ ^
= (↑¬1 (χo1 )(a1 ) → (↓2 (β ∗ (↓¬1 (χa1 )))(o2 ) → r2 (o2 , a2 )))
a1 ∈A1 o2 ∈B2
^ ^
= ((↑¬1 (χo1 )(a1 )⊗ ↓2 (β ∗ (↓¬1 (χa1 )))(o2 )) → r2 (o2 , a2 ))
a1 ∈A1 o2 ∈B2
^ _
= ( (↑¬1 (χo1 )(a1 )⊗ ↓2 (β ∗ (↓¬1 (χa1 )))(o2 )) → r2 (o2 , a2 ))
o2 ∈B2 a1 ∈A1
_
= ↑2 ( (↑¬1 (χo1 )(a1 )⊗ ↓2 (β ∗ (↓¬1 (χa1 ))))(a2 )
a1 ∈A1
_
= ↑2 ( (↑¬1 (χo1 )(a1 ) ⊗ (↑∆ (β r ))mfn (a1 ))(a2 )
a1 ∈A1
= ↑2 ((↑∆ (β r ))mfn∗ (↑¬1 (χo1 )))(a2 )
= ↓∆ ↑∆ (β r )(o1 , a2 )
⋆
= β ∗ (↓¬1 ↑¬1 (χo1 ))(a2 ) = β(o1 )(a2 )
where (⋆) follows, firstly, from the hypothesis, which states that it equals to
β ∗ (↓¬1 ↑¬1 (χo1 ))(a2 ) and, as β ∈ L-Bond(C1 , C2 ), by Lemma 5. ⊓
⊔
78 Ondrej Krı́dlo, Stanislav Krajči, and Manuel Ojeda-Aciego
7 Conclusions and future work
We have introduced an adequate generalization of the study of L-bonds as mor-
phisms among contexts, initiated in [14], by showing how the classical relation-
ships between bonds and contexts can be lifted to a more general framework.
The contribution seems to pave the way towards determining possible cat-
egories on which to model knowledge transfer and information sharing. Other
steps have been given in [12, 13] where the category of L-Chu correspondences
has been considered. However, much work still has to be done.
A thorough study of the properties of the extended categorical framework of
Chu correspondences and L-Chu correspondences is needed, in order to identify
their natural interpretation within the theory of knowledge representation.
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