Recently, the concept lattices working with the heterogeneous 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 de nition of a bond and to investigate the relationships between bonds and the particular formal contexts.
Formal concept analysis (FCA) [
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 de nition 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.
De nition 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 objects B and a set of their attributes A. Relationships between objects and their attributes are saved in the relation R. Let us de ne a pair of derivation operators ("; #) as the mappings between powersets of B and A such that { ": P(B) ! P(A) and #: P(A) ! P(B) where for any X { " (X) = fa 2 Aj(8b 2 X)(b; a) 2 Rg { # (Y ) = fb 2 Bj(8a 2 Y )(b; a) 2 Rg.
B and Y
A is Such derivation operators can be de ned as the mappings between 2-sets (borrowed from fuzzy generalization of FCA that is sometimes easier to use) { ": 2B ! 2A and #: 2A ! 2B where for any X 2 2B and Y 2 2A { " (X)(a) = Vb2B((b 2 X) ) ((b; a) 2 R)) = V { # (Y )(b) = Va2A((a 2 Y ) ) ((b; a) 2 R)) = Vab22AB((YX((ab)) )) RR((bb;;aa)))).
Pair of such derivation operators forms an antitone Galois connection between complete lattices of all subsets of B and A. Hence, the compositions of the mappings form closure operators on such complete lattices.
De nition 2. Let C = hB; A; Ri be a formal context. Any pair of sets (X; Y ) 2 2B 2A is said to be a formal concept i 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 notation Int(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 de nition De nition 3. Let Ci = hBi; Ai; Rii for i 2 f1; 2g be two formal contexts. Relation B1 A2 is said to be a bond i any row of the table is an intent of C2 and any of its column is an extent of C1. Set of all bonds between C1 and C2 will be denoted by 2-Bonds(C1; C2).
Lemma 1. Let Ci = hBi; Ai; Rii for i 2 f1; 2g be two formal contexts. Then
B1 A2 is a bond between C1 and C2 if and only if Ext(hB1; A2; i) Ext(C1) and Int(hB1; A2; i) Int(C2).
Proof. ): Let X 2 Ext(hB1; A2; i) be an arbitrary extent of any bond between formal contexts C1 and C2. Derivation operators of Ci will be denoted by ("i; #i) for i 2 f1; 2g. Derivation operators of the bond will be denoted by (" ; # ). Then there exists a set of attributes Y A2 such that # (Y )(b1) = ^ (Y (a2) )
(b1; a2)) a22A2 ( ; a2) is an extent of Ext(C1) hence there exists Z A1 =
^ (Y (a2) )#1 (Z)(b1)) a22A2 = ^
a22A2 = ^
a22A2 a12A1 = ^
a22A2 a12A1 = ^ _ a12A1 a22A2
^ a12A1 Y (a2) )
Z(a1) ) R1(b1; a1) ^ (Y (a2) ) (Z(a1) ) R1(b1; a1))) ^ ((Y (a2) ^ Z(a1)) ) R1(b1; a1))
Y (a2) ^ Z(a1) ) R1(b1; a1) =
^ (ZY (a1) ) R1(b1; a1)) a12A1 =#1 (ZY )(b1) where ZY (a1) = Wa22A2 (Y (a2) ^ Z(a1)) Hence, Ext(hB1; A2; i) Ext(C1). Similarly for intents.
(: Assume a formal context hB1; A2; i such that it holds Ext(hB1; A2; i) Ext(C1) and Int(hB1; A2; i) 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 that is a bond between C1 and C2. tu
Hence, the notion of bond can be de ned equivalently as follows. De nition 4. Let Ci = hBi; Ai; Rii for i 2 f1; 2g be two formal contexts. Formal context B = hB1; A2; i is said to be a bond between C1 and C2 if Ext(B) Ext(C1) and Int(B) Int(C2).
More about the equivalent de nition of bond could be found in [17{19]. 2.2
Direct product of two formal contexts and bonds
Let us recall the de nition and important property of direct product of two
formal contexts. More details about such topic can be found in [
Lemma 2. Let Ci = hBi; Ai; Rii be two formal contexts. Every extent of C1 C2 is a bond between C1 and C2. 2.3
In this subsection, we remind a notion of a second order formal concept [
Moreover, consider two non-empty sets of 2-contexts notated { fCi = hBi; Ti; pii : i 2 Ig { fDj = hOj; Aj; qji : j 2 J g.
Formal context of second order is a tuple
D [ Bi; fCi; i 2 Ig; [ Aj; fDj; j 2 J g; i2I j2J [ (i;j)2I J
E ri;j , where ri;j : Bi Aj ! 2 de ned as ri;j(b; a) = r(b; a) for any b 2 Bi and a 2 Aj.
In what follows, consider the below described notation. Let us have an L-set f : X ! 2 for a non-empty universe set X = Si2I Xi, where Xi1 \ Xi2 = ; for any i1; i2 2 I. Then f i : Xi ! 2 is de ned as f i(x) = f (x) for an arbitrary x 2 Xi and i 2 I.
We de ne the mappings between direct products of two sets of concept lattices (that correspond to the two sets of 2-contexts given above) in the following form: De nition 7. Let us de ne the mappings h*; +i as follows *: Y Ext(Ci) ! i2I
Y Int(Dj) and +: Y Int(Dj) ! Y Ext(Ci) j2J j2J i2I * ( )j = ^ "ij ( i), for any
2 Y Ext(Ci) + ( )i = ^ #ij ( j), for any
2 Y Int(Dj) i2I j2J i2I j2J such that ("ij; #ij) is a pair of derivation operators de ned on hBi; Aj; iji where ij = ^f 2 2-Bonds(Ci; Dj) : (8(bi; aj) 2 Bi Aj) (bi; aj) rij(bi; aj)g. 2.4
A heterogeneous extension in FCA based on the totally diversi cation of objects,
attributes and table elds has been introduced in [
! Pb;a.
Let us de ne the derivation operators of a heterogeneous formal context as a pair of mappings (%; .), whereby %: Qb2B Ub ! Qa2A Va and .: Qa2A Va ! Qb2B Ub such that { . (f )(a) = Wfv 2 Vajf (b) b;a v { % (g)(b) = Wfu 2 Ubju b;a g(a)
R(b; a)g for any f 2 Qb2B Ub
R(b; a)g for any g 2 Qa2A Va. 3
In this section we discussed why we have proposed an equivalent de nition of
bond. First, consider the classical de nition of bond. It is a binary relation (table)
between objects and attributes from di erent contexts such that its rows are
intents and columns are extents of di erent input contexts. The issue of looking
for bonds in a classical or homogeneous fuzzy case can be solved successfully
[
The solution of this issue requires the alternative de nition of a bond. Hence, new de nition of a bond focuses not only on a relation with some special properties, 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 nd the methods in e ort to equivalently modify the input heterogeneous formal contexts and to extract bonds as the extents of a direct product.
The proposed modi cation 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 from B A.
At the beginning, we will show how this modi cation looks in terms of second order formal contexts. Then we de ne new modi ed heterogeneous formal context such that its concept lattice is identical to the original.
In e ort to formalize the second order form of scaled heterogeneous formal context and its derivation operators, the de nition of the following mappings is required: De nition 9. Let (L; ) be a complete lattice. Let us de ne mappings ( )L and ( )L where { ( )L : L ! 2L such that kL(m) = (m k) for any k; m 2 L { ( )L : 2L ! L such that XL = W X for any X L.
Let us have an arbitrary f 2 Qb2B Ub. Let us denote f as a subset of Sb2B Ub de ned as f = Sb2Bfu 2 Ubju f (b)g. Similarly for any g 2 Qa2A Va.
More information about Cartesian representation of fuzzy sets could be found
in [
Now, consider a heterogeneous formal context C = hB; A; P; R; U ; V; i. A second order form of scaled heterogeneous formal context is de ned as C = * [ Ub; fhUb; Ub; ijb 2 Bg; [ Va; fhVa; Va; ija 2 Ag; R b2B a2A + whereby all external contexts are classical crisp contexts and R is a classical crisp binary relation de ned as R(u; v) = ((u b;a v) R(b; a)) for any (u; v) 2 Ub Va and any (b; a) 2 B A.
In the following, we de ne the derivation operators of such special second order formal context. First, we state some appropriate remarks and facts. Note that a relation R constrained to Ub Va for any pair (b; a) 2 B A is monotone in both arguments due to its de nition. 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 relation R constrained to Ub Va for some (b; a) 2 B A is a 2-bond between hUb; Ub; i and hVa; Va; i which will be denoted by b;a. Note that any 2 Qb2B Ext(hUb; Ub; i) has the form f for some f 2 Qb2B Ub. Consider an arbitrary f 2 Qb2B Ub and g 2 Qa2A Va. Hence, the derivation operators are de ned as follows: { %(f )(v) = Vb2B "b;a (f (b)b)(v) for any v 2 Va and a 2 A { .(g)(u) = Va2A #b;a (g(a)a)(u) for any u 2 Ub and b 2 B.
In a previous de nition, the pair of mappings ("b;a; #b;a) are derivation operators of a formal context hUb; Va; b;ai for any (b; a) 2 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 and C are isomorphic.
Proof. Consider an arbitrary f 2 Qb2B Ub. We will show that %(f ) = % (f ).
Firstly consider the fact of left-continuity of both arguments of b;a for any (b; a) 2 B A. Due to this property, one can de ne two residuums in the following way. Let (b; a) 2 B A be an arbitrary object-attribute pair and consider the arbitrary values u 2 Ub, v 2 Va and p 2 Pb;a. Then de ne { !b;a: Ub { !a;b: Va
Pb;a ! Va, such that u !b;a p = Wfv 2 Vaju b;a v Pb;a ! Ub, such that v !a;b p = Wfu 2 Ubju b;a v pg pg. % f (v) = ^ "b;a f (b) b
(v) b2B = ^ ^
b2B u2Ub = ^
^ ((u b2B u2Ub 0 = (v
f (b) !b;a R(b; a)) ^ (f (b) !b;a R(b; a)) b2B
! ^ _fw 2 Vaj(f (b) b;a w b2B _fw 2 Vaj(8b 2 B)(f (b) b;a w
a % (f )(a)) = % (f )(a) (v): ! R(b; a))g
R(b; a))g b Analogously one can obtain . (g) (u) = . (g)(b) (u). 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 context Cb, whereby the family of posets fhPb;a; ij(b; a) 2 B Ag is replaced by tu a set of 2-bonds f b;a 2 2-Bonds(hUb; Ub; i; hVa; Va; i)j(b; a) 2 B the nal form of such heterogeneous formal context is Ag. Hence,
D Cb = B; A; ; Rb; U ; V; f b;aj(b; a) 2 B AgE where { = f b;a 2 2-Bonds(hUb; Ub; i; hVa; Va; i)j(b; a) 2 B { b;a(u; v) = (u b;a v R(b; a)) Ag { Rb(b; a) = b;a 2 2-Bonds(hUb; Ub; i; hVa; Va; i) for any (b; a) 2 B { b;a : Ub Va ! 2Ub Va de ned as a Cartesian product u v = u A v.
The derivation operators of Cb are de ned as { " (f )(a) = Wfv 2 Vaj(8b 2 B)f (b) b;av { # (g)(b) = Wfu 2 Ubj(8a 2 A)u b;ag(a) Lemma 4. The concept lattices of C and Cb are identical.
Proof. Firstly consider that for any (u; v) 2 Ub following holds: b;ag for any f 2 Qb2B Ub b;ag for any g 2 Qa2A Va.
Va for any (b; a) 2 B
A the u v b;a = u v
b;a = b;a(u; v) = (u b;a v
R(b; a)): Let f 2 Qb2B Ub be arbitrary. Then " (f )(a) = _fv 2 Vaj(8b 2 B)f (b) b;av = _fv 2 Vaj(8b 2 B)f (b) b;a v =% (f )(a): b;ag R(b; a)g Analogously for # (g)(b) =. (g)(b) for any g 2 Qa2A Va. tu 5
We present a de nition of a bond between two heterogeneous formal contexts which can be formulated as follows.
De nition 10. Let Ci = hBi; Ai; Pi; Ri; Ui; Vi; ii for i 2 f1; 2g be two heterogeneous formal contexts. The heterogeneous formal context B = hB1; A2; P; R; U1; V2; i such that Ext(B) Ext(C1) and Int(B) Int(C2) is said to be a bond between two heterogeneous formal contexts C1 and C2.
Direct product of two heterogeneous formal contexts In this subsection, we de ne a direct product of two heterogeneous formal contexts. Further, we give an answer on how to nd a bond between two heterogeneous formal contexts.
De nition 11. Let Ci = hBi; Ai; Pi; Ri; Ui; Vi; ii for i 2 f1; 2g be two heterogeneous formal contexts. The heterogeneous formal context
C1 C2 = hB1
A2; B2
A1; P ; R ; U ; V ; i such that { P = f b1;a1 b2;a2 j((b1; a2); (b2; a1)) 2 (B1 A2) (B2 A1)g { where bi;ai (u; v) = (u bi;ai v Ri(bi; ai)) for any (u; v) 2 Ubi Vai for any (bi; ai) 2 Bi Ai for any i 2 f1; 2g { R ((b1; a2); (b2; a1)) = b1;a1 b2;a2 for any bi 2 Bi and ai 2 Ai for all i 2 f1; 2g { U = f 1;2 2 2-Bonds(hUb1 ; Ub1 ; i; hVa2 ; Va2 ; i)j(b1; a2) 2 B1 A2g { V = f 2;1 2 2-Bonds(hUb2 ; Ub2 ; i; hVa1 ; Va1 ; i)j(b2; a1) 2 B2 A1g 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 2 f1; 2g be two heterogeneous formal contexts. Let
R 2
Y (b1;a2)2B1 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; ; R; U1; V2; i where = f2-Bonds(hUb1 ; Ub1 ; i; hVa2 ; Va2 ; i)j(b1; a2) 2 B1 A2g is a bond between C1 and C2.
Proof. Let us have any intent of B. Then there exists f 2 Qb12B1 Ub1 such that ^ (f (b1)b1 (u1) ) R(b1; a2)(u1; v2)) (Q) for some Q 2 Q(b2;a1)2B2 A1 2-Bonds(hUb2 ; Ub2 ; i; hVa1 ; Va1 ; i) ^ (f (b1)b1 (u1) ).
(Q)(b1; a2)(u1; v2)) ^ %B (f )(a2)a2 (v2) = %B(f )(v2)
"R(b1;a2) (f (b1)b1 )(v2) = ^
b12B1 = ^ R =. = ^ = ^ b12B1 u12Ub1 b12B1 u12Ub1 = ^ = ^ = ^ = ^
^ b22B2 u22Ub2 _
_ ) b2;a2 (u2; v2) = ^
b22B2 u22Ub2 where q(b2)(u2) = _ = %C2 (q)(v2) = %C2 (q)(a2)(v2) _ _
_ f (b1)(u1)^Q(b2; a1)(u2; v1)^: b1;a1 (u1; v1) b12B1 u12Ub1 a12A1 v12Va1 Hence, %B (f ) =%C2 (q). So any intent of B is an intent of C2.
By using the following equality
(: b1;a1 (u1; v1) ) b2;a2 (u2; v2)) = (: b2;a2 (u2; v2) ) b1;a1 (u1; v1)) analogously we obtain that any extent of B is an extent of C1. Hence, B is a bond between C1 and C2. tu 6
Bonds and their L-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 conclusion, an alternative de nition of a bond provides an e cient 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 morphisms between such objects and categorical relationship to homogeneous FCA categorical description will be studied in the near future.