L-bonds represent relationships between formal contexts. We study properties of these intercontextual structures w.r.t. isotone conceptforming operators in fuzzy setting. We also focus on the direct product of two formal fuzzy contexts and show conditions under which a bond can be obtained as an intent of the product. In addition, we show that the previously studied properties of their antitone counterparts can be easily derived from the present results.
Formal Concept Analysis (FCA) has become a very active research topic, both theoretical and practical, for formally describing structural and hierarchical properties of data with “object-attribute” character. It is this wide applicability which justifies the need of a deeper knowledge of its underlying mechanisms: and one important way to obtain this extra knowledge turns out to be via generalization and abstraction.
A number of different approaches have presented towards a generalization
of the framework and scope of FCA and, nowadays, one can find papers which
extend the theory by using ideas from rough set theory [
Goguen argued in [
In this paper, we deal with an extremely general form of L-fuzzy Formal
Concept Analysis, based on categorical constructs and L-fuzzy sets. Particularly,
our approach originated in relation to a previous work [
In this paper, we study properties of L-bonds w.r.t. isotone concept-forming operators in a fuzzy setting. We also focus on the direct product of two formal fuzzy contexts and show conditions under which a bond can be obtained as an extent of the product. In addition, we show that the previously studied properties of their antitone counterparts can be easily derived from the present results. 2
We use complete residuated lattices as basic structures of truth-degrees. A complete residuated lattice is a structure L “ xL, ^, _, b, Ñ, 0, 1y such that (i) xL, ^, _, 0, 1y is a complete lattice, i.e. a partially ordered set in which arbitrary infima and suprema exist; (ii) xL, b, 1y is a commutative monoid, i.e. b is a binary operation which is commutative, associative, and a b 1 “ a for each a P L; (iii) b and Ñ satisfy adjointness, i.e. a b b ď c iff a ď b Ñ c. 0 and 1 denote the least and greatest elements. The partial order of L is denoted by ď. Throughout this work, L denotes an arbitrary complete residuated lattice.
Elements a of L are called truth degrees. Operations b (multiplication) and Ñ (residuum) play the role of (truth functions of) “fuzzy conjunction” and “fuzzy implication”. Furthermore, we define the complement of a P L as a “ a Ñ 0
An L-set (or fuzzy set) A in a universe set X is a mapping assigning to each x P X some truth degree Apxq P L where L is a support of a complete residuated lattice. The set of all L-sets in a universe X is denoted LX , or LX if the structure of L is to be emphasized.
The operations with L-sets are defined componentwise. For instance, the intersection of L-sets A, B P LX is an L-set A X B in X such that pA X Bqpxq “ Apxq^Bpxq for each x P X, etc. An L-set A P LX is also denoted tApxq{x | x P Xu. If for all y P X distinct from x1, x2, . . . , xn we have Apyq “ 0, we also write tApx1q{x1, Apx2q{x1, . . . , Apxnq{xnu.
An L-set A P LX is called crisp if Apxq P t0, 1u for each x P X. Crisp Lsets can be identified with ordinary sets. For a crisp A, we also write x P A for Apxq “ 1 and x R A for Apxq “ 0. An L-set A P LX is called empty (denoted by H) if Apxq “ 0 for each x P X. For a P L and A P LX , the L-sets a b A P LX , a Ñ A ,A Ñ a, and A in X are defined by pa b Aqpxq “ a b Apxq, pa Ñ Aqpxq “ a Ñ Apxq, pA Ñ aqpxq “ Apxq Ñ a,
(1) (2) (3) (4) An L-set A P LX is called an a-complement if A “ a Ñ B for some B P LX .
Binary L-relations (binary fuzzy relations) between X and Y can be thought of as L-sets in the universe X ˆ Y . That is, a binary L-relation I P LXˆY between a set X and a set Y is a mapping assigning to each x P X and each y P Y a truth degree Ipx, yq P L (a degree to which x and y are related by I).
The composition operators are defined by pA ˝ Bqpx, yq “ ł Apx, f q b Bpf, yq,
fPF pA Ž Bqpx, yq “ ľ Apx, f q Ñ Bpf, yq,
fPF pA Ż Bqpx, yq “ ľ Bpf, yq Ñ Apx, f q.
fPF
An L-context is a triplet xX, Y, Iy where X and Y are (ordinary) sets and I P LXˆY is an L-relation between X and Y . Elements of X are called objects, elements of Y are called attributes, I is called an incidence relation. Ipx, yq “ a is read: “The object x has the attribute y to degree a.”
Consider the following pairs of operators induced by an L-context xX, Y, Iy. First, the pair xÒ, Óy of operators Ò : LX Ñ LY and Ó : LY Ñ LX is defined by CÒpyq “ ľ Cpxq Ñ Ipx, yq, xPX
DÓpxq “ ľ Dpyq Ñ Ipx, yq. yPY Second, the pair xX, Yy of operators X : LX Ñ LY and Y : LY Ñ LX is defined by
CXpyq “ C^pyq “ ł Cpxq b Ipx, yq, xPX
ľ Ipx, yq Ñ Dpyq, yPY ľ Ipx, yq Ñ Cpxq, xPX
D_pxq “ ł Dpyq b Ipx, yq, yPY Third, the pair x^, _y of operators ^ : LX Ñ LY and _ : LY Ñ LX is defined by (6) (7) (8) (9) (10) (11) for C P LX , D P LY .
To emphasize that the operators are induced by I, we also denote the operators by xÒI , ÓI y, xXI , YI y, and x^I , _I y. Furthermore, denote the corresponding sets of fixpoints by BpXÒ, Y Ó, Iq, BpXX, Y Y, Iq, and BpX^, Y _, Iq, i.e.
BpXÒ, Y Ó, Iq “ txC, Dy P LX ˆ LY | CÒ “ D, DÓ “ Cu, BpXX, Y Y, Iq “ txC, Dy P LX ˆ LY | CX “ D, DY “ Cu,
BpX^, Y _, Iq “ txC, Dy P LX ˆ LY | C^ “ D, D_ “ Cu.
The sets of fixpoints are complete lattices, called L-concept lattices associated to I, and their elements are called formal concepts.
For a concept lattice BpXM, Y O, Iq, where xM, Oy is either of xÒ, Óy, xX, Yy, or x^, _y, denote the corresponding sets of extents and intents by ExtpXM, Y O, Iq and IntpXM, Y O, Iq. That is,
ExtpXM, Y O, Iq “ tC P LX | xC, Dy P BpXM, Y O, Iq for some Du, IntpXM, Y O, Iq “ tD P LY | xC, Dy P BpXM, Y O, Iq for some Cu, A system of L-sets V Ď LX is called an L-interior system if – V is closed under b-multiplication, i.e. for every a P L and C P V we have a b C P V ; – V is closed under union, i.e. for Cj P V (j P J ) we have ŤjPJ Cj P V .
V Ď LX is called an L-closure system if – V is closed under left Ñ-multiplication (or Ñ-shift), i.e. for every a P L and C P V we have a Ñ C P V (here, a Ñ C is defined by pa Ñ Cqpiq “ a Ñ Cpiq for i “ 1, . . . , n); – V is closed under intersection, i.e. for Cj P V (j P J ) we have ŞjPJ Cj P V . 3
This section introduces some new notions studied in this work. To begin with, we introduce the notion of weak L-bonds as a convenient generalization of bond. Definition 1. A weak L-bond between two L-contexts K1 “ xX1, Y1, I1y and K2 “ xX2, Y2, I2y w.r.t. xX, Yy is an L-relation β P LX1ˆY2 s.t.
ExtpX1X, Y2Y, βq Ď ExtpX1X, Y1Y, I1q
and IntpX1X, Y2Y, βq Ď IntpX2X, Y2Y, I2q.
This notion can be put in relation with that of i-morphism.
Definition 2. A mapping h : V Ñ W from an L-interior system V Ď LX into an L-interior system W Ď LY is called an i-morphism if it is a b- and Ž-morphism, i.e. if – hpa b Cq “ a b hpCq for each a P L and C P V ; – hpŽkPK Ckq “ ŽkPK hpCkq for every collection of Ck P V (k P K). An i-morphism h : V Ñ W is said to be an extendable i-morphism if h can be extended to an i-morphism of LX to LY , i.e. if there exists an i-morphism h1 : LX Ñ LY such that for every C P V we have h1pCq “ hpCq;
Lemma 1 ([
IntpF X, Y Y, Bq if and only if there exists A P LXˆF such that I “ A ˝ B, 4. Consider two contexts xX, Y, Iy and xX, F, Ay. Then, we have ExtpXX, Y Y, Iq Ď
ExtpXX, F Y, Aq if and only if there exists B P LF ˆY such that I “ A ˝ B. Theorem 1. The weak L-bonds between K1 “ xX1, Y1, I1y and K2 “ xX2, Y2, I2y are in one-to-one correspondence with extendable i-morphisms IntpX1X, Y1Y, I1q to IntpX2X, Y2Y, I2q.
Proof. We show procedures to obtain the i-morphism from a weak L-bond and vice versa.
“ñ”: Let β be a weak L-bond between K1 “ xX1, Y1, I1y and K2 “ xX2, Y2, I2y. By Definition 1 we have IntpX1X, Y2Y, βq Ď IntpX2X, Y2Y, I2q; thus by Lemma 1(3) there exists Si P LY1ˆY2 such that β “ I1 ˝ Si. The induced opeator XSi is an extendable i-morphism IntpX1X, Y1Y, I1q to IntpX2X, Y2Y, I2q by Lemma 1(2).
“ð”: For an extendable i-morphism f : IntpX1X, Y1Y, I1q Ñ IntpX2X, Y2Y, I2q there is an L-relation Si s.t. f pBq “ BXSi for each B P IntpX1X, Y1Y, I1q by Lemma 1(1). Then β “ I1 ˝ Si is a weak L-bond by Lemma 1(3) and Lemma 1(4).
One can check that these two procedures are mutually inverse. \[
Now, consider L-bonds w.r.t. x^, _y defined similarly as in Definition 1, i.e. an L-relation β P LX1ˆY2 s.t.
ExtpX1^, Y2_, βq Ď ExtpX1^, Y1_, I1q and IntpX1^, Y2_, βq Ď IntpX2^, Y2_, I2q.
Note that the weak L-bonds w.r.t. xX, Yy are different from L-bonds w.r.t. x^, _y as the following example shows.
Example 1. Consider L a finite chain containing a ă b with b defined as follows: #x ^ y if x “ 1 or y “ 1, x b y “ 0 otherwise, for each x, y P L. One can easily see that x b Žj yj “ Žjpx b yjq and thus an adjoint operation Ñ exists such that xL, ^, _, b, Ñ, 0, 1y is a complete residuated lattice. Namely, Ñ is given as follows: x Ñ y “ $1 if x ď y, ’ &
y if x “ 1, ’%b otherwise, for each x, y P L. Consider I1 “ `a˘ and I2 “ `b˘. One can check that, we have ExtptxuX, tyuY, I1q “ ExtptxuX, tyuY, I2q “ ttb{xu, xu and, trivially, IntptxuX, tyuY, I2q “ IntptxuX, tyuY, I2q. Thus I2 is a weak L-bond between I1 and I2 w.r.t. xX, Yy.
On the other hand, I2 is not a weak L-bond between I1 and I2 w.r.t. x^, _y since Extptxu^, tyu_, I1q “ tH, ta{xuu Ğ tH, tb{xuu “ Extptxu^, tyu_, I2q. Theorem 2. The system of all weak L-bonds is an L-interior system.
4
We provide a stronger version of the previously studied weak L-bond, naturally named strong L-bond.
Definition 3. A strong L-bond between two L-contexts K1 “ xX1, Y1, I1y and K2 “ xX2, Y2, I2y is an L-relation β P LX1ˆY2 s.t. β is a weak L-bond w.r.t. both xX, Yy and x^, _y.
The following lemma introduces equivalent definitions of strong L-bonds. Lemma 2. The following propositions are equivalent: (a) β is a strong L-bond between K1 “ xX1, Y1, I1y and K2 “ xX2, Y2, I2y. (b) β satisfies both ExtpX1^, Y2_, βq Ď ExtpX1^, Y1_, I1q and IntpX1X, Y2Y, βq Ď
IntpX2X, Y2Y, I2q. (c) β “ Se ˝ I2 “ I1 ˝ Si for some Se P LX1ˆX2 and Si P LY1ˆY2 .
Proof. (a) ô (b): By use of the Lemma 1(3) and (4).
(a) ô (c): By definitions.
In this case, this stronger notion can be related with the c-morphisms, introduced below: Definition 4. A mapping h : V Ñ W from a L-closure system V Ď LX into an L-closure system W Ď LY is called a c-morphism if it is a Ñ- and Ź-morphism, i.e. if – hpa Ñ Cq “ a Ñ hpCq for each a P L and C P V ; – hpŹkPK Ckq “ ŹkPK hpCkq for every collection of Ck P V (k P K); – if C is an a-complement then hpCq is an a-complement.
For formally establishing the relationship, the two following results are
recalled:
Lemma 3 ([
A P LXˆY such that hpCq “ C Ż A for every C P LY . 2. Let A P LXˆY , the mapping hA : LX Ñ LY defined by hApCq “ C Ż A p“
C^A q is an extendable c-morphism.
Theorem 3. The strong L-bonds between K1 “ xX1, Y1, I1y and K2 “ xX2, Y2, I2y are in one-to-one correspondence with extendable c-morphisms ExtpX2X, Y2Y, I2q to ExtpX1X, Y1Y, I1q. Proof. We show procedures to obtain the c-morphism from a strong L-bond and vice versa.
“ñ”: Let β be a strong L-bond between K1 “ xX1, Y1, I1y and K2 “ xX2, Y2, I2y. By Lemma 2 there is Se P LX1ˆX2 such that β “ Si˝I2. The induced operator YSi is an extendable c-morphism ExtpX2X, Y2Y, I2q to ExtpX1X, Y1Y, I1q by Lemma 3(2).
“ð”: For extendable c-morphism f : ExtpX2Ò, Y2Ó, I2q Ñ ExtpX1X, Y1Y, I1q there is an L-relation Se s.t. f pBq “ BYSi for each A P ExtpX2X, Y2Y, I2q by Lemma 3(1). Then β “ Se ˝ I2 is a strong L-bond by Lemma 2.
One can check that these two procedures are mutually inverse. Theorem 4. The system of all strong L-bonds is an L-interior system. Proof. Using Lemma 2 (b), it is an intersection of the L-interior systems from Theorem 2. \[ Definition 5. Let K1 “ xX1, Y1, I1y, K2 “ xX2, Y2, I2y be L-contexts. The direct product of K1 and K2 is defined as the L-context K1‘K2 “ xX2 ˆ Y1, X1 ˆ Y2, Δy with Δpxx2, y1y, xx1, y2yq “ I1px1, y1q b I2px2, y2q.
Theorem 5. The intents of K1 ‘ K2 are L-bonds between K1 and K2. Proof. We have φXpx1, y2q “ ł φpx2, y1q b Δpxx2, y1y, xx1, y2yq “ ł φpx2, y1q b I1px1, y1q b I2px2, y2q “ “ “ xx2,y1y ł ł φpx2, y1q b I1px1, y1q b I2px2, y2q y1PY1 x2PX2
x2PX2 ł I1px1, y1q b pφT ˝ I2qpy1, y2q y1PY1 ł I1px1, y1q b ł φpx2, y1q b I2px2, y2q y1PY1 “ pI1 ˝ φT ˝ I2qpx1, y2q.
Now, notice that pI1 ˝ φT q ˝ I2 “ I1 ˝ pφT ˝ I2q “ β is a strong L-bond by Lemma 2. \[ Remark 1. It is worth mentioning that not every strong L-bond is included in IntppX1 ˆ Y2qX, pX2 ˆ Y1qY, Δq since there are isotone L-bonds which are not of the form of I1 ˝ φT ˝ I2. For instance, using the same structure of truth degrees and I1 as in Example 1, obviously I1 is L-bond on K1 (i.e. between K1 and K1), but IntppX1 ˆ Y2qX, pX2 ˆ Y1qY, Δq contains only H.
The end of proof of the Theorem 5 also explains which L-bonds are intents
of K1 ‘ K2:
Corollary 1. The intents of K1 ‘ K2 are exactly those L-bonds between K1
and K2 which can be decomposed as I1 ˝ φT ˝ I2 for some φ P LX2ˆY1 .
Remark 2 (Relationship to the antitone case in [
Assuming the double negation law, we have the equality ExtpXÒ, Y Ó, Iq “ ExtpXX, Y Y, Iq. Thus, for a strong L-bond β P LX1ˆY2 “ Se ˝ I2 “ I1 ˝ Si between K1 and K2 we have β “ Se Ž I2 “ I1 Ż Si being an antitone L-bond between K1 and K2.
Remark 3. Some papers [
I1px1, y1q Ñ I2px2, y2q p“
I2px2, y2q Ñ I1px1, y1qq induces concept-forming operator φÒΔ for which we have φÒΔ px1, y2q “ “ “ “ “ “
ľ xx1,y2yPX1ˆY2
ľ xx2,y1yPX1ˆY2 ľ ľ x2PX2 y1PY1 ľ y1PY1 ľ y1PY1 ľ y1PY1 φpx2, y1q Ñ r I1px1, y1q Ñ I2px2, y2qs
I1px1, y1q Ñ pφpx2, y1q Ñ I2px2, y2qq
I1px1, y1q Ñ pφpx2, y1q Ñ I2px2, y2qq I1px1, y1q Ñ ľ pφpx2, y1q Ñ I2px2, y2qq x2PX2 I1px1, y1q Ñ pφT Ž I2qpy1, y2q pφT Ž I2qpy1, y2q Ñ I1px1, y1q “ rI1 Ż pφT Ž I2qspx1, y2q “ r I1 Ž pφT Ž I2qspx1, y2q “ r I1 Ž pφT Ž I2qspx1, y2q “ rp I1 ˝ φT q Ž I2qspx1, y2q “ r p I1 ˝ φT ˝ I2qspx1, y2q Whence an antitone L-bond is an intent of the concept lattice of K1 b K2 iff it is possible to write it as p I1 ˝ φT ˝ I2q i.e. if its complement is an intent of K1 ‘ K2.
We studied L-bonds with respect to isotone concept-forming operators, computation of L-bonds using dirrect products, and the relationship of these results to the previous results on antitone L-bonds.
The present results can be easily generalized to a setting in which extents, intents and the context relation use different structures of truth-degrees. We will bring this generalization in an extended version of the paper.
Our future research in this area includes the the study of yet another type of
(extendable) i-morphisms and c-morphisms. In [