- Email: [email protected]
Note on generating fuzzy concept lattices via Galois connections
Information Sciences 185 (2012) 128–136
Contents lists available at SciVerse ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Note on generating fuzzy concept lattices via Galois connections Jozef Pócs Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01 Košice, Slovakia
a r t i c l e
i n f o
Article history: Received 29 September 2010 Received in revised form 7 July 2011 Accepted 18 September 2011 Available online 24 September 2011
a b s t r a c t The aim of this paper is to compare an approach of creating fuzzy concept lattices proposed by Popescu with several other approaches. Particularly, we show that this approach is in some way equivalent to the approach of Krajcˇi called generalized concept lattices. We also give a straightforward generalization of Popescu’s approach to non-homogeneous cases. 2011 Elsevier Inc. All rights reserved.
Keywords: Galois connection Fuzzy concept lattice Generalized concept lattice
1. Introduction Fuzzy Formal Concept Analysis arose as an attempt for generalization of the classical Formal Concept Analysis [10] to the cases in which objects and attributes can take not only crisp but also fuzzy values. There are several approaches and methods on how to construct fuzzy concept lattices. We mention the approach of Beˇlohlávek [1–3] based on logical framework of complete residuated lattices, the work of Georgescu and Popescu to extend this framework to non-commutative logic [11–13], the approach of Krajcˇi [16] and Popescu [20]. There are also other approaches that generalized the previous ones, from which we mention the approach of Medina, Ojeda-Aciego, Ruiz-Calviño dealing with multi-adjoint concept lattices [18,19], cf. also [17], the work of Jaoua and Elloumi on Galois lattices of real relations [14], and the paper on variable threshold concept lattices by Zhang et al. [24], cf. also [4]. Most of these approaches can be characterized as searching for a Galois connection between powers of complete lattices. Note that there are also other methods for generating fuzzy concept lattices not involving Galois connections (see [6] for a relatively good overview of some existing methods). Concept analysis is also considered in other frameworks, cf. [22] or [23]. In [20] Popescu proposed a method for creating fuzzy concept lattices different from other ones. This method does not involve a ‘‘static’’ fuzzy relation between objects and attributes (i.e., an object has an attribute in a certain degree) as the input, but it involves a ‘‘dynamic’’ relation that is Galois connection expressing the mutual relationship between object and attribute. This approach has two major advantages. First, this construction does not require any logical structure on underlaying complete lattices. Further, it is very flexible and allows generalization to non-homogeneous cases, i.e., where various complete lattices are assigned to different objects and attributes. It seems very convenient to use mixed types of structures for truth values in object-attribute models, hence one of our goals is to provide a straightforward generalization of Popescu’s approach to non-homogeneous cases. Moreover, we prove that this generalization provides the most general method of creating concept lattices, which are based on Galois connections between direct products of complete lattices. Further, we describe the relationship between Popescu’s and some other approaches. Specifically, we deal with classical crisp Ganter and Wille’s [10] case, Krajcˇi’s [15] and Ben Yahia and Jaoua’s [7] one-sided concept lattices, and finally with generalized concept lattices introduced by Krajcˇi [16]. E-mail address: [email protected] 0020-0255/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.09.021
J. Pócs / Information Sciences 185 (2012) 128–136
129
Generalized concept lattices represent another approach to fuzzify concept lattices. This approach, in contrast with that of Popescu, involves a ‘‘static’’ fuzzy relation as the input, with values in partially ordered set, which is independent on the underlaying structures for objects and attributes respectively. The main aim of this paper is to show that appropriate selection of this partially ordered set make these approaches equivalent, i.e., with respect to all possible formal contexts, they produce all Galois connections between powers of two complete lattices. 2. Preliminaries In this section we give basic overview of Galois connections needed for our purposes. Note that by Galois connection we will always understand an antitone Galois connection, as it is usual in Formal Concept Analysis. We assume that the reader is familiar with the basic notions of lattice theory. Definition 2.1. Let (P, 6) and (Q, 6) be ordered sets and let
u : P ! Q and w : Q ! P be maps between these ordered sets. Such a pair (u, w) of mappings is called a Galois connection between the ordered sets if: (a) p1 6 p2 implies u(p1) P u(p2), (b) q1 6 q2 implies w(q1) P w(q2), (c) p 6 w(u(p)) and q 6 u(w(q)). These two maps are also called dually adjoint to each other. We note that
u ¼ u w u and w ¼ w u w and that the conditions (a), (b) and (c) are equivalent to the following one: (d) p 6 w(q) if and only if u(p) P q. For partially ordered sets P, Q denote by Gal(P, Q) the set of all Galois connections between P and Q. The class of all complete lattices will be denoted by CL. We will use the following well known fact (see [10,21]): Lemma 2.2. A map u : L ? M between complete lattices L and M has a dual adjoint if and only if
u
_
! xi
i2I
¼
^
uðxi Þ;
i2I
holds for any subset {xi : i 2 I} of L. In this case the dual adjoint w is uniquely determined by
wðyÞ ¼
_ fx 2 L : uðxÞ P yg:
Galois connections between complete lattices are closely related to the notion of closure operator and closure system. Let L be a complete lattice. By a closure operator in L we understand a mapping c : L ? L satisfying: (a) x 6 c(x) for all x 2 L, (b) c(x1) 6 c(x2) for x1 6 x2, (c) c(c(x)) = c(x) for all x 2 L (i.e., c is idempotent). Subset X of the complete lattice L is called closure system in L if X is closed under arbitrary meets. We note, that this condition guarantees that (X, 6) is a complete lattice, in which the infima are the same as in L, but the suprema in X may not coincide with those from L. For a closure operator c in L, the set FP(c) of all fixed points of c (i.e., FP(c) = {x 2 L : c(x) = x}) V is the closure system in L. Conversely, for closure system X in L, mapping CX : L ? L defined by CX(x) = {u 2 X : x 6u} is the closure operator in L. Moreover these correspondences are inverses of each other, i.e., FP(CX) = X for each closure system X in L and CFP(c) = c for each closure operator c in L. Next, we recall the relationship between the closure operators induced by the Galois connections. Two ordered sets P, Q are called dually isomorphic, if there is an antitone (order reversing) bijective mapping f : P ? Q such that f1 is also antitone. Let L, M 2 CL and (u, w) be Galois connection between L and M. Then mapping u w : L ? L is closure operator in L, similarly, w u : M ? M is closure operator in M. Moreover the corresponding closure systems are dually isomorphic. Conversely, suppose that X1 and X2 are closure systems in L, M respectively, and f : X1 ? X2 is dual isomorphism between complete lattices (X1, 6) and (X2, 6). Then a pair ðcX 1 f ; cX 2 f 1 Þ, where cX 1 ; cX 2 are closure operators corresponding to X1 and to X2, forms the Galois connection between L and M. Hence, any Galois connection between complete lattices induces dually isomorphic closure systems on these lattices and vice versa.
130
J. Pócs / Information Sciences 185 (2012) 128–136
In particular, we obtain that Gal(L, M) – ; for all L, M 2 CL, since {1L} and {1M} are dually isomorphic closure systems in L and M. The symbols 1L, 1M denote the greatest elements of L and M respectively. Similarly, 0L, 0M will denote the smallest elements. Galois connections between complete lattices stand in a one-to-one correspondence with so called G-ideals. Let L, M be complete lattices. h # L M is called a G-ideal of L M when (i) (x, y) 6 (a, b) and (a, b) 2 h implies (x, y) 2 h, W V V W (ii) if {(ai, bi)}i2I # h then ( i2Iai, i2Ibi) 2 h and ( i2Iai, i2Ibi) 2 h, (iii) (0L, 1M) 2 h and (1L, 0M) 2 h. Proposition 2.3 [21]. Let L and M be complete lattices. If (u, w) is a Galois connection between L and M, then
h ¼ fða; bÞ : uðaÞ P bg # L M is a G-ideal. Conversely, let h # L M be a G-ideal. If the mappings u : L ? M, w : M ? L are defined by
uðaÞ ¼
_ _ fb : ða; bÞ 2 hg ; wðbÞ ¼ fa : ða; bÞ 2 hg
then (u, w) is a Galois connection between L and M. Moreover, this correspondences between Galois connections and G-ideals are mutually inverse. Properties of Galois connections allow us to construct complete lattices which are in our special interest. Formally, let (u, w) be a Galois connection between complete lattices L and M. Denote by Lu,w a subset of L M consisting of all pairs (x, y) with u(x) = y and w(y) = x. Define the partial order on Lu,w as follows:
ðx1 ; y1 Þ 6 ðx2 ; y2 Þ if
x1 6 x2 iffy1 P y2
ðdue to condition ðdÞ of Definition 2:1Þ:
Proposition 2.4. Let (u, w) be a Galois connection between complete lattices L and M. Then (Lu,w, 6) forms a complete lattice, where
^ ðxi ; yi Þ ¼
^
i2I
i2I
xi ; u
^ i2I
!! xi
;
_ ðxi ; yi Þ ¼
w
i2I
^ i2I
! yi ;
^
! yi
i2I
for each family (xi,yi)i2I of elements from Lu,w. The proof of this proposition is based on the fact that w(u(L)) and u(w(M)) form dually isomorphic closure systems, and expressions for infima and suprema are obtained as in the case of classical concept lattices, see [10]. V W Note that if I = ;, than i2I(xi, yi) = (1L, u(1L)) and i2I(xi, yi) = (w(1M), 1M). 3. Lattices generated by formal fuzzy contexts In this section we describe a general method of generating fuzzy concept lattices proposed by Popescu [20] for the homogeneous case, i.e., there is only one structure of truth degrees for all objects and possibly another one for all attributes. In this paper the author gave the following definition of formal context: A context is a system C ¼ ðB; L; A; M; u; wÞ, where B is a set of objects, L is a complete lattice (structure of truth values for each b 2 B), A is a set of attributes, M is a complete lattice (structure of truth values for each a 2 A) and u = (ub,a)(b,a)2BA, w = (wb,a)(b,a)2BA where for each b 2 B, a 2 A(ub,a, wb,a) 2 Gal(L, M). Consequently, there are defined two mappings " : LB ? MA and ; : MA ? LB by
" ðf ÞðaÞ ¼
^ b2B
ub;a ðf ðbÞÞ; # ðgÞðbÞ ¼
^
wb;a ðgðaÞÞ
a2A
for all f 2 LB, g 2 MA, b 2 B, a 2 A. As we show later this two mappings form Galois connection between LB and MA. Now we give a straightforward generalization of this approach, also in non-homogeneous case, i.e., for each object and each attribute there is assigned possibly different structure of truth degrees. In what follows, we will adopt the notation introduced in [20]. A 6-tuple C ¼ ðB; L; A; M; u; wÞ is called formal fuzzy context if: (i) B,A – ; (B is a set of objects, A is a set of attributes), (ii) L : B ? CL, M : A ? CL (recall that CL denotes the class of all complete lattices, thus for b 2 B, L(b) represents a complete lattice with possible truth values of the object b and similarly for a 2 A), (iii) u = (ub,a)(b,a)2BA, w = (wb,a)(b,a)2BA where for each b 2 B, a 2 A (ub,a, wb,a) 2 Gal(L(b), M(a)) (the pair (ub,a, wb,a) is Galois connection between lattices L(b) and M(a)).
131
J. Pócs / Information Sciences 185 (2012) 128–136
Further, define ":
" ðf ÞðaÞ ¼
^
Q
b2B LðbÞ
!
Q
a2A MðaÞ
as follows:
ub;a ðf ðbÞÞ for all a 2 A; if f 2
b2B
^ a2A
LðbÞ:
ð1Þ
b2B
Similarly we put #:
# ðgÞðbÞ ¼
Y
Q
a2A MðaÞ
!
Q
b2B LðbÞ
wb;a ðgðaÞÞ for all b 2 B;
as follows:
if g 2
Y
MðaÞ
ð2Þ
a2A
To emphasize that given context C corresponds to given pair (",;), we will sometimes write C as the a subscript, i.e., ð"C ; #C Þ. Example 3.1. Consider the following context C ¼ ðB; L; A; M; u; wÞ. Sets B and A consist of two elements, say B = {b1,b2}, A = {a1, a2}. The mapping L maps b1 to the two element chain 2 and b2 to the lattice P(2) isomorphic to the power set of the two element set. Similarly M maps a1 to 2 and a2 to the three element chain 3. Corresponding Galois connections ðubi ;aj ; wbi ;aj Þ between L(bi) and M(aj), for {i, j} = {1, 2}, are depicted in the following table (see Fig. 1). Note that for more legibility we only indicate corresponding dual isomorphism of closure systems. This obviously uniquely determine corresponding Galois connection. Using the definition of " and ; we obtain the following mapping (see Fig. 2) between L(b1) L(b2) ffi 2 P(2) and M(a1) M(a2) ffi 2 3, which forms a Galois connection, as we show in Theorem 3.2.
Theorem 3.2. Let (B, L, A, M, u, w) be a formal fuzzy context. Then the pair (", ;) forms a Galois connection between Q a2A MðaÞ.
Fig. 1. Example of formal fuzzy context.
Fig. 2. Resulting Galois connection of Example 3.1.
Q
b2B LðbÞ
and
132
J. Pócs / Information Sciences 185 (2012) 128–136
Proof. Suppose that f1 6 f2 where f1 ; f2 2
" ðf1 ÞðaÞ ¼
^
ub;a ðf1 ðbÞÞ P
b2B
^
Q
b2B LðbÞ.
For any a 2 A, b 2 B we have ub,a(f1(b)) P ub,a(f2(b)), hence
ub;a ðf2 ðbÞÞ ¼" ðf2 ÞðaÞ;
b2B
for all a 2 A, which yields "(f1) P "(f2). The condition g1 6 g2 implies ;(g1) P ;(g2), may be proved analogously. Q Next we prove that for any f 2 b2B LðbÞ; f 6# ð" ðf ÞÞ. Let b 2 B be an arbitrary element. Then for any a 2 A
^
" ðf ÞðaÞ ¼
ub0 ;a ðf ðb0 ÞÞ 6 ub;a ðf ðbÞÞ;
0
b 2B
hence
f ðbÞ 6 wb;a ðub;a ðf ðbÞÞÞ 6 wb;a ð" ðf ÞðaÞÞ; which yields
f ðbÞ 6
^
wb;a ð" ðf ÞðaÞÞ ¼# ð" ðf ÞÞðbÞ:
a2A
Similarly, the inequality g 6 "(;(g)) for g 2
Q
a2A MðaÞ,
may be proved analogously.
h
According to the result of this theorem, for the formal fuzzy context C ¼ ðB; L; A; M; u; wÞ we will call the lattice L",; the fuzzy concept lattice corresponding to C and denote it by FCLðCÞ. Corollary 3.3. FCLðCÞ is a complete lattice where
^ ðfi ; g i Þ ¼
^
i2I
i2I
fi ; "
^
!! fi
;
i2I
_ ðfi ; g i Þ ¼
#
i2I
^
! gi ;
i2I
^
! gi
i2I
for each family (fi,gi)i2I of elements from FCLðCÞ. The following theorem is a generalization of the so-called main theorem of concept lattices. Theorem 3.4. Let W be a complete lattice. Then W ffi FCLðCÞ if and only if for all b 2 B, a 2 A there are mappings cb : L(b) ? W, la : M(a) ? W such that S W S V (a) b2Bcb(L(b)) is -dense in W and a2Ala(M(a)) is -dense in W, (b) for all b 2 B, a 2 A, x 2 L(b), y 2 M(a), cb(x) 6 la(y) iff y 6 ub,a(x) iff x 6 wb,a(y). The proof of this theorem is exactly same as in [20] (see Lemma 5.1 and Proposition 5.2). In what follows, we will show that each Galois connection between direct products of complete lattices has a decomposition. Q Let {Li : i 2 I} be a system of complete lattices. For x 2 Lj we denote by 0jx the following element of i2I Li : 0jx ðjÞ ¼ x and j 0x ðiÞ ¼ 0Li for all i 2 I,i – j. Theorem 3.5. Let (U, W) be a Galois connection between C ¼ ðB; L; A; M; u; wÞ, such that "C ¼ U and #C ¼ W.
Q
b2B LðbÞ
and
Q
a2A MðaÞ.
Then there exists a formal context
Proof. For b 2 B, a 2 A, x 2 L(b) and y 2 M(a), define
ub;a ðxÞ ¼ U 0bx ðaÞ ; wb;a ðyÞ ¼ W 0ay ðbÞ: We show that (ub,a, wb,a) forms a Galois connection between L(b) and M(a). Consider the following series of equivalent assertions:
x 6 wb;a ðyÞ iff 0bx 6 W 0ay iff U 0bx P 0ay iff ub;a ðxÞ P y: This yields, that C ¼ ðB; L; A; M; u; wÞ forms a formal context. Q W Further, let f 2 b2B LðbÞ be an arbitrary element. Then f ¼ b2B 0bfðbÞ and according to Lemma 2.2 we obtain that:
Uðf ÞðaÞ ¼ U
_ b2B
!
0bfðbÞ ðaÞ ¼
^
U 0bfðbÞ ðaÞ ¼
b2B
^ b2B
ub;a ðf ðbÞÞ ¼ "C ðf ÞðaÞ:
J. Pócs / Information Sciences 185 (2012) 128–136
133
This shows that "C ¼ U. The equality #C ¼ W can be proved similarly, or we can simply use the fact, that the dual adjoint is unique, which yields #C ¼ W, too. h Theorem 3.5 shows, that the generalization of Popescu’s approach provides the most general method of creating Galois connections between direct products of complete lattices, i.e., it produces all possible Galois connections. 4. Relation to other approaches In this section we compare several existing approaches with that proposed by Popescu and show that the result of Theorem 3.5 can be used for showing that these other approaches are the most general of this kind, i.e. all possible Galois connections between certain lattices can be obtained by this way. 4.1. Classical FCA First, we will deal with Galois connections between power sets described by Birkhoff [8], which are the cornerstone of classical Formal Concept Analysis developed by Ganter and Wille [10]. We show that classical Galois connections are particular cases of Popescu’s approach, cf. [20]. Let (B, A, I) be a formal context, i.e., B, A – ; and I # B A. There is a pair of mappings ":2B ? 2A and ; : 2A ? 2B, which forms Galois connection between power sets
X " ¼ fy 2 A : ðx; yÞ 2 I; 8x 2 Xg; Y # ¼ fx 2 B : ðx; yÞ 2 I; 8y 2 Yg: The corresponding concept lattice is denoted by BðB; A; IÞ. Denote by 2 the two element lattice. There are only two Galois connections between 2 and 2, namely ð1; 1Þ where 1ð0Þ ¼ 1; 1ð1Þ ¼ 1 and ð0; 0Þ where 0ð0Þ ¼ 1; 0ð1Þ ¼ 0. Because of avoiding some technical complications we will substitute each subset of B and A with its characteristic function, due to their isomorphism. Also, we put L(b) = 2,M(a) = 2 for each b 2 B and a 2 A and define (ub,a, wb,a) 2 Gal(2, 2) as follows:
( ðub;a ; wb;a Þ ¼
ð1; 1Þ if ðb; aÞ 2 I; ð0; 0Þ if ðb; aÞ R I:
It is evident, that the 6-tuple C ¼ ðB; L; A; M; u; wÞ is a formal context. Conversely, for a given (B, L, A, M, u,w) we define (b, a) 2 I iff ðub;a ; wb;a Þ ¼ ð1; 1Þ. Since
" ðXÞðaÞ ¼
^
^
ub;a ðXðbÞÞ ¼
b2B
1ðXðbÞÞ ^
b2B;ðb;aÞ2I
^
0ðXðbÞÞ;
b2B;ðb;aÞRI
we obtain that: "(X)(a) = 0 iff there is b 2 X such that (b, a) R I iff a R X". Similarly, ;(Y)(b) = 0 iff b R Y;. Hence, we obtain that FCLðCÞ and BðB; A; IÞ are almost the same, up to isomorphism of power sets and characteristic functions. Now using Theorem 3.5 we obtain the well known result of Everett [9], which state that each Galois connection between power sets 2B and 2A can be obtained using binary relation I # B A. 4.2. One-sided concept lattices Further, we will deal with the one-sided approach, independently described by Krajcˇi [15] and by Ben Yahia and Jaoua [7], cf. also [5]. In this case the input consists of a L-context (B, A, I) where L is complete lattice and I : B A ? L. Again, there is a pair of mappings b : 2B ? LA and a : LA ? 2B,
bðXÞðaÞ ¼
^
Iðb; aÞ;
b2X
aðgÞ ¼ fb 2 B : 8a 2 A; gðaÞ 6 Iðb; aÞg: First, we note that each Galois connection between 2 and L is uniquely determined by single element c 2 L. This follows trivially from the fact, that each such a Galois connection induce dually isomorphic closure systems and there are only two closure systems in 2. Hence, for any c 2 L denote by ~c a mapping satisfying ~cð0Þ ¼ 1L and ~cð1Þ ¼ c and denote by ~c the dual g g adjoint to ~c. Now we put ðu ; w Þ ¼ Iðb; aÞ; Iðb; aÞ . Again we will not distinguish between subsets of B and their charb;a
b;a
acteristic functions from 2B. For any X # B we obtain
134
J. Pócs / Information Sciences 185 (2012) 128–136
" ðXÞðaÞ ¼
^
g Iðb; aÞð1Þ ^
b2X
^
g Iðb; aÞð0Þ ¼
bRX
^
Iðb; aÞ ^
b2X
^
1L ¼ bðXÞðaÞ:
bRX
Conversely,
# ðgÞðbÞ ¼
^
g Iðb; aÞ ðgðaÞÞ:
a2A
g Since Iðb; aÞ ðgðaÞÞ ¼ 1 iff g(a) 6 I(b, a), we obtain: b 2 a(g) iff ;(g)(b) = 1. As a consequence of Theorem 3.5 we obtain that this method produce all possible Galois connections between 2B and LA. Next we provide a heterogeneous generalization of this one-sided approach. Consider the following formal fuzzy context (B, L, A, M, u, w) where L(b) = 2 for all b 2 B and ðub;a ; wb;a Þ ¼ ð~c; ~c Þ for c 2 M(a). In classical way we have (B, A, I) where Q I(b, a) 2 M(a) for all a 2 M. The same computations show that a pair of mappings b : 2B ! a2A MðaÞ and Q B a : a2A MðaÞ ! 2
bðXÞðaÞ ¼
^
Iðb; aÞ;
b2X
aðgÞ ¼ fb 2 B : for each a 2 A; gðaÞ 6 Iðb; aÞg; forms a Galois connection between 2B and Q between 2B and a2A MðaÞ.
Q
a2A MðaÞ.
Again, due to Theorem 3.5, this approach covers all Galois connections
4.3. Generalized concept lattices At the end of this section we will deal with generalized concept lattices introduced by Krajcˇi [16]. Let P be a poset, L and M be complete lattices. Let :L M ? P be isotone and left-continuous in both their arguments, i.e., c1 6 c2 implies c1 d 6 c2 d, d1 6 d2 implies c d1 6 c d2, W if cj d 6 p for each j 2 J then ( j2Jcj) d 6 p, W if c dj 6 p for each j 2 J then c ( j2Jdj) 6 p. In the case of P being a complete lattice, this conditions are equivalent to the following identities:
c
_
! dj
j2J
_ ¼ ðc dj Þ and j2J
_
! cj
d¼
_ ðcj dÞ:
j2J
j2J
Remark 4.1. Left-continuity of implies c 0M = 0L d = 0P for all c 2 L,d 2 M, i.e., P possesses the least element 0P. This W follows from the fact that for J = ;, any p 2 P,d 2 M we have: for all j 2 J,cj d 6 p, hence ; d = 0L d 6 p. Let B and A be non-empty sets and let R be P-fuzzy relation on their Cartesian product, i.e., R : B A ? P. Define the following mappings % : LB ? MA and . : MA ? LB:
_
fd 2 M : ð8b 2 BÞ f ðbÞ d 6 Rðb; aÞg; _ . ðgÞðbÞ ¼ fc 2 L : ð8a 2 AÞ c gðaÞ 6 Rðb; aÞg:
% ðf ÞðaÞ ¼
Again, the pair (%,.) forms Galois connection between LB and MA, see [16]. For p 2 P denote hp = {(c, d) : c d 6 p}. Lemma 4.2. For all p 2 P, hp is a G-ideal of L M. Proof. Condition (c) of G-ideal is satisfied due to Remark 4.1 and (a) follows from isotonity of . Finally, suppose that V {(ci, di)}i2I # hp, i.e., ci di 6 p for each i 2 I. Then ( j2Icj) di 6 ci di 6 p for all i 2 I, hence by left-continuity V W W V ( i2Ici) ( i2Idi) 6 p. Similarly, ( i2Ici) ( i2Idi) 6 p. h Now define ub,a:L ? M and wb,a : M ? L as follows:
_ _ fd : ðc; dÞ 2 hRðb;aÞ g ¼ fd : c d 6 Rðb; aÞg; _ _ wb;a ðdÞ ¼ fc : ðc; dÞ 2 hRðb;aÞ g ¼ fc : c d 6 Rðb; aÞg:
ub;a ðcÞ ¼
According to Proposition 2.3, pair (ub,a, wb,a) is Galois connection between L and M, corresponding to the G-ideal hR(b, a). Thus C ¼ ðB; L0 ; A; M 0 ; u; wÞ forms a formal context, where L0(b) = L for all b 2 B and M0(a) = M for all a 2 A.
J. Pócs / Information Sciences 185 (2012) 128–136
135
Lemma 4.3. %¼ "C and .¼ #C . Proof. Note, that from isotonity and left-continuity of we obtain:
if d0 6
_ _ fd : c d 6 pg then c d0 6 c fd : c d 6 pg 6 p:
Let a 2 A be an arbitrary element. Since
"C ðf ÞðaÞ ¼
^
ub;a ðf ðbÞÞ ¼
b2B
^ _ fd : f ðbÞ d 6 Rðb; aÞg ; b2B
W
we obtain that "C ðf ÞðaÞ 6 fd : f ðbÞ d 6 Rðb; aÞg for all b 2 B. Hence, by left-continuity f ðbÞ "C ðf ÞðaÞ 6 Rðb; aÞ for all b 2 B, which yields "C ðf ÞðaÞ 6% ðf ÞðaÞ. Conversely, let b 2 B. Then
fd 2 M : ð8b0 2 BÞ f ðb0 Þ d 6 Rðb0 ; aÞg # fd : f ðbÞ d 6 Rðb; aÞg; W W thus for each b 2 B, this yields %(f)(a) = {d 2 M : ("b 2 B) f(b) d 6 R(b, a)} 6 {d : f(b) d 6 R(b, a)} = ub,a(f(b)). Hence, we V obtain % ðf ÞðaÞ 6 b2B ub;a ðf ðbÞÞ ¼ "C ðf ÞðaÞ. The equality #C ¼. may be proved analogously. h Finally, we will show that this approach is the most general in some sense. We will show that every concept lattice LU,W where (U, W) is a Galois connection between LB and MA is in fact a generalized concept lattice. Theorem 4.4. Let L, M be complete lattices and B, A be nonempty sets. Then there exists a complete lattice G and a mapping : L M ! G isotone and left-continuous in both their arguments, such that for any Galois connection (U, W) between LB and MA there is R : B A ! G with U = %R and W = .R. Proof. Denote by G the set of all G-ideals of L M, ordered by inclusion. Since L M 2 G and G is closed under intersections, we obtain that G is a closure system in 2LM. Now define for any c 2 L, d 2 M,c d = g(c, d), where g(c, d) denotes the smallest G-ideal containing (c, d). Isotonity of follows from fact that if c1 6 c2 than (c1, d) 6 (c2, d), thus (c1, d) 2 g(c2, d) what implies W V W c1 d = g(c1, d) # g(c2, d) = c2d. Further, if ci d # h 2 G for all i 2 I, then ( i2Ici, i2Id) = ( i2Ici, d) 2 h, which yields W W ( i2Ici) d = g( i2Ici, d) # h. Let (U, W) be Galois connection between LB and MA. Using Theorem 3.5 we obtain a decomposition of (U, W) into (ub,a, wb,a), b 2 B, a 2 A, i.e., there is a formal context C ¼ ðB; L; A; M; u; wÞ such that U ¼ "C and W ¼ #C . Now define Rðb; aÞ ¼ hub;a ;wb;a , where hub;a ;wb;a denotes the G-ideal of L M corresponding to the Galois connection (ub,a, wb,a). In order to complete the proof, we show that for each G-ideal h of L M holds: h = {(c, d) 2 L M : c d # h}. If (c, d) 2 h then obviously g(c, d) # h, thus c d # h. Conversely, if c d = g(c, d) # h then (c, d) 2 h. According to Proposition 2.3, for each b 2 B, a 2 A
_ _ fd : ðc; dÞ 2 hub;a ;wb;a g ¼ fd : c d 6 Rðb; aÞg; _ _ wb;a ðdÞ ¼ fc : ðc; dÞ 2 hub;a ;wb;a g ¼ fc : c d 6 Rðb; aÞg:
ub;a ðcÞ ¼
Hence U ¼ "C ¼ %R ; W ¼ #C ¼ .R , as it was shown in Lemma 4.3. h 5. Conclusions We introduced a generalization of Popescu’s approach (cf. [20]) of creating fuzzy concept lattices. This generalization has two major advantages. First, it allows to construct concept lattices in non-homogeneous cases, i.e., where various complete lattices are assigned to objects and attributes. Further, it does not require any logical structure on underlaying complete lattices. This generalization can be briefly characterized as building new concept lattices from old ones. Moreover, in Theorem 3.5 it was proved that this approach is the most general one, i.e., it produces all possible Galois connections between products of complete lattices. Based on this fact it was shown that Popescu’s approach is equivalent to several other approaches, e.g. one-sided concept lattices or generalized concept lattices. It is expected that this technique (as in Theorem 4.4) will yield similar results for some other approaches of creating fuzzy concept lattices. Acknowledgments The author wishes to express his gratitude to Professor Stanislav Krajcˇi for his many helpful comments and remarks during the preparation of this paper. This work was partially supported by the Slovak VEGA Grant No. 2/0194/10 and by the Slovak Research and Development Agency under contract APVV-0035-10.
136
J. Pócs / Information Sciences 185 (2012) 128–136
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
R. Beˇlohlávek, Lattices generated by binary fuzzy relations, Tatra Mt. Math. Publ. 16 (1999) 11–19. R. Beˇlohlávek, Fuzzy Galois connections, Math. Log. Quart. 45 (4) (1999) 497–504. R. Beˇlohlávek, Lattices of fixed points of fuzzy Galois connections, Math. Log. Quart. 47 (1) (2001) 111–116. R. Beˇlohlávek, A note on variable threshold concept lattices: threshold-based operators are reducible to classical concept-forming operators, Inform. Sci. 177 (15) (2007) 3186–3191. R. Beˇlohlávek, V. Sklenárˇ, J. Zacpal, Crisply generated fuzzy concepts, in: B. Ganter, R. Godin (Eds.), International Conference on Formal Concept Analysis 2005, Lecture Notes in Computer Science vol. 3403, 2005, pp. 268–283. R. Beˇlohlávek, V. Vychodil, What is a fuzzy concept lattice? in: Proceedings of the conference on Concept Lattices and Their Applications 2005, pp. 34– 45. S. Ben Yahia, A. Jaoua, Discovering knowledge from fuzzy concept lattice, in: A. Kandel, M. Last, H. Bunke (Eds.), Data Mining and Computational Intelligence, Physica-Verlag, 2001, pp. 167–190. G. Birkhoff, Lattice Theory, third ed., Amer. Math. Soc Providence, R.I., 1967. C.J. Everett, Closure operators and Galois theory in lattices, Trans. Am. Math. Soc. 55 (1944) 514–525. B. Ganter, R. Wille, Formal concept analysis, Mathematical Foundations, Springer, Berlin, 1999. G. Georgescu, A. Popescu, Concept lattices and similarity in non-commutative fuzzy logic, Fundam. Inform. 55 (1) (2002) 23–54. G. Georgescu, A. Popescu, Non-commutative fuzzy Galois connections, Soft Comput. 7 (2003) 458–467. G. Georgescu, A. Popescu, Non-dual fuzzy connections, Arch. Math. Logic 43 (2004) 1009–1039. A. Jaoua, S. Elloumi, Galois connection, formal concepts and Galois lattice in real relations: application in a real classifier, J. Syst. Softw. 60 (2002) 149– 163. S. Krajcˇi, Cluster based efficient generation of fuzzy concepts, Neural Netw. World 13 (5) (2003) 521–530. S. Krajcˇi, A generalized concept lattice, Logic J. IGPL 13 (5) (2005) 543–550. J. Medina, M. Ojeda-Aciego, Multi-adjoint t-concept lattices, Inform. Sci. 180 (5) (2010) 712–725. J. Medina, M. Ojeda-Aciego, J. Ruiz-Calviño, On multi-adjoint concept lattices: definition and representation theorem, in: International Conference on Formal Concept Analysis 2007, Lecture Notes in Artificial Intelligence, 2007, pp. 197–209. J. Medina, M. Ojeda-Aciego, J. Ruiz-Calviño, Formal concept analysis via multi-adjoint concept lattices, Fuzzy Sets Syst. 160 (2009) 130–144. A. Popescu, A general approach to fuzzy concepts, Math. Logic Quart. 50 (3) (2004) 265–280. Z. Shmuely, The structure of Galois connections, Pacific J. Math. 54 (2) (1974) 209–225. L. Wang, X. Liu, Concept analysis via rough set and AFS algebra, Inform. Sci. 178 (21) (2008) 4125–4137. L. Wang, X. Liu, J. Cao, A new algebraic structure for formal concept analysis, Inform. Sci. 180 (24) (2010) 4865–4876. W.X. Zhang, J.M. Ma, S.Q. Fan, Variable threshold concept lattices, Inform. Sci. 177 (22) (2007) 4883–4892.
Contents lists available at SciVerse ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Note on generating fuzzy concept lattices via Galois connections Jozef Pócs Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01 Košice, Slovakia
a r t i c l e
i n f o
Article history: Received 29 September 2010 Received in revised form 7 July 2011 Accepted 18 September 2011 Available online 24 September 2011
a b s t r a c t The aim of this paper is to compare an approach of creating fuzzy concept lattices proposed by Popescu with several other approaches. Particularly, we show that this approach is in some way equivalent to the approach of Krajcˇi called generalized concept lattices. We also give a straightforward generalization of Popescu’s approach to non-homogeneous cases. 2011 Elsevier Inc. All rights reserved.
Keywords: Galois connection Fuzzy concept lattice Generalized concept lattice
1. Introduction Fuzzy Formal Concept Analysis arose as an attempt for generalization of the classical Formal Concept Analysis [10] to the cases in which objects and attributes can take not only crisp but also fuzzy values. There are several approaches and methods on how to construct fuzzy concept lattices. We mention the approach of Beˇlohlávek [1–3] based on logical framework of complete residuated lattices, the work of Georgescu and Popescu to extend this framework to non-commutative logic [11–13], the approach of Krajcˇi [16] and Popescu [20]. There are also other approaches that generalized the previous ones, from which we mention the approach of Medina, Ojeda-Aciego, Ruiz-Calviño dealing with multi-adjoint concept lattices [18,19], cf. also [17], the work of Jaoua and Elloumi on Galois lattices of real relations [14], and the paper on variable threshold concept lattices by Zhang et al. [24], cf. also [4]. Most of these approaches can be characterized as searching for a Galois connection between powers of complete lattices. Note that there are also other methods for generating fuzzy concept lattices not involving Galois connections (see [6] for a relatively good overview of some existing methods). Concept analysis is also considered in other frameworks, cf. [22] or [23]. In [20] Popescu proposed a method for creating fuzzy concept lattices different from other ones. This method does not involve a ‘‘static’’ fuzzy relation between objects and attributes (i.e., an object has an attribute in a certain degree) as the input, but it involves a ‘‘dynamic’’ relation that is Galois connection expressing the mutual relationship between object and attribute. This approach has two major advantages. First, this construction does not require any logical structure on underlaying complete lattices. Further, it is very flexible and allows generalization to non-homogeneous cases, i.e., where various complete lattices are assigned to different objects and attributes. It seems very convenient to use mixed types of structures for truth values in object-attribute models, hence one of our goals is to provide a straightforward generalization of Popescu’s approach to non-homogeneous cases. Moreover, we prove that this generalization provides the most general method of creating concept lattices, which are based on Galois connections between direct products of complete lattices. Further, we describe the relationship between Popescu’s and some other approaches. Specifically, we deal with classical crisp Ganter and Wille’s [10] case, Krajcˇi’s [15] and Ben Yahia and Jaoua’s [7] one-sided concept lattices, and finally with generalized concept lattices introduced by Krajcˇi [16]. E-mail address: [email protected] 0020-0255/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.09.021
J. Pócs / Information Sciences 185 (2012) 128–136
129
Generalized concept lattices represent another approach to fuzzify concept lattices. This approach, in contrast with that of Popescu, involves a ‘‘static’’ fuzzy relation as the input, with values in partially ordered set, which is independent on the underlaying structures for objects and attributes respectively. The main aim of this paper is to show that appropriate selection of this partially ordered set make these approaches equivalent, i.e., with respect to all possible formal contexts, they produce all Galois connections between powers of two complete lattices. 2. Preliminaries In this section we give basic overview of Galois connections needed for our purposes. Note that by Galois connection we will always understand an antitone Galois connection, as it is usual in Formal Concept Analysis. We assume that the reader is familiar with the basic notions of lattice theory. Definition 2.1. Let (P, 6) and (Q, 6) be ordered sets and let
u : P ! Q and w : Q ! P be maps between these ordered sets. Such a pair (u, w) of mappings is called a Galois connection between the ordered sets if: (a) p1 6 p2 implies u(p1) P u(p2), (b) q1 6 q2 implies w(q1) P w(q2), (c) p 6 w(u(p)) and q 6 u(w(q)). These two maps are also called dually adjoint to each other. We note that
u ¼ u w u and w ¼ w u w and that the conditions (a), (b) and (c) are equivalent to the following one: (d) p 6 w(q) if and only if u(p) P q. For partially ordered sets P, Q denote by Gal(P, Q) the set of all Galois connections between P and Q. The class of all complete lattices will be denoted by CL. We will use the following well known fact (see [10,21]): Lemma 2.2. A map u : L ? M between complete lattices L and M has a dual adjoint if and only if
u
_
! xi
i2I
¼
^
uðxi Þ;
i2I
holds for any subset {xi : i 2 I} of L. In this case the dual adjoint w is uniquely determined by
wðyÞ ¼
_ fx 2 L : uðxÞ P yg:
Galois connections between complete lattices are closely related to the notion of closure operator and closure system. Let L be a complete lattice. By a closure operator in L we understand a mapping c : L ? L satisfying: (a) x 6 c(x) for all x 2 L, (b) c(x1) 6 c(x2) for x1 6 x2, (c) c(c(x)) = c(x) for all x 2 L (i.e., c is idempotent). Subset X of the complete lattice L is called closure system in L if X is closed under arbitrary meets. We note, that this condition guarantees that (X, 6) is a complete lattice, in which the infima are the same as in L, but the suprema in X may not coincide with those from L. For a closure operator c in L, the set FP(c) of all fixed points of c (i.e., FP(c) = {x 2 L : c(x) = x}) V is the closure system in L. Conversely, for closure system X in L, mapping CX : L ? L defined by CX(x) = {u 2 X : x 6u} is the closure operator in L. Moreover these correspondences are inverses of each other, i.e., FP(CX) = X for each closure system X in L and CFP(c) = c for each closure operator c in L. Next, we recall the relationship between the closure operators induced by the Galois connections. Two ordered sets P, Q are called dually isomorphic, if there is an antitone (order reversing) bijective mapping f : P ? Q such that f1 is also antitone. Let L, M 2 CL and (u, w) be Galois connection between L and M. Then mapping u w : L ? L is closure operator in L, similarly, w u : M ? M is closure operator in M. Moreover the corresponding closure systems are dually isomorphic. Conversely, suppose that X1 and X2 are closure systems in L, M respectively, and f : X1 ? X2 is dual isomorphism between complete lattices (X1, 6) and (X2, 6). Then a pair ðcX 1 f ; cX 2 f 1 Þ, where cX 1 ; cX 2 are closure operators corresponding to X1 and to X2, forms the Galois connection between L and M. Hence, any Galois connection between complete lattices induces dually isomorphic closure systems on these lattices and vice versa.
130
J. Pócs / Information Sciences 185 (2012) 128–136
In particular, we obtain that Gal(L, M) – ; for all L, M 2 CL, since {1L} and {1M} are dually isomorphic closure systems in L and M. The symbols 1L, 1M denote the greatest elements of L and M respectively. Similarly, 0L, 0M will denote the smallest elements. Galois connections between complete lattices stand in a one-to-one correspondence with so called G-ideals. Let L, M be complete lattices. h # L M is called a G-ideal of L M when (i) (x, y) 6 (a, b) and (a, b) 2 h implies (x, y) 2 h, W V V W (ii) if {(ai, bi)}i2I # h then ( i2Iai, i2Ibi) 2 h and ( i2Iai, i2Ibi) 2 h, (iii) (0L, 1M) 2 h and (1L, 0M) 2 h. Proposition 2.3 [21]. Let L and M be complete lattices. If (u, w) is a Galois connection between L and M, then
h ¼ fða; bÞ : uðaÞ P bg # L M is a G-ideal. Conversely, let h # L M be a G-ideal. If the mappings u : L ? M, w : M ? L are defined by
uðaÞ ¼
_ _ fb : ða; bÞ 2 hg ; wðbÞ ¼ fa : ða; bÞ 2 hg
then (u, w) is a Galois connection between L and M. Moreover, this correspondences between Galois connections and G-ideals are mutually inverse. Properties of Galois connections allow us to construct complete lattices which are in our special interest. Formally, let (u, w) be a Galois connection between complete lattices L and M. Denote by Lu,w a subset of L M consisting of all pairs (x, y) with u(x) = y and w(y) = x. Define the partial order on Lu,w as follows:
ðx1 ; y1 Þ 6 ðx2 ; y2 Þ if
x1 6 x2 iffy1 P y2
ðdue to condition ðdÞ of Definition 2:1Þ:
Proposition 2.4. Let (u, w) be a Galois connection between complete lattices L and M. Then (Lu,w, 6) forms a complete lattice, where
^ ðxi ; yi Þ ¼
^
i2I
i2I
xi ; u
^ i2I
!! xi
;
_ ðxi ; yi Þ ¼
w
i2I
^ i2I
! yi ;
^
! yi
i2I
for each family (xi,yi)i2I of elements from Lu,w. The proof of this proposition is based on the fact that w(u(L)) and u(w(M)) form dually isomorphic closure systems, and expressions for infima and suprema are obtained as in the case of classical concept lattices, see [10]. V W Note that if I = ;, than i2I(xi, yi) = (1L, u(1L)) and i2I(xi, yi) = (w(1M), 1M). 3. Lattices generated by formal fuzzy contexts In this section we describe a general method of generating fuzzy concept lattices proposed by Popescu [20] for the homogeneous case, i.e., there is only one structure of truth degrees for all objects and possibly another one for all attributes. In this paper the author gave the following definition of formal context: A context is a system C ¼ ðB; L; A; M; u; wÞ, where B is a set of objects, L is a complete lattice (structure of truth values for each b 2 B), A is a set of attributes, M is a complete lattice (structure of truth values for each a 2 A) and u = (ub,a)(b,a)2BA, w = (wb,a)(b,a)2BA where for each b 2 B, a 2 A(ub,a, wb,a) 2 Gal(L, M). Consequently, there are defined two mappings " : LB ? MA and ; : MA ? LB by
" ðf ÞðaÞ ¼
^ b2B
ub;a ðf ðbÞÞ; # ðgÞðbÞ ¼
^
wb;a ðgðaÞÞ
a2A
for all f 2 LB, g 2 MA, b 2 B, a 2 A. As we show later this two mappings form Galois connection between LB and MA. Now we give a straightforward generalization of this approach, also in non-homogeneous case, i.e., for each object and each attribute there is assigned possibly different structure of truth degrees. In what follows, we will adopt the notation introduced in [20]. A 6-tuple C ¼ ðB; L; A; M; u; wÞ is called formal fuzzy context if: (i) B,A – ; (B is a set of objects, A is a set of attributes), (ii) L : B ? CL, M : A ? CL (recall that CL denotes the class of all complete lattices, thus for b 2 B, L(b) represents a complete lattice with possible truth values of the object b and similarly for a 2 A), (iii) u = (ub,a)(b,a)2BA, w = (wb,a)(b,a)2BA where for each b 2 B, a 2 A (ub,a, wb,a) 2 Gal(L(b), M(a)) (the pair (ub,a, wb,a) is Galois connection between lattices L(b) and M(a)).
131
J. Pócs / Information Sciences 185 (2012) 128–136
Further, define ":
" ðf ÞðaÞ ¼
^
Q
b2B LðbÞ
!
Q
a2A MðaÞ
as follows:
ub;a ðf ðbÞÞ for all a 2 A; if f 2
b2B
^ a2A
LðbÞ:
ð1Þ
b2B
Similarly we put #:
# ðgÞðbÞ ¼
Y
Q
a2A MðaÞ
!
Q
b2B LðbÞ
wb;a ðgðaÞÞ for all b 2 B;
as follows:
if g 2
Y
MðaÞ
ð2Þ
a2A
To emphasize that given context C corresponds to given pair (",;), we will sometimes write C as the a subscript, i.e., ð"C ; #C Þ. Example 3.1. Consider the following context C ¼ ðB; L; A; M; u; wÞ. Sets B and A consist of two elements, say B = {b1,b2}, A = {a1, a2}. The mapping L maps b1 to the two element chain 2 and b2 to the lattice P(2) isomorphic to the power set of the two element set. Similarly M maps a1 to 2 and a2 to the three element chain 3. Corresponding Galois connections ðubi ;aj ; wbi ;aj Þ between L(bi) and M(aj), for {i, j} = {1, 2}, are depicted in the following table (see Fig. 1). Note that for more legibility we only indicate corresponding dual isomorphism of closure systems. This obviously uniquely determine corresponding Galois connection. Using the definition of " and ; we obtain the following mapping (see Fig. 2) between L(b1) L(b2) ffi 2 P(2) and M(a1) M(a2) ffi 2 3, which forms a Galois connection, as we show in Theorem 3.2.
Theorem 3.2. Let (B, L, A, M, u, w) be a formal fuzzy context. Then the pair (", ;) forms a Galois connection between Q a2A MðaÞ.
Fig. 1. Example of formal fuzzy context.
Fig. 2. Resulting Galois connection of Example 3.1.
Q
b2B LðbÞ
and
132
J. Pócs / Information Sciences 185 (2012) 128–136
Proof. Suppose that f1 6 f2 where f1 ; f2 2
" ðf1 ÞðaÞ ¼
^
ub;a ðf1 ðbÞÞ P
b2B
^
Q
b2B LðbÞ.
For any a 2 A, b 2 B we have ub,a(f1(b)) P ub,a(f2(b)), hence
ub;a ðf2 ðbÞÞ ¼" ðf2 ÞðaÞ;
b2B
for all a 2 A, which yields "(f1) P "(f2). The condition g1 6 g2 implies ;(g1) P ;(g2), may be proved analogously. Q Next we prove that for any f 2 b2B LðbÞ; f 6# ð" ðf ÞÞ. Let b 2 B be an arbitrary element. Then for any a 2 A
^
" ðf ÞðaÞ ¼
ub0 ;a ðf ðb0 ÞÞ 6 ub;a ðf ðbÞÞ;
0
b 2B
hence
f ðbÞ 6 wb;a ðub;a ðf ðbÞÞÞ 6 wb;a ð" ðf ÞðaÞÞ; which yields
f ðbÞ 6
^
wb;a ð" ðf ÞðaÞÞ ¼# ð" ðf ÞÞðbÞ:
a2A
Similarly, the inequality g 6 "(;(g)) for g 2
Q
a2A MðaÞ,
may be proved analogously.
h
According to the result of this theorem, for the formal fuzzy context C ¼ ðB; L; A; M; u; wÞ we will call the lattice L",; the fuzzy concept lattice corresponding to C and denote it by FCLðCÞ. Corollary 3.3. FCLðCÞ is a complete lattice where
^ ðfi ; g i Þ ¼
^
i2I
i2I
fi ; "
^
!! fi
;
i2I
_ ðfi ; g i Þ ¼
#
i2I
^
! gi ;
i2I
^
! gi
i2I
for each family (fi,gi)i2I of elements from FCLðCÞ. The following theorem is a generalization of the so-called main theorem of concept lattices. Theorem 3.4. Let W be a complete lattice. Then W ffi FCLðCÞ if and only if for all b 2 B, a 2 A there are mappings cb : L(b) ? W, la : M(a) ? W such that S W S V (a) b2Bcb(L(b)) is -dense in W and a2Ala(M(a)) is -dense in W, (b) for all b 2 B, a 2 A, x 2 L(b), y 2 M(a), cb(x) 6 la(y) iff y 6 ub,a(x) iff x 6 wb,a(y). The proof of this theorem is exactly same as in [20] (see Lemma 5.1 and Proposition 5.2). In what follows, we will show that each Galois connection between direct products of complete lattices has a decomposition. Q Let {Li : i 2 I} be a system of complete lattices. For x 2 Lj we denote by 0jx the following element of i2I Li : 0jx ðjÞ ¼ x and j 0x ðiÞ ¼ 0Li for all i 2 I,i – j. Theorem 3.5. Let (U, W) be a Galois connection between C ¼ ðB; L; A; M; u; wÞ, such that "C ¼ U and #C ¼ W.
Q
b2B LðbÞ
and
Q
a2A MðaÞ.
Then there exists a formal context
Proof. For b 2 B, a 2 A, x 2 L(b) and y 2 M(a), define
ub;a ðxÞ ¼ U 0bx ðaÞ ; wb;a ðyÞ ¼ W 0ay ðbÞ: We show that (ub,a, wb,a) forms a Galois connection between L(b) and M(a). Consider the following series of equivalent assertions:
x 6 wb;a ðyÞ iff 0bx 6 W 0ay iff U 0bx P 0ay iff ub;a ðxÞ P y: This yields, that C ¼ ðB; L; A; M; u; wÞ forms a formal context. Q W Further, let f 2 b2B LðbÞ be an arbitrary element. Then f ¼ b2B 0bfðbÞ and according to Lemma 2.2 we obtain that:
Uðf ÞðaÞ ¼ U
_ b2B
!
0bfðbÞ ðaÞ ¼
^
U 0bfðbÞ ðaÞ ¼
b2B
^ b2B
ub;a ðf ðbÞÞ ¼ "C ðf ÞðaÞ:
J. Pócs / Information Sciences 185 (2012) 128–136
133
This shows that "C ¼ U. The equality #C ¼ W can be proved similarly, or we can simply use the fact, that the dual adjoint is unique, which yields #C ¼ W, too. h Theorem 3.5 shows, that the generalization of Popescu’s approach provides the most general method of creating Galois connections between direct products of complete lattices, i.e., it produces all possible Galois connections. 4. Relation to other approaches In this section we compare several existing approaches with that proposed by Popescu and show that the result of Theorem 3.5 can be used for showing that these other approaches are the most general of this kind, i.e. all possible Galois connections between certain lattices can be obtained by this way. 4.1. Classical FCA First, we will deal with Galois connections between power sets described by Birkhoff [8], which are the cornerstone of classical Formal Concept Analysis developed by Ganter and Wille [10]. We show that classical Galois connections are particular cases of Popescu’s approach, cf. [20]. Let (B, A, I) be a formal context, i.e., B, A – ; and I # B A. There is a pair of mappings ":2B ? 2A and ; : 2A ? 2B, which forms Galois connection between power sets
X " ¼ fy 2 A : ðx; yÞ 2 I; 8x 2 Xg; Y # ¼ fx 2 B : ðx; yÞ 2 I; 8y 2 Yg: The corresponding concept lattice is denoted by BðB; A; IÞ. Denote by 2 the two element lattice. There are only two Galois connections between 2 and 2, namely ð1; 1Þ where 1ð0Þ ¼ 1; 1ð1Þ ¼ 1 and ð0; 0Þ where 0ð0Þ ¼ 1; 0ð1Þ ¼ 0. Because of avoiding some technical complications we will substitute each subset of B and A with its characteristic function, due to their isomorphism. Also, we put L(b) = 2,M(a) = 2 for each b 2 B and a 2 A and define (ub,a, wb,a) 2 Gal(2, 2) as follows:
( ðub;a ; wb;a Þ ¼
ð1; 1Þ if ðb; aÞ 2 I; ð0; 0Þ if ðb; aÞ R I:
It is evident, that the 6-tuple C ¼ ðB; L; A; M; u; wÞ is a formal context. Conversely, for a given (B, L, A, M, u,w) we define (b, a) 2 I iff ðub;a ; wb;a Þ ¼ ð1; 1Þ. Since
" ðXÞðaÞ ¼
^
^
ub;a ðXðbÞÞ ¼
b2B
1ðXðbÞÞ ^
b2B;ðb;aÞ2I
^
0ðXðbÞÞ;
b2B;ðb;aÞRI
we obtain that: "(X)(a) = 0 iff there is b 2 X such that (b, a) R I iff a R X". Similarly, ;(Y)(b) = 0 iff b R Y;. Hence, we obtain that FCLðCÞ and BðB; A; IÞ are almost the same, up to isomorphism of power sets and characteristic functions. Now using Theorem 3.5 we obtain the well known result of Everett [9], which state that each Galois connection between power sets 2B and 2A can be obtained using binary relation I # B A. 4.2. One-sided concept lattices Further, we will deal with the one-sided approach, independently described by Krajcˇi [15] and by Ben Yahia and Jaoua [7], cf. also [5]. In this case the input consists of a L-context (B, A, I) where L is complete lattice and I : B A ? L. Again, there is a pair of mappings b : 2B ? LA and a : LA ? 2B,
bðXÞðaÞ ¼
^
Iðb; aÞ;
b2X
aðgÞ ¼ fb 2 B : 8a 2 A; gðaÞ 6 Iðb; aÞg: First, we note that each Galois connection between 2 and L is uniquely determined by single element c 2 L. This follows trivially from the fact, that each such a Galois connection induce dually isomorphic closure systems and there are only two closure systems in 2. Hence, for any c 2 L denote by ~c a mapping satisfying ~cð0Þ ¼ 1L and ~cð1Þ ¼ c and denote by ~c the dual g g adjoint to ~c. Now we put ðu ; w Þ ¼ Iðb; aÞ; Iðb; aÞ . Again we will not distinguish between subsets of B and their charb;a
b;a
acteristic functions from 2B. For any X # B we obtain
134
J. Pócs / Information Sciences 185 (2012) 128–136
" ðXÞðaÞ ¼
^
g Iðb; aÞð1Þ ^
b2X
^
g Iðb; aÞð0Þ ¼
bRX
^
Iðb; aÞ ^
b2X
^
1L ¼ bðXÞðaÞ:
bRX
Conversely,
# ðgÞðbÞ ¼
^
g Iðb; aÞ ðgðaÞÞ:
a2A
g Since Iðb; aÞ ðgðaÞÞ ¼ 1 iff g(a) 6 I(b, a), we obtain: b 2 a(g) iff ;(g)(b) = 1. As a consequence of Theorem 3.5 we obtain that this method produce all possible Galois connections between 2B and LA. Next we provide a heterogeneous generalization of this one-sided approach. Consider the following formal fuzzy context (B, L, A, M, u, w) where L(b) = 2 for all b 2 B and ðub;a ; wb;a Þ ¼ ð~c; ~c Þ for c 2 M(a). In classical way we have (B, A, I) where Q I(b, a) 2 M(a) for all a 2 M. The same computations show that a pair of mappings b : 2B ! a2A MðaÞ and Q B a : a2A MðaÞ ! 2
bðXÞðaÞ ¼
^
Iðb; aÞ;
b2X
aðgÞ ¼ fb 2 B : for each a 2 A; gðaÞ 6 Iðb; aÞg; forms a Galois connection between 2B and Q between 2B and a2A MðaÞ.
Q
a2A MðaÞ.
Again, due to Theorem 3.5, this approach covers all Galois connections
4.3. Generalized concept lattices At the end of this section we will deal with generalized concept lattices introduced by Krajcˇi [16]. Let P be a poset, L and M be complete lattices. Let :L M ? P be isotone and left-continuous in both their arguments, i.e., c1 6 c2 implies c1 d 6 c2 d, d1 6 d2 implies c d1 6 c d2, W if cj d 6 p for each j 2 J then ( j2Jcj) d 6 p, W if c dj 6 p for each j 2 J then c ( j2Jdj) 6 p. In the case of P being a complete lattice, this conditions are equivalent to the following identities:
c
_
! dj
j2J
_ ¼ ðc dj Þ and j2J
_
! cj
d¼
_ ðcj dÞ:
j2J
j2J
Remark 4.1. Left-continuity of implies c 0M = 0L d = 0P for all c 2 L,d 2 M, i.e., P possesses the least element 0P. This W follows from the fact that for J = ;, any p 2 P,d 2 M we have: for all j 2 J,cj d 6 p, hence ; d = 0L d 6 p. Let B and A be non-empty sets and let R be P-fuzzy relation on their Cartesian product, i.e., R : B A ? P. Define the following mappings % : LB ? MA and . : MA ? LB:
_
fd 2 M : ð8b 2 BÞ f ðbÞ d 6 Rðb; aÞg; _ . ðgÞðbÞ ¼ fc 2 L : ð8a 2 AÞ c gðaÞ 6 Rðb; aÞg:
% ðf ÞðaÞ ¼
Again, the pair (%,.) forms Galois connection between LB and MA, see [16]. For p 2 P denote hp = {(c, d) : c d 6 p}. Lemma 4.2. For all p 2 P, hp is a G-ideal of L M. Proof. Condition (c) of G-ideal is satisfied due to Remark 4.1 and (a) follows from isotonity of . Finally, suppose that V {(ci, di)}i2I # hp, i.e., ci di 6 p for each i 2 I. Then ( j2Icj) di 6 ci di 6 p for all i 2 I, hence by left-continuity V W W V ( i2Ici) ( i2Idi) 6 p. Similarly, ( i2Ici) ( i2Idi) 6 p. h Now define ub,a:L ? M and wb,a : M ? L as follows:
_ _ fd : ðc; dÞ 2 hRðb;aÞ g ¼ fd : c d 6 Rðb; aÞg; _ _ wb;a ðdÞ ¼ fc : ðc; dÞ 2 hRðb;aÞ g ¼ fc : c d 6 Rðb; aÞg:
ub;a ðcÞ ¼
According to Proposition 2.3, pair (ub,a, wb,a) is Galois connection between L and M, corresponding to the G-ideal hR(b, a). Thus C ¼ ðB; L0 ; A; M 0 ; u; wÞ forms a formal context, where L0(b) = L for all b 2 B and M0(a) = M for all a 2 A.
J. Pócs / Information Sciences 185 (2012) 128–136
135
Lemma 4.3. %¼ "C and .¼ #C . Proof. Note, that from isotonity and left-continuity of we obtain:
if d0 6
_ _ fd : c d 6 pg then c d0 6 c fd : c d 6 pg 6 p:
Let a 2 A be an arbitrary element. Since
"C ðf ÞðaÞ ¼
^
ub;a ðf ðbÞÞ ¼
b2B
^ _ fd : f ðbÞ d 6 Rðb; aÞg ; b2B
W
we obtain that "C ðf ÞðaÞ 6 fd : f ðbÞ d 6 Rðb; aÞg for all b 2 B. Hence, by left-continuity f ðbÞ "C ðf ÞðaÞ 6 Rðb; aÞ for all b 2 B, which yields "C ðf ÞðaÞ 6% ðf ÞðaÞ. Conversely, let b 2 B. Then
fd 2 M : ð8b0 2 BÞ f ðb0 Þ d 6 Rðb0 ; aÞg # fd : f ðbÞ d 6 Rðb; aÞg; W W thus for each b 2 B, this yields %(f)(a) = {d 2 M : ("b 2 B) f(b) d 6 R(b, a)} 6 {d : f(b) d 6 R(b, a)} = ub,a(f(b)). Hence, we V obtain % ðf ÞðaÞ 6 b2B ub;a ðf ðbÞÞ ¼ "C ðf ÞðaÞ. The equality #C ¼. may be proved analogously. h Finally, we will show that this approach is the most general in some sense. We will show that every concept lattice LU,W where (U, W) is a Galois connection between LB and MA is in fact a generalized concept lattice. Theorem 4.4. Let L, M be complete lattices and B, A be nonempty sets. Then there exists a complete lattice G and a mapping : L M ! G isotone and left-continuous in both their arguments, such that for any Galois connection (U, W) between LB and MA there is R : B A ! G with U = %R and W = .R. Proof. Denote by G the set of all G-ideals of L M, ordered by inclusion. Since L M 2 G and G is closed under intersections, we obtain that G is a closure system in 2LM. Now define for any c 2 L, d 2 M,c d = g(c, d), where g(c, d) denotes the smallest G-ideal containing (c, d). Isotonity of follows from fact that if c1 6 c2 than (c1, d) 6 (c2, d), thus (c1, d) 2 g(c2, d) what implies W V W c1 d = g(c1, d) # g(c2, d) = c2d. Further, if ci d # h 2 G for all i 2 I, then ( i2Ici, i2Id) = ( i2Ici, d) 2 h, which yields W W ( i2Ici) d = g( i2Ici, d) # h. Let (U, W) be Galois connection between LB and MA. Using Theorem 3.5 we obtain a decomposition of (U, W) into (ub,a, wb,a), b 2 B, a 2 A, i.e., there is a formal context C ¼ ðB; L; A; M; u; wÞ such that U ¼ "C and W ¼ #C . Now define Rðb; aÞ ¼ hub;a ;wb;a , where hub;a ;wb;a denotes the G-ideal of L M corresponding to the Galois connection (ub,a, wb,a). In order to complete the proof, we show that for each G-ideal h of L M holds: h = {(c, d) 2 L M : c d # h}. If (c, d) 2 h then obviously g(c, d) # h, thus c d # h. Conversely, if c d = g(c, d) # h then (c, d) 2 h. According to Proposition 2.3, for each b 2 B, a 2 A
_ _ fd : ðc; dÞ 2 hub;a ;wb;a g ¼ fd : c d 6 Rðb; aÞg; _ _ wb;a ðdÞ ¼ fc : ðc; dÞ 2 hub;a ;wb;a g ¼ fc : c d 6 Rðb; aÞg:
ub;a ðcÞ ¼
Hence U ¼ "C ¼ %R ; W ¼ #C ¼ .R , as it was shown in Lemma 4.3. h 5. Conclusions We introduced a generalization of Popescu’s approach (cf. [20]) of creating fuzzy concept lattices. This generalization has two major advantages. First, it allows to construct concept lattices in non-homogeneous cases, i.e., where various complete lattices are assigned to objects and attributes. Further, it does not require any logical structure on underlaying complete lattices. This generalization can be briefly characterized as building new concept lattices from old ones. Moreover, in Theorem 3.5 it was proved that this approach is the most general one, i.e., it produces all possible Galois connections between products of complete lattices. Based on this fact it was shown that Popescu’s approach is equivalent to several other approaches, e.g. one-sided concept lattices or generalized concept lattices. It is expected that this technique (as in Theorem 4.4) will yield similar results for some other approaches of creating fuzzy concept lattices. Acknowledgments The author wishes to express his gratitude to Professor Stanislav Krajcˇi for his many helpful comments and remarks during the preparation of this paper. This work was partially supported by the Slovak VEGA Grant No. 2/0194/10 and by the Slovak Research and Development Agency under contract APVV-0035-10.
136
J. Pócs / Information Sciences 185 (2012) 128–136
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
R. Beˇlohlávek, Lattices generated by binary fuzzy relations, Tatra Mt. Math. Publ. 16 (1999) 11–19. R. Beˇlohlávek, Fuzzy Galois connections, Math. Log. Quart. 45 (4) (1999) 497–504. R. Beˇlohlávek, Lattices of fixed points of fuzzy Galois connections, Math. Log. Quart. 47 (1) (2001) 111–116. R. Beˇlohlávek, A note on variable threshold concept lattices: threshold-based operators are reducible to classical concept-forming operators, Inform. Sci. 177 (15) (2007) 3186–3191. R. Beˇlohlávek, V. Sklenárˇ, J. Zacpal, Crisply generated fuzzy concepts, in: B. Ganter, R. Godin (Eds.), International Conference on Formal Concept Analysis 2005, Lecture Notes in Computer Science vol. 3403, 2005, pp. 268–283. R. Beˇlohlávek, V. Vychodil, What is a fuzzy concept lattice? in: Proceedings of the conference on Concept Lattices and Their Applications 2005, pp. 34– 45. S. Ben Yahia, A. Jaoua, Discovering knowledge from fuzzy concept lattice, in: A. Kandel, M. Last, H. Bunke (Eds.), Data Mining and Computational Intelligence, Physica-Verlag, 2001, pp. 167–190. G. Birkhoff, Lattice Theory, third ed., Amer. Math. Soc Providence, R.I., 1967. C.J. Everett, Closure operators and Galois theory in lattices, Trans. Am. Math. Soc. 55 (1944) 514–525. B. Ganter, R. Wille, Formal concept analysis, Mathematical Foundations, Springer, Berlin, 1999. G. Georgescu, A. Popescu, Concept lattices and similarity in non-commutative fuzzy logic, Fundam. Inform. 55 (1) (2002) 23–54. G. Georgescu, A. Popescu, Non-commutative fuzzy Galois connections, Soft Comput. 7 (2003) 458–467. G. Georgescu, A. Popescu, Non-dual fuzzy connections, Arch. Math. Logic 43 (2004) 1009–1039. A. Jaoua, S. Elloumi, Galois connection, formal concepts and Galois lattice in real relations: application in a real classifier, J. Syst. Softw. 60 (2002) 149– 163. S. Krajcˇi, Cluster based efficient generation of fuzzy concepts, Neural Netw. World 13 (5) (2003) 521–530. S. Krajcˇi, A generalized concept lattice, Logic J. IGPL 13 (5) (2005) 543–550. J. Medina, M. Ojeda-Aciego, Multi-adjoint t-concept lattices, Inform. Sci. 180 (5) (2010) 712–725. J. Medina, M. Ojeda-Aciego, J. Ruiz-Calviño, On multi-adjoint concept lattices: definition and representation theorem, in: International Conference on Formal Concept Analysis 2007, Lecture Notes in Artificial Intelligence, 2007, pp. 197–209. J. Medina, M. Ojeda-Aciego, J. Ruiz-Calviño, Formal concept analysis via multi-adjoint concept lattices, Fuzzy Sets Syst. 160 (2009) 130–144. A. Popescu, A general approach to fuzzy concepts, Math. Logic Quart. 50 (3) (2004) 265–280. Z. Shmuely, The structure of Galois connections, Pacific J. Math. 54 (2) (1974) 209–225. L. Wang, X. Liu, Concept analysis via rough set and AFS algebra, Inform. Sci. 178 (21) (2008) 4125–4137. L. Wang, X. Liu, J. Cao, A new algebraic structure for formal concept analysis, Inform. Sci. 180 (24) (2010) 4865–4876. W.X. Zhang, J.M. Ma, S.Q. Fan, Variable threshold concept lattices, Inform. Sci. 177 (22) (2007) 4883–4892.