Block Relations in Fuzzy Setting

Block relations in fuzzy setting

Jan Konecny, Michal Krupka Department of Computer Science

Palack´y University in Olomouc 17. listopadu 12, CZ-77146 Olomouc

Czech Republic

jan.konecny@upol.cz, michal.krupka@upol.cz

Abstract. One of the main problems in FCA (especially in fuzzy setting) is to reduce a concept lattice to appropriate size to make it graspable and understand- able. We generalize known results on the correspondence of block relations of formal contexts and complete tolerances on concept lattices to fuzzy setting. Us- ing an illustrative example we show that the results can be used for reduction of fuzzy concept lattices.

Keywords: Fuzzy concept lattice, Tolerance, Block relation, Factorization

1 Introduction

The problem of the size of concept lattices is recognized as one of the most important problems of Formal Concept Analysis (FCA). The main aim of this paper is to describe approximation of formal concepts by fuzzy tolerances in fuzzy setting.

In [3, 2], a method of factorization of fuzzy concept lattices is presented. A similar- ity thresholdais supplied and the method outputs a factor lattice instead of the whole concept lattice which might be large. The elements of the factor lattice are maximal blocks of concepts from the whole concept lattice which are pairwise similar to degree at leasta. We generalize the results to a more general family of similarities (complete fuzzy tolerances) than those induced by a single threshold. We also show that the factor lattice can be computed from a special kind of superrelations of the incidence relation, called fuzzy block relations; and the relationship between complete fuzzy tolerances and fuzzy block relations. That way we generalize a Wille’s results on (crisp) toler- ances and block relations in [17] (see also [7]).

In [13], Meschke proposed a method of approximation of (crisp) concepts by toler- ances and block relations. We use some of his ideas (namely the selection of important objects and attributes) in an illustrative example at the end of this paper.

The organization of the paper is as follows. In Section 2 we recall the fundamental notions and results on the basic fuzzy structures used in the paper. Section 3 is devoted to study of fuzzy tolerance relations: we define complete tolerance relations on com- pletely lattice ordered fuzzy sets and show fundamental properties of the corresponding factor sets. In Section 4 we describe fuzzy block relations and prove the main result

?Supported by grant no. P103/10/1056 of the Czech Science Foundation.

of the paper: the correspondence between fuzzy block relations and factor structures of concept lattices. Section 5 contains an illustrative example.

Due to space limitations, we abbreviate, resp. omit, some proofs.

2 Preliminaries

We recall basic notions and facts of residuated lattices, fuzzy sets and fuzzy relations, fuzzy Galois connections and fuzzy concept lattices.

2.1 Residuated lattices and fuzzy sets

We use complete residuated lattices as basic structures of truth degrees. A complete residuated lattice [3, 9, 16] is a structureL=hL,∧,∨,⊗,→,0,1isuch that

(i) hL,∧,∨,0,1iis a complete lattice, i.e., a partially ordered set in which arbitrary infima and suprema exist;

(ii) hL,⊗,1iis a commutative monoid, i.e.,⊗is a binary operation which is commuta- tive, associative, anda⊗1=afor eacha∈L;

(iii) ⊗and→satisfy adjointness, i.e.,a⊗b≤ciffa≤b→c.

0 and 1 denote the least and greatest elements. The partial order ofLis denoted by≤.

Throughout the paper,Ldenotes an arbitrary complete residuated lattice.

ElementsaofLare called truth degrees.⊗and→(truth functions of) “fuzzy con- junction” and “fuzzy implication”.

Common examples of complete residuated lattices include those defined on a unit interval, (i.e.,L= [0,1]) or on a finite chain in the unit interval, e.g.L={0,¹_n, . . . ,ⁿ⁻¹_n ,1},

∧and∨being minimum and maximum,⊗being a left-continuous t-norm with the cor- responding→.

The three most important pairs of adjoint operations on the unit interval are Łukasiewicz: a⊗b=max(a+b−1,0)

a→b=min(1−a+b,1) G¨odel:

a⊗b=min(a,b) a→b=

1a≤b botherwise Goguen (product):

a⊗b=a·b a→b=

1 a≤b

b

a otherwise

AnL-set (or fuzzy set)Ain a universe setX is a mapping assigning to eachx∈X some truth degreeA(x)∈LwhereLis the support of a complete residuated lattice. The set of allL-sets in a universeXis denotedL^X.

The operations withL-sets are defined componentwise. For instance, the intersec- tion ofL-setsA,B∈L^X is anL-setA∩BinX such that(A∩B)(x) =A(x)∧B(x)for eachx∈X, etc. AnL-setA∈L^Xis also denoted{^A(x)/x|x∈X}. If for ally∈Xdistinct fromx₁,x₂, . . . ,x_nwe haveA(y) =0, we also write{^A(x1)/x₁,^A(x2)/x₁, . . . ,^A(xn)/x_n}.

BinaryL-relations (binary fuzzy relations) betweenX andY can be thought of as L-sets in the universeX×Y. That is, a binaryL-relationI∈L^X×Y between a setXand a setYis a mapping assigning to eachx∈Xand eachy∈Y a truth degreeI(x,y)∈L(a degree to whichxandyare related byI). AnL-setA∈L^Xis called crisp ifA(x)∈ {0,1}

for eachx∈X. CrispL-sets can be identified with ordinary sets. For a crispA, we also writex∈Afor A(x) =1 andx6∈A for A(x) =0. An L-setA∈L^X is called empty (denoted by /0) if A(x) =0 for eachx∈X. Fora∈L andA∈L^X,a⊗A∈L^X and a→A∈L^X are defined by

(a⊗A)(x) =a⊗A(x)and(a→A)(x) =a→A(x).

For anL-setA∈L^X anda∈L, thea-cut of Ais a crisp subset^aA⊆X such that x∈^aAiffa≤A(x). This definition applies also to binaryL-relations, whosea-cuts are interpreted as classical (crisp) binary relations.

For universeXwe defineL-relationgraded subsethood L^X×L^X→Lby:

S(A,B) = ^{^}

x∈X

A(x)→B(x) (1)

Graded subsethood generalizes the classical subsethood relation⊆; indeed, in the crisp case (i.e.L={0,1}) (1) becomesS(A,B) =1 iff∀x∈X :x∈Aimpliesy∈B. Note that unlike⊆,Sis a binaryL-relation onL^X. Described verbally,S(A,B)represents a degree to whichAis a subset ofB. In particular, we writeA⊆BiffS(A,B) =1. As a consequence, we haveA⊆BiffA(x)≤B(x)for eachx∈X.

Further we set

A≈^XB=S(A,B)∧S(B,A). (2)

The valueA≈^XBis interpreted as the degree to which the setsAandBare similar.

A binaryL-relationRon a setXis calledreflexiveifR(x,x) =1 for anyx∈X,sym- metricifR(x,y) =R(y,x)for anyx,y∈X, andtransitiveifR(x,y)⊗R(y,z)≤R(x,z)for anyx,y,z∈X.Ris called anL-tolerance, if it is reflexive and symmetric,L-equivalence if it is reflexive, symmetric and transitive. IfR is anL-equivalence such that for any x,y∈X from R(x,y) =1 it followsx=y, then R is called an L-equality on X. L- equalities are often denoted by≈. The similarity≈^X ofL-sets (2) is anL-equality on L^X.

Let∼be anL-equivalence onX,A∈L^XanL-set. We say thatAiscompatible with

∼(orextensional with respect to∼) if the following condition holds for anyx,x⁰∈X: A(x)⊗(x∼x⁰)≤A(x⁰).

For each A∈L^X there exists a minimal (with respect to inclusion of L-sets) L-set C≈(A)∈L^X, compatible with∼, and such thatA⊆C∼(A). It holds

C∼(A)(x) = ^_

x⁰∈X

A(x⁰)⊗(x⁰∼x). (3)

We say that a binaryL-relationRonX iscompatible with∼if for eachx,x⁰,y,y⁰∈X, R(x,y)⊗(x∼x⁰)⊗(y∼y⁰)≤R(x⁰,y⁰).

For a mapping f:X →Y, whereY is endowed with anL-equality ≈, andL-set A∈L^X we define theimage of A with respect to f as anL-setf(A)∈L^Y satisfying

f(A)(y) = ^_

x∈X

A(x)⊗(f(x)≈y).

f(A)is compatible with≈and f(A) =C≈(B), where Bis is anL-set inY given by B(y) =^W_x∈f−1({y})A(x).

In the following we use well-known properties of residuated lattices and fuzzy struc- tures which can be found e.g. in [3, 9].

2.2 L-ordered sets

In this section, we recall basic definitions and results of the theory ofL-ordered sets.

Basic references are [1, 3] and the references therein.

AnL-orderon a setUwith anL-equality relation≈is a binaryL-relationonU which is compatible with≈, reflexive, transitive and satisfies(uv)∧(vu)≤u≈v for anyu,v∈U(antisymmetry). The tupleU=hhU,≈i,iis called anL-ordered set.

An immediate consequence of the definition is that for anyu,v∈Uit holds

u≈v= (uv)∧(vu). (4)

IfU=hhU,≈i,iis anL-ordered set, then the tuplehU,¹i, where¹is the 1- cut of, is a (partially) ordered set. We sometimes write≤instead of¹and use the symbols∧,^Vresp.∨,^Wfor denoting infima resp. suprema inhU,¹i.

For twoL-ordered setsU=hhU,≈_Ui,_UiandV=hhV,≈_Vi,_Vi, a mapping f: U →V is called an isomorphism of Uand V, if it is a bijection and (u_1U u₂) = (f(u₁)_V f(u₂))for anyu₁,u₂∈V.UandVare then calledisomorphic.

For anL-ordered setUand anL-setV∈L^Uwe defineL-setsUV,LV∈L^U by LV(u) = ^{^}

v∈U

V(v)→(uv), UV(u) = ^{^}

v∈U

V(v)→(vu).

Using basic rules of fuzzy logic,LV (resp.UV) can be interpreted as the set of ele- ments which are smaller (resp. greater) than or equal to elements ofV.LV (resp.UV) is calledthe lower cone(resp.the upper cone) ofU.

For anyL-setV ∈L^U there exists at most one elementu∈U such thatLV(u)∧ U(LV)(u) =1 (resp.UV∧L(UV)(u) =1) [1, 3]. If there is such an element, we call itthe infimum of V (resp.the supremum of V) and denote infV (resp. supV); otherwise we say, that the infimum (resp. supremum) does not exist.

Ifu,v∈U,v≤u, then theL-set[v,u] =U{v} ∩L{u}is called anintervalinU. It holds inf[v,u] =v, sup[v,u] =u.

AnL-ordered setUis calledcompletely latticeL-ordered, if for eachV∈L^U, both infVand supV exist.

LetU be an L-ordered set. For a∈L andu∈U we set a→u=inf{^a/u} and a⊗u=sup{^a/u}.a→uis called the a-shift of u,a⊗uis calledthe a-multiple of u.

Uis calledshift complete(resp.multiple complete) ifa→u(resp.a⊗u) exists for any a∈L,u∈U.

The notions of shift and multiple coincide with the notions ofcotensorandtensor [10, 15, 18] and shift-complete resp. multiple-completeL-ordered sets are cotensored resp. tensoredΩ-categories from [15, 18]. In this paper, we use basic properties of shifts and multiples, which can be found in these papers and also in [12]. We summarize some of them below.

For anyu,v∈Uanda∈Lit holdsv=a→uiff for eachw∈U,

(wv) =a→(wu). (5)

Similarly,v=a⊗uiff for eachw∈U,

(vw) =a→(uw). (6)

Shifts and multiples satisfy the following adjointness condition: Ifa→vanda⊗u both exist then(u(a→v)) = ((a⊗u)v).

The following theorem has been proved in [18] (Propositions 3.12, 3.13) and shows how shifts and multiples can be efficiently used for proving that anL-ordered set is completely latticeL-ordered.

Theorem 1. Vis a completely latticeL-ordered set iffVis shift complete and multiple complete andhV,¹ iis a complete lattice.

2.3 Isotone L-Galois connections

We introduce the notion of isotoneL-Galois connections. For details, see [8].

AnisotoneL-Galois connectionbetweenL-ordered setsUandVis a pairhf,gi, where f:U→V,g:V→Uare mappings such that for eachu∈U,v∈V it holds

(f(u)v) = (ug(v)). (7)

A pair hu,vi, where u∈U andv∈V, is called a fixpoint of hf,gi if f(u) =vand g(v) =u. More generally,the degree to whichhu,viis a fixpoint ofhf,giis defined by

Fix_hf,gi(hu,vi) = (f(u)≈v)∧(g(v)≈u). (8) Fix_hf,giis anL-set inU×Vand is calledtheL-set of fixpoints ofhf,gi.

Theorem 2 (basic properties of isotone L-Galois connections).Lethf,gibe an iso- toneL-Galois connection betweenL-ordered setsUandV. Then

(a) u≤g(f(u))for each u∈U , f(g(v))≤v for each v∈V .

(b) f and g are increasing:(u₁u₂)≤(f(u₁)f(u₂))and(v₁v₂)≤(g(v₁) g(v₂)).

(c) f(g(f(u))) = f(u), g(f(g(v))) =g(v).

(d) If U⁰∈L^Uthen f(infU⁰)≤inff(U⁰)and f(supU⁰)≥supf(U⁰). If V⁰∈L^Vthen g(infV⁰)≤infg(V⁰)and g(supV⁰)≥supg(V⁰).

(e) IfUandVare completely latticeL-ordered sets and K∈L^U×V, then

S(K,Fix_hf,gi)≤Fix_hf,gi(hinfprU(K),f(g(infprV(K)))i), (9) S(K,Fix_hf,gi)≤Fix_hf,gi(hg(f(suppr_U(K))),suppr_V(K)i), (10) where pr_U:U×V→U and pr_V:U×V →V are Cartesian projections.

Proof. Omitted. Some of the properties are proved in [8].

For anL-ordered setU, an isotoneL-Galois connectionhf,gibetweenUandUis calledan inflationary Galois connection onUif f is deflationary andginflationary:

f(u)≤u and g(u)≥u (11)

for eachu∈U. Notice that if one of the conditions (11) holds true, then the second one follows from (7).

2.4 Composition Operators and Concept Lattices Associated toI

We use three relational composition operators,◦,/, and.. The composition operators are defined by

(A◦B)(x,y) =^W_z∈ZA(x,z)⊗B(z,y), (12) (A/B)(x,y) =^V_z∈ZA(x,z)→B(z,y), (13) (A.B)(x,y) =^V_z∈ZB(z,y)→A(x,z). (14) Note that these operators were extensively studied by Bandler and Kohout, see e.g.

[11] to which we refer for an overview of knowledge processing applications. One may easily see that.can be defined in terms of/and vice versa. They have natural verbal descriptions. For instance,(A◦B)(x,y)is the truth degree of the proposition “there is a factorzsuch thatzapplies to the objectxand the attributeyis a manifestation ofz”;

(A/B)(x,y)is the truth degree of “for every factorz, ifzapplies to the objectxthen the attributeyis a manifestation ofz”. Note also that forL={0,1},A◦Bcoincides with the well-known composition of binary relations.

AformalL-context(or just context) is a triplehX,Y,IiwhereXandYare sets whose elements are called objects and attributes, respectively andI is anL-relation between X andY. Consider the following pairs of operators betweenL^X andL^Y induced by an L-relationI∈L^X×Y:

A^↑(y) =^V_x∈XA(x)→I(x,y), B^↓(x) =^V_y∈YB(y)→I(x,y), (15) A^∩(y) =^W_x∈XA(x)⊗I(x,y), B^∪(x) =^V_y∈YI(x,y)→B(y), (16) A^∧(y) =^V_x∈XI(x,y)→A(x), B^∨(x) =^W_y∈YB(y)⊗I(x,y), (17) for A∈L^X,B∈L^Y. We call (15) antitone concept-forming operators and (16), (17) isotone concept-forming operators (these are less common if FCA). Furthemore, denote the set of fixpoints ofh^↑,^↓ibyB(X^↑,Y^↓,I). We have

B(X^↑,Y^↓,I) ={hA,Bi |A^↑=B,B^↓=A},

B(X^↑,Y^↓,I)is a completely latticeL-ordered set with theL-equality≈andL-order defined by

hA₁,B1i ≈ hA₂,B2i= (A₁≈^XA2) (=B1≈^YB2), (18) hA₁,B₁i hA₂,B₂i=S(A₁,A₂) (=S(B₂,B₁)), (19)

and is called the L-concept lattice associated to I. Its elements are calledformalL- concepts.L-concept lattices are the fundamental structures of formal concept analysis in fuzzy setting [7, 3]. For a formalL-concepthA,Bi,AandBare called theextentand theintentand they represent the collection of objects and attributes to which the formal concept applies. The sets of all extents and intents of the respective concept lattices are denoted by Ext(X^↑,Y^↓,I)and Int(X^↑,Y^↓,I), respectively. It may be shown that

Ext(X^↑,Y^↓,I) ={A∈L^X|A=A^↑↓}, Int(X^↑,Y^↓,I) ={B∈L^Y|B=B^↓↑}.

The above-defined operators and their sets of fixpoints have been extensively studied, see e.g. [1, 3, 8, 14].

Example 1. LetLbe the 6-element Łukasiewicz chain (i.e.,L={0,0.2,0.4,0.6,0.8,1}) andhX,Y,Iibe a formalL-context, whereX={BV,LH,MD,TSS},Y={c1,c2,c3,c4, c5}, andIis depicted in Fig. 1. Elements ofXare four selected movies by director David Lynch, elements ofY are five film critics and values ofIare ratings the critics assigned to the movies, taken fromwww.metacritic.comand rescaled to the six-element scale.

The context is our central example in this paper and we describe it deeper and work with it in Section 5. For now, we use it just to give an example of anL-concept lattice.

Note that our data are suitable for interpreting by means of fuzzy logic. A movie rating (usually given by number of “stars” or by a percentage) can be interpreted as an answer to the question: “Do you like this movie?”, given in degrees. In most cases, it is not possible to answer the above question with simple “Yes” or “No”, and, in the same time, there are no doubts about the answer (especially, if given by a film critic).

c1 c2 c3 c4 c5

BV 0.4 0.4 0.2 0.4 0.6

LH 0.8 0.8 0.8 1 1

MD 0.8 0.4 1 0.8 1

TSS 0.4 0.4 0.8 0.6 0.4

Fig. 1. Formal context of movies (“Blue Velvet” (BV), “Lost Highway” (LH), ”Mulhol- land Drive” (MD), and “The Straight Story” (TSS)), reviewers (David Sterritt (c1), Desson Thomson (c2), Jonathan Rosenbaum (c3), Owen Gleiberman (c4) and Roger Ebert (c5)) and their ratings on the 6-element Łukasievicz chainL={0,0.2,0.4,0.6,0.8,1}. Data taken from www.metacritic.comon March 20, 2011.

TheL-concept latticeB(X^↑,Y^↓,I)is depicted in Fig. 2. When displayingL-concept lattices, we use labeled Hasse diagrams to include all the information on extents and intents. For anyx∈X,y∈Yand formalL-concepthA,Biwe haveA(x)≥aandB(y)≥b if and only if there is a formal concepthA₁,B₁i ≤ hA,Bi, labeled by^a/xand a formal concepthA₂,B₂i ≥ hA,Bi, labeled by^b/y. We use labelsxresp.yinstead of¹/xresp.

1/yand omit redundant labels (i.e., if a concept has both the labels ^a/xand^b/xthen we keep only that with the greater degree; dually for attributes). The whole structure ofL-ordered set onB(X^↑,Y^↓,I)can be determined from the labeled diagram using the results from [1] (see also [3]).

0.6/TSS

0.4/c1,^0.4/c2,^0.2/c3,^0.4/c4,^0.4/c5

BV,^0.6/c5 ^0.4/c3,^0.6/c4

0.6/c1 ^0.6/c3

0.6/c2 ^0.8/c5 ^0.8/c4 TSS,^0.8/c3

0.8/BV ^0.8/c1

c3

0.8/c2 c4 ^0.8/TSS

c5

0.6/BV

LH c1 MD

0.8/MD

0.6/MD 0.4/BV,c2

0.2/BV,^0.8/LH,^0.4/MD,^0.4/TSS,

Fig. 2. L-concept lattice of movies, critics and ratings (from Fig. 1).

3 Complete L-tolerances

In this section we develop a generalization of the known results on factorization of complete lattices by complete tolerances [6, 17, 7] to fuzzy setting. We give a definition of a completeL-tolerance on a completely latticeL-ordered set, introduce a structure of anL-ordered set on the corresponding factor set and show that together with this structure, the factor set is a completely latticeL-ordered set (Theorem 7). In addition, we show that completeL-tolerances on a completely latticeL-ordered set are in one- to-one correspondence with inflationaryL-Galois connections (Theorem 8).

For anL-tolerance∼on a setX, anL-setB∈L^Xis called ablock of theL-tolerance

∼if for eachx₁,x₂∈Xit holdsB(x₁)⊗B(x₂)≤(x₁∼x₂). A blockBis calledmaximal

if for each blockB⁰fromB⊆B⁰it followsB=B⁰. The set of all maximal blocks of∼ always exists by Zorn’s lemma, is calledthe factor set of X by∼and denoted byX/∼.

Further we set for eachx∈X,JxK∼(y) =x∼y, obtaining anL-setJxK∼calledthe class of∼determined by x.

AnL-tolerance∼on a completely latticeL-ordered setU=hhU,≈i,iis called completeif it is compatible with≈and for eachR⊆ ∼it holds

sup(pr₁(R))∼sup(pr₂(R)) =inf(pr₁(R))∼inf(pr₂(R)) =1, (20) wherepr₁(resp.pr₂) is the projectionU×U→Uto the first (resp. second) component.

Note that in the classical case the condition (20) becomes the well-known condition of completeness of∼[17, 7]: Any subsetR⊆ ∼is a set of pairs{hu_j,v_ji | j∈J}such that for each j∈Jit holdsu_j∼v_jand sup(pr₁(R)) =^W_j∈Ju_j, sup(pr₂(R)) =^W_j∈Jv_j and similarly for infima.

For the rest of this section we suppose∼is a completeL-tolerance on a completely latticeL-ordered setU=hhU,≈i,i.

Lemma 1. If u∼v=1then(a⊗u)∼(a⊗v) =1and(a→u)∼(a→v) =1.

Proof. Follows directly from the definition.

Lemma 2. If∼is a complete L-tolerance on a complete latticeL-ordered set U= hhU,≈i,ithen so is a→ ∼, for each a∈L.

Proof. Since∼is compatible with≈then foru,u⁰,v,v⁰∈Uwe have(u≈u⁰)⊗(a→ (u⁰∼v⁰))⊗(v⁰≈v)≤a→((u≈u⁰)⊗(u⁰∼v⁰)⊗(v⁰≈v))≤a→(u∼v),proving a→ ∼is compatible with≈.

LetR⊆a→ ∼. We have for eachu,v∈U,(u R v)≤a→(u∼v), which is equivalent toa⊗(u R v)≤(u∼v). Thus,a⊗R⊆ ∼and we can use (20). We have sup(pr₁(a⊗ R)) =sup(a⊗pr₁(R)) =a⊗sup(pr₁(R)), inf(pr₁(a⊗R)) =inf(a⊗pr₁(R)) =a→ inf(pr₁(R))and similarly forpr₂. Now the result follows from Lemma 1.

Theorem 3. If∼is a completeL-tolerance onU, then for each a∈L the a-cut^a∼is a complete tolerance on the complete latticehU,¹i.

Proof. Fora=1 the assertion follows easily from the definition of completeL-tolerances.

The rest follows from Lemma 2.

We set for anyu∈U,

u^∼=supJuK^∼, u∼=infJuK^∼. (21) Theorem 4. The pairh∼,^∼iis an inflationary isotoneL-Galois connection onU.

Proof. We show that foru,v∈Uit holds(v∼u)≤(vu^∼), the proof of the opposite inequality is similar. Denote(v_∼u) =a. SinceU{^a/v_∼}(u) =1 (by the definition of upper cone), then we havea⊗v∼≤u. Now,(a⊗v)∼(a⊗v∼) =1 (Lemma 1) andu∼(u∨(a⊗v)) =1. Thus,u∨(a⊗v)≤u^∼ which means that(a⊗v)≤u^∼or vu^∼≥a. The fact thath_∼,^∼iis inflationary is trivial.

Lemma 3. Let u,v∈U be such that v≤u, u∼v=1. If w1≤u, then(vw1)≤(u∼ w1)and if v≤w2, then(w₂u)≤(w₂∼v).

Proof. By completeness of∼,(u∨w₁)∼(v∨w₁) =1. Thus,

(vw₁) = (v∨w₁)≈w₁= (u∼(v∨w₁))⊗((v∨w₁)≈w₁)≤(u∼w₁).

Similarly forw₂.

Theorem 5. For each u∈U the classJuK^∼is equal to the interval[u∼,u^∼]inU.

Proof. The inclusionJuK∼⊆[u∼,u^∼]follows from basic properties of cones (for ex- ample,JuK∼⊆L UJuK∼=L{u^∼}).

We prove the opposite inclusion. Forw∈Udenoteu∼w=aandwu^∼=b. We have[u_∼,u^∼](w) =a∧b. By Lemma 3,(u_∼(w∧u))≤((w∧u)∼u)and((w∨u) u^∼)≤((w∨u)∼u). Butu∼(w∧u) =a and(w∨u)u^∼=b which means that [u_∼,u^∼](w) =a∧b≤((w∧u)∼u)∧((w∨u)∼u). Using Theorem 3 we easily obtain that the right-hand side is less than or equal tow∼u.

Theorem 6. Maximal blocks of ∼are exactly L-intervals [v,u] where hu,vi are fix- points ofh∼,^∼i.

Proof. Letw₁,w₂∈U. We have

[v,u](w₁)⊗[v,u](w₂) = ((vw₁)∧(w₁u))⊗((vw₂)∧(w₂u))

≤((vw1)⊗(w2u))∧((w₁u)⊗(vw2))

= ((vw₁)⊗(w_2∼v))∧((w₁u)⊗(uw^∼₂))

≤(w_2∼w1)∧(w₁w^∼₂) =Jw2K∼(w₁) = (w₁∼w2), showing[v,u]is a block.

Conversely, ifW∈L^Uis a block then so is the interval[infW,supW]which contains W as a subset. Among all intervals[v,u], the maximal blocks are those satisfyingv^∼=u andu∼=v.

We define two binaryL-relations≈andonU/∼by

[v₁,u1]≈[v₂,u2] =v1≈v2 (=u1≈u2), (22) [v₁,u₁][v₂,u₂] =v₁v₂ (=u₁u₂) (23) (the equalities in the parentheses follow directly from (4), Theorem 4, and Theorem 2 (b))

Theorem 7. The tuplehhU/∼,≈i,iis a completely latticeL-ordered set.

Proof. Sketch: we use Theorem 1, (5), (6).

Lethf,gibe an inflationary Galois connection onU. Set for eachu,v∈U

u∼_hf,giv= (f(u)v)∧(vg(u)). (24)

Theorem 8. ∼_hf,giis anL-complete tolerance, satisfying u_∼h

f,gi=f(u), u^∼hf,gi=g(u), for each u∈U . The assignment hf,gi 7→ ∼_hf,gi is a bijection between the set of all inflationary isotoneL-Galois connectionsUand the set of allL-complete tolerances onU.

Proof. Sketch: We use basic properties of isotoneL-Galois connections (Theorem 2) and the definition of anL-complete tolerance onU.

4 Block L-relations

This section introduces blockL-relations onL-formal contexts, their properties and the relationship to completeL-tolerances onL-concept lattices. The main results are con- tained in Theorems 11 and 12, where we show that block L-relations of any formal L-context are in one-to-one correspondence with completeL-tolerances on the associ- atedL-concept lattice and that theL-concept lattice of each of the block relations is isomorphic to the originalL-concept lattice, factorized by the corresponding complete L-tolerance.

We define block relations as follows:BlockL-relationofI∈L^X×Y is anL-relation J⊇Isuch that Ext(X^↑,Y^↓,J)⊆Ext(X^↑,Y^↓,I)and Int(X^↑,Y^↓,J)⊆Int(X^↑,Y^↓,I).

In crisp setting, [17] defines block relation as a relationJ⊇Iwhere each row is an intent ofI and each column is an extent ofI. Lemma 4 says that blockL-relation is a proper generalization of crisp block relation. The reason we define the notion that way is to allow an analogous definition in the isotone case.

Lemma 4. J∈L^X×Y, J⊇I is a blockL-relation of I iff{x}^↑I∈Int(X^↑,Y^↓,I)for each x∈X and{y}^↓I∈Ext(X^↑,Y^↓,I)for each y∈Y .

The following theorem provide characterization of blockL-relations.

Theorem 9. Let I∈L^X×Y be anL-relation. The following statements are equivalent:

(a) J is a block relation of I.

(b) J=I.S_i with S_i∈L^Y×Y and for the induced mapping ^∧Si we have B^∧Si ∈ Int(X^↑,Y^↓,I)and B⊆B^∧Si for each B∈Int(X^↑,Y^↓,I).

(c) J=S_e/I with S_e∈L^X×X and for the induced mapping ^∪Se we have A^∪Se ∈ Ext(X^↑,Y^↓,I)and A⊆A^∪Se for each A∈Ext(X^↑,Y^↓,I).

Proof. (sketch, equivalence of (a) and (b)): We have Ext(X^↑,Y^↓,J)⊆Ext(X^↑,Y^↓,I) iff there exists a matrix Si such that J=I.Si by [4, Theorem 7]. We have A^↑J = A^↑I.Si = (A^↑I)^∧Si forA∈L^X. SinceA^↑Iis any intentBin Int(X^↑,Y^↓,I)andA^↑J=B^∧Si ∈ Int(X^↑,Y^↓,I)we obtainB^∧Si ∈Int(X^↑,Y^↓,I)for each B∈Int(X^↑,Y^↓,I). By (16) and (14) we have thatJ(x,y)is equal to{x}^↑I∧Si(y)and with the propertyB⊆B^∧Si we get I⊆Jproving thatJis a blockL-relation.

Theorem 10. Let I∈L^X×Y be anL-relation between X and Y .

(a) The set of all blockL-relations J of I is anL-closure system (i.e. it is closed under^Vand→).

(b) The set of all Se(from Theorem 9) is anL-interior system (i.e. it is closed under Wand⊗).

(c) The set of all S_i(from Theorem 9) is anL-interior system.

Proof. (a) LetJ_ibe blockL-relations ofI. LetJ=^V_iJ_iand letB∈Int(X^↑,Y^↓,J), hence B=A^↑J for someA∈L^X. By definition of^↑J and properties of residuated lattices we have

A^↑J(y) =^{^}

x∈X

A(x)→J(x,y) =^{^}

x∈X

A(x)→^{^}

i

J_i(x,y) =^{^}

i

^

x∈X

A(x)→J_i(x,y) =^{^}

i

A^↑Ji.

Thus we haveA^↑J=^T_iA^↑Ji ∈Int(X^↑,Y^↓,I). Similarly,B^↓J =^T_iB^↓Ji ∈Ext(X^↑,Y^↓,I), A^↑a→Ji =a→A^↑Ji ∈Int(X^↑,Y^↓,I), andB^↓a→Ji =^T_ia→B^↓Ji ∈Ext(X^↑,Y^↓,I).

Finally, fromI⊆J_iwe haveI⊆^V_iJ_iandI⊆a→J_i. Whence^V_iJ_iandI⊆a→J_i are blockL-relations proving that set of all blockL-relations J of I is an L-closure system.

(b) and (c) can be proved similarly.

The induced operators^∩,^∪of the matricesS_eandS_ihave following properties:

Lemma 5. Let I∈L^X×Y be anL-relation between X and Y and let J be its blockL- relation:

(a) A^∪Se=A^↑I↓Jfor any A∈Ext(X^↑,Y^↓,I), B^∧Si =B^↓I↑J for any B∈Int(X^↑,Y^↓,I).

(b)hA^∪Se,A^{∪Se∩Se↑I}i ∈B(X^↑,Y^↓,J)and hB^{∧Si∨Si↓I},B^∧Sii ∈B(X^↑,Y^↓,J)for each hA,Bi ∈B(X^↑,Y^↓,I).

(c) A^∩Se↑I=A^↑I∧Si =A^↑J for any A∈L^Xand B^∨Si↓I=B^↓I∪Se=B^↓Jfor any B∈L^Y. LetI∈L^X×Y be anL-relation between X andY and let J be its block relation.

Denote by^θJ a mapping^θJ :B(X^↑,Y^↓,I)→B(X^↑,Y^↓,I)defined by

hA,Bi^θJ =hA^∪Se,A^∪Se↑Ii (25) and denote by_θJ a mapping_θJ :B(X^↑,Y^↓,I)→B(X^↑,Y^↓,I)defined by

hA,Bi_θJ=hB^∧Si↓I,B^∧Sii (26) Notice, that different blockL-relationsJof anL-relationIinduce different mappings

θ_J,_θJ. In what follows we omit subscript in the notation ^θJ, _θJ and write just ^θ,_θ. Obviously, for eachhA,Bi ∈B(X^↑,Y^↓,I)we havehA,Bi_θ≤ hA,Bi ≤ hA,Bi^θ.

Now, we explain the relationship between block L-relations and compatible L- tolerances.

Theorem 11. (a) Let J⊇I be a block L-relation. Then the mappings^θ,_θ form an inflationaryL-Galois connection onB(X^↑,Y^↓,I).

(b) For any inflationaryL-Galois connectionhf,gionB(X^↑,Y^↓,I)theL-relation J∈L^X×Y given by

J(x,y) =f(h{x}^↑I↓I,{x}^↑Ii) h{y}^↓I,{y}^↓I↑Ii (27) ( =h{x}^↑I↓I,{x}^↑Ii g(h{y}^↓I,{y}^↓I↑Ii) )

is a blockL-relation ofhX,Y,Iisuch that its mappings^θ and_θ are equal to the map- pings f and g, respectively.

Proof. (a) SinceJis a blockL-relation thenA^↑J is an intent ofhX,Y,Iiandf(hA,Bi)∈ B(X^↑,Y^↓,I). Similarly forg.

The pairh^↑J,^↓Jiis an antitoneL-Galois connection betweenL^X andL^Y. Thus, for eachA∈L^X andB∈L^Y we haveS(A,B^↓J) =S(B,A^↑J). Now,

f(hA₁,B1i) hA₂,B2i=S(B₂,A^↑₁^J) =S(A₁,B^↓₂^J) =hA₁,B₁i g(hA₂,B2i) provinghf,giis an isotoneL-Galois connection. SinceI⊆JthenA^↑I⊆A^↑J and f is deflationary.

(b) First we need to show that each object intent ofhX,Y,Jiis an intent ofhX,Y,Ii and each attribute extent ofhX,Y,Jiis an extent ofhX,Y,Ii. We shall prove the part for extents. For eachx∈X we have

{x}^↑J(y) =J(x,y) =S(f_X({x}^↑I↓I),{y}^↓I) = ^{^}

x⁰∈X

f_X({x}^↑I↓I)(x⁰)→ {y}^↓I(x⁰)

= ^{^}

x⁰∈X

f_X({x}^↑I↓I)(x⁰)→I(x⁰,y) =f_X({x}^↑I↓I)^↑I(y)

and{x}^↑J is an intent ofhX,Y,Ii. That proves that Ext(X^↑,Y^↓,J)⊆Ext(X^↑,Y^↓,I); sim- ilarly can be shown that Int(X^↑,Y^↓,J)⊆Int(X^↑,Y^↓,J). From properties of inflation- aryL-Galois connections and antitone Galois connections, we have f_X({x}^↑I↓I)^↑I(y)≥ {x}^↑I↓I↑I(y) ={x}^↑I(y); henceJ⊇I. The equality of^θ,_θ andf,gis immediate.

Theorem 11 and Theorem 8 generalize [7, Theorem 15]. The following theorem represents the main result of this paper.

Theorem 12. (a) CompleteL-tolerances are in one-to-one correspondence with block L-relations.

(b) If completeL-tolerance∼and blockL-relation J are in that correspondence, thenB(X^↑,Y^↓,J)is isomorphic to the lattice of blocks ofL-tolerance∼.

Proof. The one-to-one correspondence follows from Theorem 11 and the fact that dif- ferent block relation produce different mappings^θ,_θ. The isomorphism ofB(X^↑,Y^↓,J) and the lattice of∼then follows from definition of^θ,_θand their equality to mappings

f,gof∼.

5 Illustrative Example

In this section, we use results developed in the previous sections to factorize the L- concept latticeB(X^↑,Y^↓,I)from Example 1.

Our aim is to reduce the size of theL-concept lattice B(X^↑,Y^↓,I)by factoriza- tion by completeL-tolerances. First we take the approach from [2]. This approach is based on a choice of a thresholda∈Land using thea-cut^a≈of theL-equality≈on B(X^↑,Y^↓,I)for factorization. Thea-cut ^a≈is a complete tolerance on the complete latticehB(X^↑,Y^↓,I),¹i and, as noted in [5], the factor lattice is isomorphic to the crisp part of the concept latticeB(X^↑,Y^↓,a→I). As it can be easily seen, these re- sults are a special case of the results of this paper. Namely, one can take the complete

L-tolerancea→ ≈onB(X^↑,Y^↓,I)(the fact that it is indeed a completeL-tolerance eas- ily follows from Lemma 2) and construct the associated blockL-relationJ⊇I, which is equal toa→I. The factorizedL-concept latticeB(X^↑,Y^↓,I)/^a≈is isomorphic to B(X^↑,Y^↓,a→I).

As an example, we present in Fig. 3 results we obtained for our formal context of movies, critics and ratings with values of the thresholdaequal to 0.4 and 0.3.

0.8/TSS 0.6/c1,^0.6/c2,^0.4/c3,^0.6/c4,^0.6/c5

BV,^0.8/c5 ^0.6/c3,^0.8/c4

0.8/c1 ^0.8/c3

0.8/c2 TSS,c3

0.8/BV c1

c5 c4

MD

0.6/BV,c2 ^0.6/MD

0.4/BV,LH,^0.6/MD,^0.6/TSS

0.8/c1,^0.8/c2,^0.6/c3,^0.8/c4,^0.8/c5

BV,^0.8/c5 ^0.8/c3,c4

c1 TSS,c3

0.8/BV,c2 MD

0.6/BV,LH,^0.8/MD,^0.8/TSS

Fig. 3. L-concept lattice of movies, critics and ratings (from Fig. 1), factorized with respect to the thresholda=0.8 (left) anda=0.6 (right).

We also tried a more sophisticated approach, based on [13]. The author’s method is based on using a complete tolerance on a complete lattice induced by an interior and closure operator. As an example, the author uses a complete tolerance on a (crisp) concept lattice, obtained by selecting subsetsX⁰⊆X andY⁰⊆Y of important objects and important attributes, respectively.

Although a proper fuzzification of the results from [13] remains to be developed, it is possible to try some experiments. We provide a short outline of our method here and leave the details to a forthcoming paper.

For a formalL-context hX,Y,Ii we select two L-sets X⁰∈L^X andY⁰∈L^Y and interpret them asL-sets of “important objects” and “important attributes”, respectively.

Thus, for an objectx∈X, the valueX⁰(x)is the degree to whichxis important, and similarly for attributes. Further we define twoL-relationsI_X⁰,I_Y⁰∈L^X×Y by

I_X⁰(x,y) =X⁰(x)→I(x,y), I_Y⁰(x,y) =Y⁰(y)→I(x,y).

As it can be easily seen from (15), Int(X^↑,Y^↓,I_X⁰)⊆Int(X^↑,Y^↓,I)and Ext(X^↑,Y^↓,I_Y⁰)⊆ Ext(X^↑,Y^↓,I). Thus, theL-relationsI_X⁰andI_Y⁰select some intents and extents of formal

0.4/c1,^0.4/c2,^0.2/c3,^0.4/c4,^0.4/c5

BV,^0.6/c5 ^0.4/c3,^0.6/c4

0.6/c1,^0.6/c2 ^0.6/c3

0.8/BV ^0.8/c4 TSS,^0.8/c3

0.8/c2

0.6 c4

/BV,c5 ^0.8/TSS,c3

0.6/TSS LH,c1

0.4/BV,c2 MD

0.2/BV,^0.8/LH,^0.8/MD,^0.4/TSS

0.6/TSS c1,c5

0.4/c1,^0.4/c2,^0.4/c3,^0.6/c4,^0.4/c5

BV,^0.6/c1,^0.6/c5

0.6/c2,^0.6/c4 TSS

0.8/TSS,c3 0.8/BV,^0.8/c1,^0.8/c5

0.8/c1

0.8/c5 c4

0.6/BV MD

LH,^0.8/MD

0.4/BV,^0.8/LH,^0.6/MD,^0.4/TSS,c2

Fig. 4. L-concept lattice of movies, critics and ratings (form Fig. 1), factorized with respect to an L-set of important objects X⁰ ={^0.8/BV,LH,MD,TSS} and L-set of important at- tributes Y⁰={c1,^0.8/c2,c3,c4,c5} (left) and with respect to an L-set of important objects X⁰={BV,LH,MD,TSS}andL-set of important attributesY⁰={c1,^0.6/c2,c3,c4,c5}(right).

concepts fromB(X^↑,Y^↓,I). These intents and extents are interpreted as “important”.

Now, letJ_X⁰_,Y⁰∈L^X×Ybe the minimal (with respect toL-set inclusion) blockL-relation ofIsuch that both intent and extent of each formalL-concept fromB(X^↑,Y^↓,J_X⁰_,Y⁰)are important (J_X0,Y⁰always exists because of Theorem 10(a); details are omitted). Using the results of this paper (Theorem 12) we obtain thatJ_X⁰_,Y⁰ induces a completeL-tolerance on theL-concept latticeB(X^↑,Y^↓,I)and the corresponding factor completely lattice L-ordered set is isomorphic to theL-concept latticeB(X^↑,Y^↓,J_X⁰_,Y⁰).

Note that this approach contains the approach from [2] as a special case. The factor completely latticeL-ordered setB(X^↑,Y^↓,I)/^a≈is isomorphic toB(X^↑,Y^↓,J_X⁰_,Y⁰)for X⁰={^a/x|x∈X}andY⁰={^a/y|y∈Y}.

We apply the above considerations to our example. Suppose we consider the film BV less important than the other films (perhaps because we have not seen BV) and the critic c2 less important than the other critics (because we do not like his opinion on MD). More precisely, setX⁰={^a/BV,LH,MD,TSS} andY⁰={c1,^b/c2,c3,c4,c5}, wherea,b∈L. In Fig. 4 we can see the resulting concept lattices in two cases: first a=b=0.8 and seconda=1 andb=0.6.

6 Conclusions

We presented a generalization of the theory of complete tolerances on complete lattices and the relationship of block relations on formal contexts and complete tolerances on the corresponding concept lattices [6, 17, 7] to fuzzy setting. Our results can be used

for reducing the size of fuzzy concept lattices by means of factorization and offer a greater degree of variability than the known approach from [2]. In the future we will focus namely on the investigation of block relations in the isotone case and develop- ing the theory of approximations in fuzzy concept lattices which would give a proper theoretical background to our experiments from Sec. 5.

References

1. Belohlavek, R.: Concept lattices and order in fuzzy logic. Ann. Pure Appl. Log. 128(1-3), 277–298 (2004)

2. Belohlavek, R.: Similarity relations in concept lattices. J. Log. Comput. 10(6), 823–845 (2000)

3. Belohlavek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publishers, Norwell, USA (2002)

4. Belohlavek, R., Konecny, J.: Operators and spaces associated to matrices with grades and their decompositions, to appear, submitted to Fundamenta Informaticae

5. Belohlavek, R., Outrata, J., Vychodil, V.: Direct factorization by similarity of fuzzy concept lattices by factorization of input data. In: Yahia, S.B., Nguifo, E.M., Belohlavek, R. (eds.) CLA. Lecture Notes in Computer Science, vol. 4923, pp. 68–79. Springer (2006)

6. Cz´edli, G.: Factor lattices by tolerances. Acta Sci. Math. 44, 35–42 (1982)

7. Ganter, B., Wille, R.: Formal Concept Analysis – Mathematical Foundations. Springer (1999)

8. Georgescu, G., Popescu, A.: Non-dual fuzzy connections. Arch. Math. Log. 43(8), 1009–

1039 (2004)

9. H´ajek, P.: Metamathematics of Fuzzy Logic (Trends in Logic). Springer (November 2001) 10. Kelly, G.M.: Basic Concepts of Enriched Category Theory. Cambridge University Press

(1982)

11. Kohout, L. J.; Bandler, W.: Relational-product architectures for information processing. In- formation Sciences 37(1-3), 25–37 (1985)

12. Krupka, M.: An alternative version of the main theorem of fuzzy concept lattices. In: Trappl, R. (ed.) Cybernetics and Systems 2010. pp. 9–14. Austrian Society for Cybernetic Studies, Vienna (2010), extended version submitted

13. Meschke, C.: Approximations in concept lattices. In: Kwuida, L., Sertkaya, B. (eds.) Formal Concept Analysis, Lecture Notes in Computer Science, vol. 5986, pp. 104–123. Springer Berlin / Heidelberg (2010)

14. Pollandt, S.: Fuzzy Begriffe: Formale Begriffsanalyse von unscharfen Daten. Springer–

Verlag, Berlin–Heidelberg (1997)

15. Stubbe, I.: Categorical structures enriched in a quantaloid: tensored and coten- sored categories. Theory Appl. Categ. 16, No. 14, 283–306 (electronic) (2006), http://www.ams.org/mathscinet-getitem?mr=2223039

16. Ward, M., Dilworth, R.P.: Residuated lattices. Transactions of the American Mathematical Society 45, 335–354 (1939)

17. Wille, R.: Complete tolerance relations of concept lattices. In: Eigenthaler, G., et al. (eds.) Contributions to General Algebra, vol. 3, pp. 397–415. H¨older-Pichler-Tempsky, Wien (1985)

18. Zhao, H., Zhang, D.: Many valued lattices and their representations. Fuzzy Sets and Systems 159(1), 81–94 (Jan 2008)