We describe properties of compositions of isotone bonds between L-fuzzy contexts over different complete residuated lattices and we show that L-fuzzy contexts as objects and isotone bonds as arrows form a category. c paper author(s), 2015. Published in Sadok Ben Yahia, Jan Konecny (Eds.): CLA 2015, pp. 205-216, ISBN 978-2-9544948-0-7, Blaise Pascal University, LIMOS laboratory, Clermont-Ferrand, 2015. Copying permitted only for private and academic purposes.
In Formal Concept Analysis, bonds represent relationships between formal
contexts. One of the motivations for introducing this notion is to provide a tool
for studying mappings between formal contexts, corresponding to the behavior
of Galois connections between their corresponding concept lattices. The notions
of bonds, scale measures and informorphisms were studied by [
In our previous works, we studied generalizations of bonds into an L-fuzzy
setting in [
In the present paper we continue our study [
The structure of the paper is as follows. First, in Section 2 we recall basic notions required in the rest of the paper. Section 3.1 considers weak homogeneous L-bonds w.r.t. isotone concept-forming operators and their compositions. Section 3.2 then generalizes the results to the setting of formal fuzzy contexts over different structure of truth degrees. Finally, we summarize our results and outline our future research in this area in Section 4. 2 2.1
Residuated lattices, fuzzy sets, and fuzzy relations 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”.
An L-set (or L-fuzzy set) A in a universe set X is a mapping assigning to each x P X some truth degree Apxq P L. The set of all L-sets in a universe X is denoted LX .
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 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 a-multiplication a b A and a-shift a Ñ A are L-sets defined by pa b Aqpxq “ a b Apxq, pa Ñ Aqpxq “ a Ñ Apxq.
Binary L-relations (binary L-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).
For an L-relation I P LXˆY we define its transpose as the L-relation IT P LY ˆX given by ITpy, xq “ Ipx, yq for each x P X, y P Y .
Various composition operators for binary L-relations were extensively studied
by [
fPF pA Ż Bqpx, yq “ ľ Bpf, yq Ñ Apx, f q.
fPF
Note also that for L “ t0, 1u, A˝B coincides with the well-known composition of binary relations.
We will occasionally use some of the following properties concerning the
associativity of several composition operators, see [
Theorem 1. The operator ˝ from above has the following properties concerning composition.
– Associativity: – Distributivity: 2.2
Formal fuzzy concept analysis
R ˝ pS ˝ T q “ pR ˝ Sq ˝ T. pď Riq ˝ S “ ďpRi ˝ Sq, and i i
R ˝ pď Siq “ ďpR ˝ Siq.
i i An L-context is a triplet xX, Y, Iy where X and Y are (ordinary nonempty) 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 pair xX, Yy of operators X : LX Ñ LY and Y : LY Ñ LX induced by an L-context xX, Y, Iy:
AXpyq “ ł Apxq b Ipx, yq,
BYpxq “ xPX ľ Ipx, yq Ñ Bpyq. yPY for all A P LX and B P LY . When we consider concept-forming operators induced by multiple L-relations, we write the inducing L-relation as the subscript of the symbols of the operators. For example, the pair of concept-forming operators induced by L-relation I are written as xXI , YI y.
Remark 1. Notice that the pair of concept-forming operators can be interpreted as instances of the composition operators between relations. Applying the isomorphisms L1ˆX – LX and LY ˆ1 – LY whenever necessary, one could write them, alternatively, as
AX “ A ˝ I and Furthermore, denote the set of fixed points of xX, Yy by BXYpX, Y, Iq, i.e.
The set of fixed points endowed with ď, defined by
xA1, B1y ď xA2, B2y if A1 Ď A2 (equivalently B2 Ď B1)
is a complete lattice [
ExtXYpX, Y, Iq “ tA P LX | xA, By P BXYpX, Y, Iq for some Bu,
IntXYpX, Y, Iq “ tB P LY | xA, By P BXYpX, Y, Iq for some Au.
Equivalently, we can characterize ExtXYpX, Y, Iq and IntXYpX, Y, Iq as follows (6) (7) (8) (9)
We will need the following lemma from [
Lemma 1. Consider L-contexts xX, Y, Iy, xX, F, Ay, and xF, Y, By.
Y ,
In this paper we assume only weak homogeneous L-bonds w.r.t. xX, Yy. In what follows, we omit the words ‘weak homogeneous’ and the pair of conceptforming operators and call them just ‘L-bonds’.
We will utilize the following characterization of L-bonds.
Lemma 2 ([
Remark 2. Note that due to results on fuzzy relational equations we have that
the L-relation Se from Lemma 2 is equal to β Ż I2T (see [
Firstly, we describe compositions of L-bonds and show that they form a category. Later we generalize the results to setting of isotone bonds between fuzzy contexts over different complete residuated lattices. 3.1
Setting with uniform structures of truth degrees We start with the notion of composition of L-bonds.
Definition 2. Let β1 be an L-bond from xX1, Y1, I1y to xX2, Y2, I2y and β2 be an L-bond from xX2, Y2, I2y to xX3, Y3, I3y. Define composition of β1 and β2 as the L-relation pβ1 Ż I2Tq ˝ β2 P LX1ˆY3 and denote it β1 ‚ β2.
Theorem 2. The composition of L-bonds is an L-bond.
Proof. Let β1 be an L-bond from xX1, Y1, I1y to xX2, Y2, I2y and β2 be an Lbond from xX2, Y2, I2y to xX3, Y3, I3y. By Lemma 2 there are Se P LX1ˆX2 , Se1 P LX2ˆX3 such that β1 “ Se ˝ I2, β2 “ Se1 ˝ I3. By Definition 2 and Remark 2 we have β1 ‚ β2 “ pβ1 Ż I2Tq ˝ β2
“ Se ˝ Se1 ˝ I3.
Hence we have
by Lemma 1 (a). Note that the mapping YSe maps extents of I2 to extents of I1 by Lemma 2 and that BYβ2 is extent of I2 for any B P IntXYpX3, Y3, I3q by (8). Thus we have
hence
The equalities (10) and (11) imply that β1 ‚ β2 is an L-bond.
Lemma 3. Let β be an L-bond from L-context xX1, Y1, I1y to L-context xX2, Y2, I2y. For any L-set A P LX1 we have that AXI1 YI1 Xβ “ AXβ .
Proof. Let A be an arbitrary L-set from LX1 . Then
AXI1 YI1 Xβ Ě AXβ since p´qXβ is isotone and AXI1 YI1 Ě A “ AXβYβXβ “ AXβYβXI2 YI2 Xβ due to definition of L-bond Ě AXI1 YI1 Xβ since the mapping p´qXI1 YI1 Xβ is isotone Hence AXI1 YI1 Xβ “ AXβ . (10) (11) \[ \[ The equality from Lemma 3 written in relational form is A˝β “ pA˝I1qŻI1Tq˝β; we use that to prove the following theorem.
Theorem 3. Composition of L-bonds is associative.
Proof. Let β1 be an L-bond from xX1, Y1, I1y to xX2, Y2, I2y, β2 be an L-bond from xX2, Y2, I2y to xX3, Y3, I3y, and β3 be an L-bond from xX3, Y3, I3y to xX4, Y4, I4y. We have pβ1 ‚ β2q ‚ β3 “ pppβ1 Ż I2Tq ˝ β2q Ż I3Tq ˝ β3 “ ppSe ˝ β2q Ż I3Tq ˝ β3 “ ppSe ˝ pSe1 ˝ I3qq Ż I3Tq ˝ β3 “ pppSe ˝ Se1q ˝ I3q Ż I3Tq ˝ β3 “ pSe ˝ Se1q ˝ β3 “ Se ˝ pSe1 ˝ β3q “ Se ˝ pβ2 ‚ β3q “ β1 ‚ pβ2 ‚ β3q by Remark 2 and Definition 2. by Definition 2 by Remark 2 by Lemma 2
by (3) by Lemma 3 by (3) \[
We obtain a category of L-contexts and L-bonds.
Theorem 4. The structure of L-contexts and L-bonds forms a category: Objects are L-contexts, Arrows are L-bonds where identity arrow of any formal L-context xX, Y, Iy is its incidence relation I,4 composition of arrows β1 ‚ β2 is given by Definition 2.
Remark 3. The category is equivalent to category of attribute-oriented concept
lattices and isotone Galois connections. That is analogous to results in [
Setting with different structures of truth degrees
In this section we generalize the previous results into a setting in which fuzzy
contexts are defined over different complete residuated lattices. To do that we need
to explore compositions of underlying morphisms called residuation-preserving
pl, kq-connections between complete residuated lattices.
pl, kq-connections and their compositions
Firstly, let us recall definition and basic properties of the pl, kq-connections
introduced in [
Definition 3 ([
‚ c b ‚ d ‚
We call xλ, κy an pl, kq-connection from L1 to L2. An pl, kq-connection from L1 to L2 is called residuation-preserving if
κpk b2 pλpaq Ñ2 λpbqqq “ κλpaq Ñ1 κλpbq holds true for any a, b P L2.
Theorem 5 ([
(equivalently, if a2 ď2 b2) and the adjoint pair is given by xa1, a2y Ñ xb1, b2y “ xa1 Ñ1 b1, k b2 pa2 Ñ2 b2qy
“ xa1 Ñ1 b1, k b2 ppk Ñ2 a2q Ñ2 pk Ñ2 b2qqy, xa1, a2y b xb1, b2y “ xl Ñ1 pl b1 a1 b1 b1q, a2 b2 pk Ñ2 b2qy
“ xl Ñ1 pl b1 a1 b1 b1q, pk Ñ2 a2q b2 b2y
is a complete residuated lattice.
(12)
(13)
(14)
(15)
(16)
(17)
Figure 1 shows an example of pl, kq-connection. We refer the reader to [
Now we define composition of pl, kq-connections and show that it is an pl, kqconnection as well. In addition, the composition preserves residuation-preservation, that means that composition of residuation-preserving pl, kq-connections is a residuation-preserving pl, kq-connection as well.
Theorem 6. Let xλ1, κ1y be an pl1, k2q-connection from L1 to L2 and xλ2, κ2y be an pk2, j3q-connection from L2 to L3. Then the pair of mappings λ : L1 Ñ L3, κ : L3 Ñ L1, defined by λpa1q “ λ2pk2 Ñ2 λ1pa1qq, κpa3q “ κ1pk2 b2 κ2pa3qq for each a1 P L1 and a2 P L2, is an pl1, j3q-connection from L1 to L3. Proof. First, we prove that κλpa1q “ l1 Ñ1 pl1 b1 a1q for each a1 P L1 and λκpa2q “ j3 b3 pj3 Ñ3 a3q for each a3 P L3. For each a1 P L1, we have Similarly, we have for each a3 P L3 κλpa1q “ κ1pk2 b2 κ2pλ2pk2 Ñ2 λ1pa1qqqq “ κ1pk2 b2 pk2 Ñ2 pk2 b pk2 Ñ2 λ1pa1qqqqq “ κ1pk2 b2 pk2 Ñ2 λ1pa1qqq “ κ1pλ1pκ1pλ1pa1qqqq “ κ1pλ1pa1qq “ l1 Ñ1 pl1 b1 a1q. λκpa3q “ λ2pk2 Ñ2 λ1pκ1pk2 b2 κ2pa3qqqq “ λ2pk2 Ñ2 pk2 b2 pk2 Ñ2 pk2 b2 κ2pa3qqqqq “ λ2pk2 Ñ2 pk2 b2 κ2pa3qqqqq “ λ2pκ2pλ2pκ2pa3qqqq “ λ2pκ2pa3qq “ j3 b3 pj3 Ñ3 a3q.
Since κλpa1q “ l1 Ñ1 pl1 b1 a1q ě1 a1 and λκpa3q “ j3 b3 pj3 b3 a3q ď3 a3 we only need to show monotony to prove that xλ, κy is an isotone Galois connection: For each a1, b1 P L1 we have a1 ď1 b1 implies λ1pa1q ď2 λ1pb1q since λ1 is monotone, implies k2 Ñ2 λ1pa1q ď2 k2 Ñ2 λ1pb1q since Ñ2 is monotone
in its second argument, implies λ2pk2 Ñ2 λ1pa1qq ď3 λ2pk2 Ñ2 λ1pb1qq since λ2 is monotone. Thus a1 ď1 b1 implies λpa1q ď3 λpb1q for each a1, b1 P L1. Similarly, one can show that a3 ď3 b3 implies κpa3q ď1 κpb3q. (18) Theorem 7. Let xλ1, κ1y be a residuation-preserving pl1, k2q-connection from L1 to L2 and xλ2, κ2y be a residuation-preserving pk2, j3q-connection from L2 to L3. Then the pair of mappings λ : L1 Ñ L3, κ : L3 Ñ L1, defined by (18), is a residuation-preserving pl1, j3q-connection from L1 to L3.
Proof. For each a1, b1 P L1 we have κλpa1q Ñ1 κλpb1q “ “ pl1 Ñ1 pl1 b1 a1qq Ñ1 pl1 Ñ1 pl1 b1 b1qq “ κ1λ1pa1q Ñ1 κ1λ1pb1q “ κ1pk2 b2 pλ1pa1q Ñ2 λ1pb1qqq “ κ1pk2 b2 pλ1κ1λ1pa1q Ñ2 λ1κ1λ1pb1qqq “ κ1pk2 b2 ppk2 b2 pk2 Ñ2 λ1pa1qqq Ñ2 pk2 b2 pk2 Ñ2 λ1pb1qqqqq “ κ1pk2 b2 ppk2 b2 pk2 Ñ2 pk2 b2 pk2 Ñ2 λ1pa1qqqqq Ñ2 pk2 b2 pk2 Ñ2 λ1pb1qqqqq “ κ1pk2 b2 ppk2 Ñ2 pk2 b2 pk2 Ñ2 λ1pa1qqqq Ñ2 pk2 Ñ2 pk2 b2 pk2 Ñ2 λ1pb1qqqqq “ κ1pk2 b2 pκ2λ2pk2 Ñ2 λ1pa1qq Ñ2 κ2λ2pk2 Ñ2 λ1pb1qqqq “ κ1pk2 b2 κ2pj3 b3 pλ2pk2 Ñ2 λ1pa1qq Ñ3 λ2pk2 Ñ2 λ1pb1qqqqq “ κ1pk2 b2 κ2pj3 b3 pλpa1q Ñ3 λpb1qqqq “ κpj3 b3 pλpa1q Ñ3 λpb1qqq.
We call xλ, κy from (18) a composition of xλ1, κ1y and xλ2, κ2y and we denote it as xλ1, κ1y ‚ xλ2, κ2y “ xλ1 ‚ λ2, κ1 ‚ κ2y. Now we show, that the composition of pl, kq-connections is associative.
Theorem 8. Let xλ1, κ1y be an pl1, k2q-connection from L1 to L2, xλ2, κ2y be a pk2, j3q-connection from L2 to L3, and xλ3, κ3y be a pj3, m4q-connection from L3 to L4. Then
xλ1, κ1y ‚ pxλ2, κ2y ‚ xλ3, κ3yq “ pxλ1, κ1y ‚ xλ2, κ2yq ‚ xλ3, κ3y.
Proof. We have for each a P L1 pλ1 ‚ pλ2 ‚ λ3qqpa1q “ pλ2 ‚ λ3qpk2 Ñ2 λ1pa1qq “ λ3pj3 Ñ λ2pk2 Ñ2 λ1pa1qqq “ λ3pj3 Ñ pλ1 ‚ λ2qpa1qq “ ppλ1 ‚ λ2q ‚ λ3qpa1q and similarly for the κ-part.
Theorem 9. The following structure forms a category.
Objects are pairs xL, ey, where L is a complete residuated lattices and e P L. Arrows from xL1, ly to xL2, ky are pl, kq-connections from L1 to L2, where \[ \[ identity arrow on any xL, ey is pe, eq-connection xλ, κy where λpaq “ e b a and κpaq “ e Ñ a for each a P L.
composition of arrows is as defined in (18).
If we use just residuation-preserving pl, kq-connections we obtain a sub-category.
Now, we can explore bonds based on residuation-preserving pl, kq-connections. Definition 4. Let L1, L2 be complete residuated lattices, xλ, κy be residuationpreserving pl, kq-connection from L1 to L2, and let xX1, Y1, I1y and xX2, Y2, I2y be L1-context and L2-context, respectively. We call β P LxXλ1,κˆyY2 a xλ, κy-bond from xX1, Y1, I1y to xX2, Y2, I2y if the following inclusions hold.
The concept-forming operators xM, Oy induced by xλ, κy-bond β from xX1, Y1, I1y to xX2, Y2, I2y are given by5 (19) (20) (21) \[ AMβ “ λpAqXproj2pβq ,
BOβ “ κpBqYproj1pβq .
Theorem 10. Let xX1, Y1, I1y be an L1-context, xX2, Y2, I2y be an L2-context, and xλ, κy an pl, kq-connection from L1 to L2. Then β P Lxλ,κy is a xλ, κy-bond from xX1, Y1, I1y to xX2, Y2, I2y if and only if it is a Lxλ,κy-bond w.r.t. xX, Yy from xX1, Y1, xκλpI1q, λpI1qyy to xX2, Y2, xκpI2q, λκpI2qyy.
Proof. Directly from the definition and (21).
For what follows we will need the following product of fuzzy relations. Let xλ1, κ1y be pl1, k2q-connection from L1 to L2, xλ2, κ2y be pk2, m3q-connection from L2 to L3, and I P LxXλˆ1,Yκ1y, J P LxYλˆ2,Zκ2y. Then I b J P LxXλˆ1‚Zλ2,κ1‚κ2y is defined as
I b J “ xκ1pKq, λ2pk2 Ñ2 Kqy where K “ proj2pIq ˝2 proj1pJ q (22)
LxYλ1,ˆκYy2 such that and ˝2 is composition of L2-relations (1).
Lemma 4. Let xX1, Y1, I1y be an L1-context, xX2, Y2, I2y be an L2-context, and xλ, κy an pl, kq-connection from L1 to L2. (a) An Lxλ,κy-relation β for which exist Lxλ,κy-relations Se P LxXλ1,κˆyX2 and Si P β “ xκλpI1q, λpI1qy b Si “ Se b xκpI2q, λκpI2qy is a xλ, κy-bond from xX1, Y1, I1y to xX2, Y2, I2y. 5 proj1, proj2 denote projection of first and second component of a pair, respectively. (b) Each xλ, κy-bond β from xX1, Y1, I1y to xX2, Y2, I2y satisfies that there is Se P LxXλ1,κˆyX2 such that
β “ Se b xκpI2q, λκpI2qy.
Proof. From Theorem 10 and Lemma 1.
Theorem 11. Let xλ1, κ1y be an pl1, k2q-connection from L1 to L2, xλ2, κ2y be an pk2, j3q-connection from L2 to L3, β1 be xλ1, κ1y-bond from xX1, Y1, I1y to xX2, Y2, I2y, and β2 be xλ2, κ2y-bond from xX2, Y2, I2y to xX3, Y3, I3y.
β “ Se b β2, where Se “ β1 Ż xκ1pI2Tq, λ1κ1pI2Tqy, is a xλ1 ‚ λ2, κ1 ‚ κ2y-bond from xX1, Y1, I1y to xX3, Y3, I3y.
Let us denote β from (23) as β “ β1 ‚ β2 and call it a composition of isotone xλ, κy-bonds. Now we show associativity of this composition.
Theorem 12. Let xλ1, κ1y be an pl1, k2q-connection from L1 to L2, xλ2, κ2y be an pk2, j3q-connection from L2 to L3, xλ3, κ3y be an pj3, m4q-connection from L3 to L4, and βi be xλi, κiy-bond from xXi, Yi, Iiy to xXi`1, Yi`1, Ii`1y. Then β1 ‚ pβ2 ‚ β3q “ pβ1 ‚ β2q ‚ β3.
Proof. Follows from Theorem 3, Theorem 8, and Theorem 10. \[
Finally, we can state that L-contexts over different structures of truth degrees and bonds between them form a category.
Theorem 13. Objects are pairs xK, ey, where K is a L-context and e P L. Arrows between xK1, ly and xK2, ky, where K1 is an L1-context, K2 is an L2-context and l P L1, k P L2, are xλ, κy-bonds, where xλ, κy is an pl, kqconnection. identity arrow for a pair xK, ey of L-context xX, Y, Iy and e is xλ, κybond I with xλ, κy are pe, eq-connections xλ, κy where λpxq “ e Ñ a and κpxq “ e b a for each a P L. composition of arrows β1 ‚ β2 is given by (23). (23) 4
Our future research in this area includes addressing the following issues:
– Antitone bonds between fuzzy contexts over different complete residuated
lattices were studied in [
Jan Konecny is supported by grant No. 15-17899S, “Decompositions of Matrices with Boolean and Ordinal Data: Theory and Algorithms”, of the Czech Science Foundation.
Ondrej Kr´ıdlo is supported by grant VEGA 1/0073/15 by the Ministry of Education, Science, Research and Sport of the Slovak republic and University Science Park TECHNICOM for Innovation Applications Supported by Knowledge Technology, ITMS: 26220220182, supported by the Research & Development Operational Programme funded by the ERDF.