Isotone L-bonds Jan Konecny1 and Manuel Ojeda-Aciego2 1 University Palacky Olomouc, Czech Republic‹ 2 Dept. Matemática Aplicada, Univ. Málaga, Spain‹‹ Abstract. L-bonds represent relationships between formal contexts. We study properties of these intercontextual structures w.r.t. isotone concept- forming 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. 1 Introduction 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 general- ization 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 [21, 15, 14], possibility theory [8], fuzzy set theory [1, 2], the multi-adjoint framework [16, 17, 19] or heterogeneous approaches [6, 18]. Goguen argued in [10] that concepts should be studied transversally, tran- scending the natural boundaries between sciences and humanities, and proposed category theory as a unifying language capable of merging different apparently disparate approaches. Krotzsch [13] suggested a categorical treatment of mor- phisms, understood as fundamental structuring blocks, in order to model, among other applications, data translation, communication, and distributed computing. 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 [12] on the notion of Chu correspondences between formal contexts, which led to obtaining information about the structure of L-bonds in such a generalized framework. ‹ Supported by the ESF project No. CZ.1.07/2.3.00/20.0059, the project is cofinanced by the European Social Fund and the state budget of the Czech Republic. ‹‹ Supported by Spanish Ministry of Science and FEDER funds through project TIN09- 14562-C05-01 and Junta de Andalucı́a project P09-FQM-5233. c paper author(s), 2013. Published in Manuel Ojeda-Aciego, Jan Outrata (Eds.): CLA 2013, pp. 153–162, ISBN 978–2–7466–6566–8, Laboratory L3i, University of La Rochelle, 2013. Copying permitted only for private and academic purposes. 154 Jan Konecny and Manuel Ojeda-Aciego 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 Preliminaries We use complete residuated lattices as basic structures of truth-degrees. A com- plete 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 arbi- trary 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“aÑ0 (1) 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 tApx1 q{x1 , Apx2 q{x1 , . . . , Apxn q{xn u. An L-set A P LX is called crisp if Apxq P t0, 1u for each x P X. Crisp L- sets 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, (2) pa Ñ Aqpxq “ a Ñ Apxq, (3) pA Ñ aqpxq “ Apxq Ñ a, (4) Apxq “ Apxq Ñ 0. (5) Isotone L-bonds 155 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 ł pA ˝ Bqpx, yq “ Apx, f q b Bpf, yq, (6) f PF ľ pA Ž Bqpx, yq “ Apx, f q Ñ Bpf, yq, (7) f PF ľ pA Ż Bqpx, yq “ Bpf, yq Ñ Apx, f q. (8) f PF 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 ľ ľ C Ò pyq “ Cpxq Ñ Ipx, yq, DÓ pxq “ Dpyq Ñ Ipx, yq. (9) xPX yPY Second, the pair xX, Yy of operators X : LX Ñ LY and Y : LY Ñ LX is defined by ł ľ C X pyq “ Cpxq b Ipx, yq, DY pxq “ Ipx, yq Ñ Dpyq, (10) xPX yPY Third, the pair x^ , _ y of operators ^ : LX Ñ LY and _ : LY Ñ LX is defined by ľ ł C ^ pyq “ Ipx, yq Ñ Cpxq, D_ pxq “ Dpyq b Ipx, yq, (11) xPX yPY for C P LX , D P LY . To emphasize that the operators are induced by I, we also denote the opera- tors by xÒI , ÓI y, xXI , YI y, and x^I , _I y. Furthermore, denote the corresponding sets of fixpoints by BpX Ò , Y Ó , Iq, BpX X , Y Y , Iq, and BpX ^ , Y _ , Iq, i.e. BpX Ò , Y Ó , Iq “ txC, Dy P LX ˆ LY | C Ò “ D, DÓ “ Cu, BpX X , Y Y , Iq “ txC, Dy P LX ˆ LY | C X “ 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. 156 Jan Konecny and Manuel Ojeda-Aciego For a concept lattice BpX M , 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 ExtpX M , Y O , Iq and IntpX M , Y O , Iq. That is, ExtpX M , Y O , Iq “ tC P LX | xC, Dy P BpX M , Y O , Iq for some Du, IntpX M , Y O , Iq “ tD P LY | xC, Dy P BpX M , 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 abC PV; Ť – V is closed under union, i.e. for Cj P V (j P J) we have jPJ 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); Ş – V is closed under intersection, i.e. for Cj P V (j P J) we have jPJ Cj P V . 3 Weak L-bonds 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 , I1 y and K2 “ xX2 , Y2 , I2 y w.r.t. xX, Yy is an L-relation β P LX1 ˆY2 s.t. ExtpX1X , Y2Y , βq Ď ExtpX1X , Y1Y , I1 q and IntpX1X , Y2Y , βq Ď IntpX2X , Y2Y , I2 q. 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 Ž an L-interior system W Ď L is called an i-morphism if it is a b- and Y into -morphism, i.e. if Žb Cq “ a bŽ – hpa hpCq for each a P L and C P V ; – hp kPK Ck q “ kPK hpCk q 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 h1 pCq “ hpCq; The following results will be used hereafter. Lemma 1 ([5]). 1. For V Ď LX , if h : V Ñ LY is an extendable i-morphism then there exists an L-relation A P LXˆY such that hpCq “ C ˝ A for every C P LY . Isotone L-bonds 157 2. Let A P LXˆY , the mapping hA : LX Ñ LY defined by hA pCq “ C˝A “ C XA is an extendable i-morphism. 3. Consider two contexts xX, Y, Iy and xF, Y, By. Then, we have IntpX X , Y Y , Iq Ď 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 ExtpX X , Y Y , Iq Ď ExtpX X , 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 , I1 y and K2 “ xX2 , Y2 , I2 y are in one-to-one correspondence with extendable i-morphisms IntpX1X , Y1Y , I1 q to IntpX2X , Y2Y , I2 q. 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 , I1 y and K2 “ xX2 , Y2 , I2 y. By Definition 1 we have IntpX1X , Y2Y , βq Ď IntpX2X , Y2Y , I2 q; thus by Lemma 1(3) there exists Si P LY1 ˆY2 such that β “ I1 ˝ Si . The induced opeator XSi is an extendable i-morphism IntpX1X , Y1Y , I1 q to IntpX2X , Y2Y , I2 q by Lemma 1(2). “ð”: For an extendable i-morphism f : IntpX1X , Y1Y , I1 q Ñ IntpX2X , Y2Y , I2 q there is an L-relation Si s.t. f pBq “ B XSi for each B P IntpX1X , Y1Y , I1 q 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 LX1 ˆY2 s.t. ExtpX1^ , Y2_ , βq Ď ExtpX1^ , Y1_ , I1 q and IntpX1^ , Y2_ , βq Ď IntpX2^ , Y2_ , I2 q. 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, xby “ 0 otherwise, Ž Ž for each x, y P L. One can easily see that x b j yj “ j px b yj q 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: $ &1 if x ď y, ’ x Ñ 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 , I1 q “ ExtptxuX , tyuY , I2 q “ ttb{xu, xu and, trivially, IntptxuX , tyuY , I2 q “ IntptxuX , tyuY , I2 q. 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_ , I1 q “ tH, ta{xuu Ğ tH, tb{xuu “ Extptxu^ , tyu_ , I2 q. 158 Jan Konecny and Manuel Ojeda-Aciego Theorem 2. The system of all weak L-bonds is an L-interior system. Proof. From properties of i-morphism. \ [ 4 Strong L-bonds 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 , I1 y and K2 “ xX2 , Y2 , I2 y is an L-relation β P LX1 ˆY2 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 , I1 y and K2 “ xX2 , Y2 , I2 y. (b) β satisfies both ExtpX1^ , Y2_ , βq Ď ExtpX1^ , Y1_ , I1 q and IntpX1X , Y2Y , βq Ď IntpX2X , Y2Y , I2 q. (c) β “ Se ˝ I2 “ I1 ˝ Si for some Se P LX1 ˆX2 and Si P LY1 ˆY2 . 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, intro- duced below: Definition 4. A mapping h : V Ñ W from a L-closure system V Ź Ď LX into an L-closure system W Ď L is called a c-morphism if it is a Ñ- and -morphism, Y i.e. if – hpaŹÑ Cq “ a Ñ ŹhpCq for each a P L and C P V ; – hp kPK Ck q “ kPK hpCk q 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 re- called: Lemma 3 ([4]). 1. If h : V Ñ LY is an extendable c-morphism then there exists an L-relation 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 hA pCq “ C Ż A p“ C ^A q is an extendable c-morphism. Theorem 3. The strong L-bonds between K1 “ xX1 , Y1 , I1 y and K2 “ xX2 , Y2 , I2 y are in one-to-one correspondence with extendable c-morphisms ExtpX2X , Y2Y , I2 q to ExtpX1X , Y1Y , I1 q. Isotone L-bonds 159 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 , I1 y and K2 “ xX2 , Y2 , I2 y. By Lemma 2 there is Se P LX1 ˆX2 such that β “ Si ˝I2 . The induced operator YSi is an extendable c-morphism ExtpX2X , Y2Y , I2 q to ExtpX1X , Y1Y , I1 q by Lemma 3(2). “ð”: For extendable c-morphism f : ExtpX2Ò , Y2Ó , I2 q Ñ ExtpX1X , Y1Y , I1 q there is an L-relation Se s.t. f pBq “ B YSi for each A P ExtpX2X , Y2Y , I2 q 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 , I1 y, K2 “ xX2 , Y2 , I2 y be L-contexts. The direct product of K1 and K2 is defined as the L-context K1 ‘K2 “ xX2 ˆ Y1 , X1 ˆ Y2 , ∆y with ∆pxx2 , y1 y, xx1 , y2 yq “ I1 px1 , y1 q b I2 px2 , y2 q. Theorem 5. The intents of K1 ‘ K2 are L-bonds between K1 and K2 . Proof. We have ł φX px1 , y2 q “φpx2 , y1 q b ∆pxx2 , y1 y, xx1 , y2 yq ł “ φpx2 , y1 q b I1 px1 , y1 q b I2 px2 , y2 q xx2 ,y1 y ł ł “ φpx2 , y1 q b I1 px1 , y1 q b I2 px2 , y2 q y1 PY1 x2 PX2 ł ł “ I1 px1 , y1 q b φpx2 , y1 q b I2 px2 , y2 q y1 PY1 x2 PX2 ł “ I1 px1 , y1 q b pφT ˝ I2 qpy1 , y2 q y1 PY1 “ pI1 ˝ φT ˝ I2 qpx1 , y2 q. Now, notice that pI1 ˝ φT q ˝ I2 “ I1 ˝ pφT ˝ I2 q “ 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 Y2 qX , pX2 ˆ Y1 qY , ∆q 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 Y2 qX , pX2 ˆ Y1 qY , ∆q contains only H. The end of proof of the Theorem 5 also explains which L-bonds are intents of K1 ‘ K2 : 160 Jan Konecny and Manuel Ojeda-Aciego 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 LX2 ˆY1 . Remark 2 (Relationship to the antitone case in [12]). Assuming the double negation law, we have the equality ExtpX Ò , Y Ó , Iq “ ExtpX X , Y Y , Iq. Thus, for a strong L-bond β P LX1 ˆY2 “ 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 [9, 12] have considered direct products in the crisp and the fuzzy settings, respectively, for the antitone case. In [12] conditions are spec- ified under which antitone L-bonds are present in the concept lattice of the direct product. Corollary 1 and Remark 2 provide a simplification of these con- ditions. The concept lattice of a direct product K1 b K2 defined as in [12] i.e. K1 b K2 “ xX2 ˆ Y1 , X1 ˆ Y2 , ∆y with ∆pxx2 , y1 y, xx1 , y2 yq “ I1 px1 , y1 q Ñ I2 px2 , y2 q p“ I2 px2 , y2 q Ñ I1 px1 , y1 qq induces concept-forming operator φÒ∆ for which we have ľ φÒ∆ px1 , y2 q “ φpx2 , y1 q Ñ r I1 px1 , y1 q Ñ I2 px2 , y2 qs xx1 ,y2 yPX1 ˆY2 ľ “ I1 px1 , y1 q Ñ pφpx2 , y1 q Ñ I2 px2 , y2 qq xx2 ,y1 yPX1 ˆY2 ľ ľ “ I1 px1 , y1 q Ñ pφpx2 , y1 q Ñ I2 px2 , y2 qq x2 PX2 y1 PY1 ľ ľ “ I1 px1 , y1 q Ñ pφpx2 , y1 q Ñ I2 px2 , y2 qq y1 PY1 x2 PX2 ľ “ I1 px1 , y1 q Ñ pφT Ž I2 qpy1 , y2 q y1 PY1 ľ “ pφT Ž I2 qpy1 , y2 q Ñ I1 px1 , y1 q y1 PY1 “ rI1 Ż pφT Ž I2 qspx1 , y2 q “ r I1 Ž pφT Ž I2 qspx1 , y2 q “ r I1 Ž pφT Ž I2 qspx1 , y2 q “ rp I1 ˝ φT q Ž I2 qspx1 , y2 q “ r p I1 ˝ φT ˝ I2 qspx1 , y2 q 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 ˝ I2 q i.e. if its complement is an intent of K1 ‘ K2 . 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