Looking for bonds between nonhomogeneous formal contexts
Ondrej Kr´ıdlo, Lubomir Antoni, Stanislav Krajˇci University of Pavol Jozef ˇSaf´arik, Koˇsice, Slovakia?
Abstract. Recently, the concept lattices working with the heteroge- neous structures have been fruitfully applied in a fuzzy formal concept analysis. We present a situation under nonhomogeneous formal contexts and explore the bonds in a such nonhomogeneous case. This issue requires to formulate the alternative definition of a bond and to investigate the relationships between bonds and the particular formal contexts.
Keywords: bond, heterogeneous formal context, second order formal context
1 Introduction
Formal concept analysis (FCA) [16] as an applied lattice theory allows us to explore the meaningful groupings of objects with respect to common attributes.
In general, FCA is an interesting research area that provides theoretical foun- dations, fruitful methods, algorithms and underlying applications in many areas and has been investigated in relation to various disciplines and integrated ap- proaches [13,15]. The feasible attempts and generalizations are investigated, one can see dual multi-adjoint concept lattices working with adjoint triples [27–29], interval-valuedL-fuzzy concept lattices [1], heterogeneous concept lattices [2, 3], connectional concept lattices [12, 32, 33]. Classical bonds and their generaliza- tions acting on residuated lattices were analyzed from a broader perspective in [17, 21, 24].
In this paper, we deal with an alternative notion of the bonds and with a problem of looking for bonds in a nonhomogeneous formal contexts. In particular, Section 2 recalls the basic notions of a concept lattice, notion of a bond, its equivalent definition and preliminaries of a second order formal context and a heterogeneous formal context. Section 3 describes the idea of a looking for bonds in a nonhomogeneous case. Sections 4 and 5 provide the solution of this issue in terms of a second order formal context and heterogeneous formal context.
?This work was partly supported by grant VEGA 1/0832/12 and by the Slovak Re- search and Development Agency under contract APVV-0035-10 “Algorithms, Au- tomata, and Discrete Data Structures”.
2 Preliminaries
Definition 1. Let B and A be the nonempty sets, R⊆B×A be an arbitrary binary relation. Triple hB, A, Ri is said to be a formal context with a set of objectsB and a set of their attributesA. Relationships between objects and their attributes are saved in the relationR. Let us define a pair of derivation operators (↑,↓)as the mappings between powersets ofB andAsuch that
– ↑:P(B)→ P(A)and↓:P(A)→ P(B)where for any X⊆B andY ⊆A is – ↑(X) ={a∈A|(∀b∈X)(b, a)∈R}
– ↓(Y) ={b∈B|(∀a∈Y)(b, a)∈R}.
Such derivation operators can be defined as the mappings between 2-sets (bor- rowed from fuzzy generalization of FCA that is sometimes easier to use)
– ↑: 2B→2A and↓: 2A→2B where for any X ∈2B andY ∈2A – ↑(X)(a) =V
b∈B((b∈X)⇒((b, a)∈R)) =V
b∈B(X(b)⇒R(b, a)) – ↓(Y)(b) =V
a∈A((a∈Y)⇒((b, a)∈R)) =V
a∈A(Y(a)⇒R(b, a)).
Pair of such derivation operators forms an antitone Galois connection be- tween complete lattices of all subsets of B and A. Hence, the compositions of the mappings form closure operators on such complete lattices.
Definition 2. Let C=hB, A, Ribe a formal context. Any pair of sets(X, Y)∈ 2B×2A is said to be a formal concept iff X =↓ (Y) and Y =↑ (X). Object part of any concept is called extent and attribute part is called intent. Set of all extents of formal context C will be denoted by Ext(C). The notationInt(C) stands for the set of all intents of C.
All concepts ordered by set inclusion of extents (or equivalently by dual of intent inclusion) form a complete lattice structure.
2.1 Notion of bond and its equivalent definition
Definition 3. Let Ci=hBi, Ai, Riifori∈ {1,2} be two formal contexts. Rela- tionβ⊆B1×A2 is said to be a bondiff any row of the table is an intent ofC2
and any of its column is an extent ofC1. Set of all bonds betweenC1 andC2 will be denoted by 2-Bonds(C1,C2).
Lemma 1. Let Ci = hBi, Ai, Rii for i ∈ {1,2} be two formal contexts. Then β ⊆B1×A2is a bond betweenC1andC2if and only ifExt(hB1, A2, βi)⊆Ext(C1) andInt(hB1, A2, βi)⊆Int(C2).
Proof. ⇒: LetX ∈Ext(hB1, A2, βi) be an arbitrary extent of any bond between formal contextsC1andC2. Derivation operators ofCi will be denoted by (↑i,↓i)
fori∈ {1,2}. Derivation operators of the bond will be denoted by (↑β,↓β). Then there exists a set of attributesY ⊆A2such that
↓β (Y)(b1) = ^
a2∈A2
(Y(a2)⇒β(b1, a2))
β(−, a2) is an extent of Ext(C1) hence there existsZ ⊆A1
= ^
a2∈A2
(Y(a2)⇒↓1(Z)(b1))
= ^
a2∈A2
Y(a2)⇒ ^
a1∈A1
Z(a1)⇒R1(b1, a1)
= ^
a2∈A2
^
a1∈A1
(Y(a2)⇒(Z(a1)⇒R1(b1, a1)))
= ^
a2∈A2
^
a1∈A1
((Y(a2)∧Z(a1))⇒R1(b1, a1))
= ^
a1∈A1
_
a2∈A2
Y(a2)∧Z(a1)
⇒R1(b1, a1)
= ^
a1∈A1
(ZY(a1)⇒R1(b1, a1))
=↓1(ZY)(b1) whereZY(a1) =W
a2∈A2(Y(a2)∧Z(a1)) Hence, Ext(hB1, A2, βi)⊆Ext(C1). Similarly for intents.
⇐: Assume a formal contexthB1, A2, βisuch that it holds Ext(hB1, A2, βi)⊆ Ext(C1) and Int(hB1, A2, βi)⊆Int(C2). From the simple fact that any row of any context is its intent and any column is its extent and from the previous inclusions, we obtain thatβ is a bond betweenC1 andC2. ut
Hence, the notion of bond can be defined equivalently as follows.
Definition 4. LetCi=hBi, Ai, Riifori∈ {1,2}be two formal contexts. Formal context B = hB1, A2, βi is said to be a bond between C1 and C2 if Ext(B) ⊆ Ext(C1) andInt(B)⊆Int(C2).
More about the equivalent definition of bond could be found in [17–19].
2.2 Direct product of two formal contexts and bonds
Let us recall the definition and important property of direct product of two formal contexts. More details about such topic can be found in [21, 26].
Definition 5. Let Ci = hBi, Ai, Rii be two formal contexts. Formal context C1∆C2=hB1×A2, B2×A1, R1∆R2iwhere
(R1∆R2)((b1, a2),(b2, a1)) =R1(b1, a1)∨R2(b2, a2)
=¬R1(b1, a1)⇒R2(b2, a2)
=¬R2(b2, a2)⇒R1(b1, a1)
for any (bi, ai) ∈ Bi×Ai for all i ∈ {1,2} is said to be a direct product of formal contextsC1 andC2.
Lemma 2. LetCi=hBi, Ai, Riibe two formal contexts. Every extent ofC1∆C2
is a bond between C1 andC2.
2.3 Second order formal contexts
In this subsection, we remind a notion of a second order formal concept [24].
Definition 6. Consider two non-empty index setsIandJ and a formal context hS
i∈IBi,S
j∈JAj, ri, whereby
– Bi1∩Bi2 =∅ for any i1, i2∈I,i16=i2, – Aj1∩Aj2 =∅for any j1, j2∈J,j16=j2, – r:S
i∈IBi×S
j∈JAj→2.
Moreover, consider two non-empty sets of2-contexts notated – {Ci=hBi, Ti, pii:i∈I}
– {Dj=hOj, Aj, qji:j∈J}.
Formal context of second order is a tuple D [
i∈I
Bi, {Ci;i∈I}, [
j∈J
Aj, {Dj;j ∈J}, [
(i,j)∈I×J
ri,j
E ,
whereri,j :Bi×Aj→2 defined asri,j(b, a) =r(b, a)for any b∈Bianda∈Aj. In what follows, consider the below described notation. Let us have anL-set f : X → 2 for a non-empty universe set X = S
i∈IXi, where Xi1 ∩Xi2 =∅ for any i1, i2∈I. Thenfi :Xi →2 is defined asfi(x) =f(x) for an arbitrary x∈Xi andi∈I.
We define the mappings between direct products of two sets of concept lat- tices (that correspond to the two sets of 2-contexts given above) in the following form:
Definition 7. Let us define the mappingsh⇑,⇓ias follows
⇑:Y
i∈I
Ext(Ci)→Y
j∈J
Int(Dj) and ⇓:Y
j∈J
Int(Dj)→Y
i∈I
Ext(Ci)
⇑(Φ)j=^
i∈I
↑ij (Φi), for any Φ∈Y
i∈I
Ext(Ci)
⇓(Ψ)i= ^
j∈J
↓ij (Ψj), for any Ψ ∈Y
j∈J
Int(Dj)
such that(↑ij,↓ij)is a pair of derivation operators defined onhBi, Aj, ρijiwhere ρij =^
{β ∈2-Bonds(Ci,Dj) : (∀(bi, aj)∈Bi×Aj)β(bi, aj)≥rij(bi, aj)}.
2.4 Heterogeneous formal contexts
A heterogeneous extension in FCA based on the totally diversification of objects, attributes and table fields has been introduced in [3]. In the following, we remind the definition of a heterogeneous formal context and its derivation operators.
Definition 8. Heterogeneous formal context is a tupleC=hB, A,P, R,U,V,i, where
– B andA are non-empty sets,
– P={hPb,a,≤Pb,ai: (b, a)∈B×A} is a system of posets,
– R is a mapping from B×A such that R(b, a) ∈ Pb,a for any b ∈ B and a∈A,
– U = {hUb,≤Ubi : b ∈ B} and V = {hVa,≤Vai : a ∈ A} are systems of complete latices,
– = {◦b,a : (b, a) ∈ B ×A} is a system of isotone and left-continuous mappings◦b,a :Ub×Va −→Pb,a.
Let us define the derivation operators of a heterogeneous formal context as a pair of mappings(%,.), whereby %:Q
b∈BUb→Q
a∈AVa and.:Q
a∈AVa→ Q
b∈BUb such that – .(f)(a) =W
{v∈Va|f(b)◦b,av≤R(b, a)}for any f ∈Q
b∈BUb – %(g)(b) =W{u∈Ub|u◦b,ag(a)≤R(b, a)} for anyg∈Q
a∈AVa.
3 Problem description and sketch of solution
In this section we discussed why we have proposed an equivalent definition of bond. First, consider the classical definition of bond. It is a binary relation (table) between objects and attributes from different contexts such that its rows are intents and columns are extents of different input contexts. The issue of looking for bonds in a classical or homogeneous fuzzy case can be solved successfully [17, 21].
The solution of this issue requires the alternative definition of a bond. Hence, new definition of a bond focuses not only on a relation with some special prop- erties, but also on a bond as a formal context, whereby its concept lattice is connected to concept lattices of input contexts in some sense. As a consequence, a generalization for heterogeneous bonds is possible. One can find the methods in effort to equivalently modify the input heterogeneous formal contexts and to extract bonds as the extents of a direct product.
The proposed modification runs as follows. Each individual pair that includes a ”conjunction” ◦b,a and a value of the poset Pb,a is replaced by a bond from 2-Bonds(hUb, Ub,≤i,hVa, Va,≥i). This completely covers the Galois connection between the complete lattices of any object–attribute pair fromB×A.
At the beginning, we will show how this modification looks in terms of sec- ond order formal contexts. Then we define new modified heterogeneous formal context such that its concept lattice is identical to the original.
4 Second order form of scaled heterogeneous formal context
In effort to formalize the second order form of scaled heterogeneous formal con- text and its derivation operators, the definition of the following mappings is required:
Definition 9. Let(L,≤)be a complete lattice. Let us define mappings(−)Land (−)L where
– (−)L:L→2L such that kL(m) = (m≤k)for any k, m∈L – (−)
L: 2L→Lsuch that XL =WX for any X⊆L.
Let us have an arbitrary f ∈Q
b∈BUb. Let us denotef as a subset ofS
b∈BUb defined as f =S
b∈B{u∈Ub|u≤f(b)}. Similarly for any g∈Q
a∈AVa.
More information about Cartesian representation of fuzzy sets could be found in [10].
Now, consider a heterogeneous formal contextC = hB, A,P, R,U,V,i. A second order form of scaled heterogeneous formal context is defined as
C=
* [
b∈B
Ub,{hUb, Ub,≤i|b∈B}, [
a∈A
Va,{hVa, Va,≥i|a∈A}, R +
,
whereby all external contexts are classical crisp contexts andRis a classical crisp binary relation defined asR(u, v) = ((u◦b,av)≤R(b, a)) for any (u, v)∈Ub×Va
and any (b, a)∈B×A.
In the following, we define the derivation operators of such special second order formal context. First, we state some appropriate remarks and facts. Note that a relationRconstrained toUb×Va for any pair (b, a)∈B×Ais monotone in both arguments due to its definition. Similarly, consider the fact that any extent of hUb, Ub,≤i and any intent of hVa, Va,≥i is a principal down-set of a corresponding complete lattice (i.e. there exists an element in this complete lattice such that all lower or equal elements are in the extent or in the intent).
Hence, a relationR constrained toUb×Va for some (b, a)∈B×Ais a 2-bond betweenhUb, Ub,≤iandhVa, Va,≥iwhich will be denoted byρb,a. Note that any Φ∈Q
b∈BExt(hUb, Ub,≤i) has the formf for somef ∈Q
b∈BUb. Consider an arbitrary f ∈Q
b∈BUb and g ∈ Q
a∈AVa. Hence, the derivation operators are defined as follows:
– %(f)(v) =V
b∈B ↑b,a(f(b)b)(v) for anyv∈Va anda∈A – .(g)(u) =V
a∈A↓b,a(g(a)a)(u) for anyu∈Ub andb∈B.
In a previous definition, the pair of mappings (↑b,a,↓b,a) are derivation op- erators of a formal context hUb, Va, ρb,aifor any (b, a)∈B×A. For the sake of brevity, we use the shortened notation (−)b instead of (−)Ub and similarly (−)a instead of (−)Va.
Lemma 3. The concept lattices of C andC are isomorphic.
Proof. Consider an arbitraryf ∈Q
b∈BUb. We will show that%(f) =%(f).
Firstly consider the fact of left-continuity of both arguments of◦b,a for any (b, a)∈B×A. Due to this property, one can define two residuums in the following way. Let (b, a)∈B×Abe an arbitrary object-attribute pair and consider the arbitrary valuesu∈Ub,v∈Va andp∈Pb,a. Then define
– →b,a:Ub×Pb,a→Va, such thatu→b,ap=W{v∈Va|u◦b,av≤p}
– →a,b:Va×Pb,a→Ub, such thatv→a,bp=W{u∈Ub|u◦b,av≤p}.
% f
(v) = ^
b∈B
↑b,a
f(b)b
(v)
= ^
b∈B
^
u∈Ub
f(b)b(u)⇒ρb,a(u, v)
= ^
b∈B
^
u∈Ub
((u≤f(b))⇒(u◦b,av≤R(b, a)))
= ^
b∈B
^
u∈Ub;u6≤f(b)
1∧ ^
u∈Ub;u≤f(b)
((u≤f(b))⇒(u◦b,av≤R(b, a)))
= ^
b∈B
^
u∈Ub;u≤f(b)
(u◦b,av≤R(b, a))
= ^
b∈B
(f(b)◦b,av≤R(b, a))
= ^
b∈B
(v≤f(b)→b,a R(b, a))
= v≤ ^
b∈B
(f(b)→b,aR(b, a))
!
= v≤ ^
b∈B
_{w∈Va|(f(b)◦b,aw≤R(b, a))}
!
= v≤_
{w∈Va|(∀b∈B)(f(b)◦b,aw≤R(b, a))}
= (v≤%(f)(a)) =%(f)(a)a(v).
Analogously one can obtain.(g) (u) =.(g)(b)b(u). ut 4.1 Back to heterogeneous formal contexts
Now, we look at heterogeneous formal context introduced in Subsection 2.3. A second order formal context C can be seen as a special heterogeneous formal contextC, whereby the family of posetsb {hPb,a,≤i|(b, a)∈B×A}is replaced by
a set of 2-bonds{ρb,a∈2-Bonds(hUb, Ub,≤i,hVa, Va,≤i)|(b, a)∈B×A}. Hence, the final form of such heterogeneous formal context is
Cb=D
B, A, ρ,R,b U,V,{×b,a|(b, a)∈B×A}E
where
– ρ={ρb,a ∈2-Bonds(hUb, Ub,≤i,hVa, Va,≤i)|(b, a)∈B×A}
– ρb,a(u, v) = (u◦b,av≤R(b, a))
– R(b, a) =b ρb,a∈2-Bonds(hUb, Ub,≤i,hVa, Va,≤i) for any (b, a)∈B×A – ×b,a:Ub×Va→2Ub×Va defined as a Cartesian productu×v=u×v.
The derivation operators ofCbare defined as – ↑(f)(a) =W
{v∈Va|(∀b∈B)f(b)×b,av⊆ρb,a}for any f ∈Q
b∈BUb
– ↓(g)(b) =W
{u∈Ub|(∀a∈A)u×b,ag(a)⊆ρb,a}for anyg∈Q
a∈AVa. Lemma 4. The concept lattices of C andCbare identical.
Proof. Firstly consider that for any (u, v)∈Ub×Va for any (b, a)∈B×Athe following holds:
u×v⊆ρb,a=u×v⊆ρb,a
=ρb,a(u, v)
= (u◦b,av≤R(b, a)).
Letf ∈Q
b∈BUbbe arbitrary. Then
↑(f)(a) =_
{v∈Va|(∀b∈B)f(b)×b,av⊆ρb,a}
=_
{v∈Va|(∀b∈B)f(b)◦b,av≤R(b, a)}
=%(f)(a).
Analogously for↓(g)(b) =.(g)(b) for anyg∈Q
a∈AVa. ut
5 Bonds between heterogeneous formal contexts
We present a definition of a bond between two heterogeneous formal contexts which can be formulated as follows.
Definition 10. LetCi=hBi, Ai,Pi, Ri,Ui,Vi,iifori∈ {1,2}be two heteroge- neous formal contexts. The heterogeneous formal contextB=hB1, A2,P, R,U1,V2,i such thatExt(B)⊆Ext(C1)and Int(B)⊆Int(C2)is said to be a bond between two heterogeneous formal contextsC1 andC2.
5.1 Direct product of two heterogeneous formal contexts
In this subsection, we define a direct product of two heterogeneous formal con- texts. Further, we give an answer on how to find a bond between two heteroge- neous formal contexts.
Definition 11. Let Ci =hBi, Ai,Pi, Ri,Ui,Vi,iifori∈ {1,2} be two hetero- geneous formal contexts. The heterogeneous formal context
C1∆C2=hB1×A2, B2×A1,P∆, R∆,U∆,V∆,×i such that
– P∆={ρb1,a1∆ρb2,a2|((b1, a2),(b2, a1))∈(B1×A2)×(B2×A1)}
– whereρbi,ai(u, v) = (u◦bi,aiv≤Ri(bi, ai))for any(u, v)∈Ubi×Vai for any (bi, ai)∈Bi×Ai for any i∈ {1,2}
– R∆((b1, a2),(b2, a1)) = ρb1,a1∆ρb2,a2 for any bi ∈ Bi and ai ∈ Ai for all i∈ {1,2}
– U∆={γ1,2∈2-Bonds(hUb1, Ub1,≤i,hVa2, Va2,≥i)|(b1, a2)∈B1×A2} – V∆={γ2,1∈2-Bonds(hUb2, Ub2,≤i,hVa1, Va1,≥i)|(b2, a1)∈B2×A1} is said to be a direct product of two heterogeneous formal contexts.
Lemma 5. Let Ci = hBi, Ai,Pi, Ri,Ui,Vi,ii for i ∈ {1,2} be two heteroge- neous formal contexts. Let
R∈ Y
(b1,a2)∈B1×A2
2-Bonds(hUb1, Ub1,≤i,hVa2, Va2,≥i)
be an extent of the direct product C1∆C2. Then a heterogeneous formal context B=hB1, A2, ρ, R,U1,V2,×i where
ρ={2-Bonds(hUb1, Ub1,≤i,hVa2, Va2,≥i)|(b1, a2)∈B1×A2} is a bond between C1 andC2.
Proof. Let us have any intent ofB. Then there existsf ∈Q
b1∈B1Ub1 such that
%B(f)(a2)a2(v2) =%B(f)(v2)
= ^
b1∈B1
↑R(b1,a2)(f(b1)b1)(v2)
= ^
b1∈B1
^
u1∈Ub1
(f(b1)b1(u1)⇒R(b1, a2)(u1, v2)) R=.∆(Q) for some Q∈Q
(b2,a1)∈B2×A12-Bonds(hUb2, Ub2,≤i,hVa1, Va1,≥i)
= ^
b1∈B1
^
u1∈Ub1
(f(b1)b1(u1)⇒.∆(Q)(b1, a2)(u1, v2))
= ^
b1∈B1
^
u1∈Ub1
f(b1)b1(u1)⇒ ^
(b2,a1)∈B2×A1
↓ρb1,a1∆ρb2,a2 (Q(b2, a1))(u1, v2)
= ^
b1∈B1
^
u1∈Ub1
f(b1)b1(u1)⇒
^
b2∈B2
^
a1∈A1
^
(u2,v1)∈Ub2×Va1
Q(b2, a1)(u2, v1)⇒(ρb1,a1∆ρb2,a2)((u1, v2),(u2, v1))
= ^
b1∈B1
^
u1∈Ub1
f(b1)b1(u1)⇒
^
b2∈B2
^
a1∈A1
^
u2∈Ub2
^
v1∈Va1
Q(b2, a1)(u2, v1)⇒(¬ρb1,a1(u1, v1)⇒ρb2,a2(u2, v2))
= ^
b1∈B1
^
b2∈B2
^
a1∈A1
^
u2∈Ub2
^
v1∈Va1
^
u1∈Ub1
f(b1)b1(u1)⇒
Q(b2, a1)(u2, v1)⇒(¬ρb1,a1(u1, v1)⇒ρb2,a2(u2, v2))
= ^
b2∈B2
^
u2∈Ub2
_
b1∈B1
_
u1∈Ub1
_
a1∈A1
_
v1∈Va1
f(b1)b1(u1)∧Q(b2, a1)(u2, v1)∧ ¬ρb1,a1(u1, v1)
⇒ρb2,a2(u2, v2)
= ^
b2∈B2
^
u2∈Ub2
(q(b2)b2(u2)⇒ρb2,a2(u2, v2))
=%C
2(q)(v2) =%C2 (q)(a2)(v2) where
q(b2)(u2) = _
b1∈B1
_
u1∈Ub1
_
a1∈A1
_
v1∈Va1
f(b1)(u1)∧Q(b2, a1)(u2, v1)∧¬ρb1,a1(u1, v1)
Hence,%B(f) =%C2 (q). So any intent ofB is an intent ofC2. By using the following equality
(¬ρb1,a1(u1, v1)⇒ρb2,a2(u2, v2)) = (¬ρb2,a2(u2, v2)⇒ρb1,a1(u1, v1)) analogously we obtain that any extent ofBis an extent ofC1. Hence,Bis a bond
betweenC1andC2. ut
6 Conclusion
Bonds and theirL-fuzzy generalizations represent a feasible way to explore the relationships between formal contexts. In this paper we have investigated the notion of a bond with respect to the heterogeneous formal contexts. In conclu- sion, an alternative definition of a bond provides an efficient tool to work with
the nonhomogeneous data and one can further explore this uncharted territory in formal concept analysis.
Categorical properties of heterogeneous formal contexts and bonds as mor- phisms between such objects and categorical relationship to homogeneous FCA categorical description will be studied in the near future.
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