On Generalization of Fuzzy Concept Lattices Based on Change of Underlying Fuzzy Order

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On Generalization of Fuzzy Concept Lattices Based on Change of Underlying Fuzzy Order

Pavel Martinek

Department of Computer Science, Palacky University, Olomouc Tomkova 40, CZ-779 00 Olomouc, Czech Republic

pavel.martinek@upol.cz

Abstract. The paper presents a generalization of the main theorem of fuzzy concept lattices. The theorem is investigated from the point of view of fuzzy logic. There are various fuzzy order types which differ by incorporated relation of antisymmetry. This paper focuses on fuzzy order which uses fuzzy antisymmetry defined by means of multiplication operation and fuzzy equality.

Keywords.Fuzzy order, fuzzy concept lattice 1 Introduction

A notion of fuzzy order has been derived from the classical one by fuzzification the three underlying relations. This led to various versions, at the beginning versions utilizing the classical relation of equality (see e.g. [9]); later versions are more general by introducing fuzzy similarity (or fuzzy equality) instead of the classical equality (see e.g. [4]). Fuzzy similarity is based on idea that relationship between objectsAandBshould be transformed to a similar relationship between objectsA⁰andB⁰ wheneverA⁰,B⁰ are similar toA,B, respectively.

In [3], one of the later definitions of fuzzy order was used to formulate and prove a fuzzy logic extension of the main theorem of concept lattices. The aim of this paper is to enlarge validity of the theorem to more general fuzzy order.

2 Preliminaries

First, we recall some basic notions. It is known that in fuzzy logic an important structure of truth values is represented by a complete residuated lattice (see e.g.

[5], [6], [7]).

Definition 1. A residuated lattice is an algebra L =hL,∧,∨,⊗,→,0,1i such that

– hL,∧,∨,0,1iis a lattice with the least element0and the greatest element1, – hL,⊗,1iis a commutative monoid,

c Radim Belohlavek, Sergei O. Kuznetsov (Eds.): CLA 2008, pp. 207–215, ISBN 978–80–244–2111–7, Palack´y University, Olomouc, 2008.

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– ⊗and →form so-called adjoint pair, i.e.a⊗b≤ciffa≤b→cholds for alla, b, c∈L.

Residuated latticeLis called complete if hL,∧,∨iis a complete lattice.

Throughout the paper, L will denote a complete residuated lattice. An L- set (or fuzzy set) A in a universe set X is any mapping A : X → L, A(x) being interpreted as the truth degree of the fact that “xbelongs toA”. ByL^X we denote the set of all L-sets inX. A binary L-relation is defined obviously.

Operations onLextend pointwise toL^X, e.g. (A∨B)(x) =A(x)∨B(x) for any A, B∈L^X. As is usual, we writeA∪Binstead ofA∨B, etc.

L-equality (or fuzzy equality) is a binary L-relation ≈ ∈ L^X×X such that (x≈x) = 1 (reflexivity), (x≈y) = (y≈x) (symmetry), (x≈y)⊗(y≈z)≤ (x≈ z) (transitivity), and (x ≈y) = 1 implies x= y. We say that a binary L-relation R ∈ L^X×Y is compatible with respect to ≈_X and ≈_Y if R(x, y)⊗ (x≈_X x⁰)⊗(y ≈_Y y⁰) ≤R(x⁰, y⁰) for any x, x⁰ ∈ X, y, y⁰ ∈ Y. Analogously anL-setA∈L^Xiscompatiblewith respect to≈_XifA(x)⊗(x≈_Xx⁰)≤A(x⁰) for anyx, x⁰∈X. GivenA, B∈L^X, in agreement with [5] we define the subsethood degree S(A, B) of AinBby S(A, B) = V

x∈XA(x) →B(x). For A∈ L^X and a∈L, the set^aA={x∈X;A(x)≥a}is called thea-cut ofA. Analogously, for R∈L^X×Y anda∈L, we denote^aR={(x, y)∈X×Y;R(x, y)≥a}. Forx∈X anda ∈L, {a/x} is theL-set defined by{a/x}(x) =a and{a/x}(y) = 0 for y6=x.

Definition 2. An L-order on a set X with an L-equality ≈ is a binary L-relation which is compatible with respect to ≈ and satisfies the following conditions for allx, y, z∈X:

(xx) = 1 (reflexivity), (xy)⊗(yx)≤(x≈y) (antisymmetry), (xy)⊗(yz)≤(xz) (transitivity).

(Cf. T-E-ordering from [4].) Since in residuated latticesx⊗y≤x∧yfor every x, y∈X, our relation is more general than L-order of [3] or [2] where antisymmetry is expressed by condition (xy)∧(yx)≤(x≈y).

Note that

(xy)⊗(yx)≤(x≈y)≤(xy)∧(yx). (1) Indeed, the first inequality represents antisymmetry and the second one follows from compatibility (see proof of Lemma 4 in [3]):

(x≈y) = (xx)⊗(x≈x)⊗(x≈y)≤(xy) and analogously, (x≈y)≤ (yx).

The inequalities (1) represent the fact thatL-equality must satisfy the interval confinement as follows:

(x≈y)∈[(xy)⊗(yx), (xy)∧(yx)].

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On the other hand, the definition of L-order by [3] (which must satisfy the condition (x_[3] y)∧(y_[3]x)≤(x≈y)) leads to firm binding ofL-equality with the upper bound of previous interval (see Lemma 4 of [3]), i.e.

(x≈y) = (x_[3]y)∧(y_[3]x).

Now, we can interpret the relationship betweenL-order andL-equality defined either in [3] and in this paper as follows. By the definition,L-order is dependent on a givenL-equality. However, if we change point of view and have a look to the inverse “dependence”, we can see that

– by [3], anL-equality is binded with any correspondingL-order firmly, – in our paper, an L-equality has certain freedom (with respect to a corre-

spondingL-order).

This will play an important role during generalizing results achieved in [3].

If is anL-order on a setX with anL-equality≈, we call the pair X= hhX,≈i,ianL-ordered set. In agreement with [3], we say thatL-ordered sets hhX,≈_Xi,_XiandhhY,≈_Yi,_Yiareisomorphicif there is a bijective mapping h:X→Y such that (x≈_X x⁰) = (h(x)≈_Y h(x⁰)) and (x_Xx⁰) =

(h(x)_Y h(x⁰)) hold for anyx, x⁰∈X.

Note that in case of firm binding ofL-equality andL-order by [3] (see the note above), preservation of theL-order by the bijectionhimplies also preservation of theL-equality. Clearly this is not true forL-order defined in this paper.

3 Some properties of fuzzy ordered sets

In this section, we describe some notions and properties related to fuzzy ordered sets which represent appropriate generalizations of notions and facts known from classical (partial) ordered sets. These generalizations were introduced mainly in [2] and [3] (the fact that originally they used less general definition ofL-order is unimportant).

Definition 3. For an L-ordered set hhX,≈i,i and A ∈ L^X we define the L-setsL(A) andU(A) inX by

L(A)(x) = ^

x⁰∈X

(A(x⁰)→(xx⁰)) for all x∈X,

U(A)(x) = ^

x⁰⁰∈X

(A(x⁰⁰)→(x⁰⁰x)) for all x∈X.

L(A)and U(A)are called the lower cone and upper cone ofA, respectively.

These L-sets can be described as the L-sets of elements which are smaller (greater) than all elements ofA. We will abbreviateU(L(A)) byU L(A),L(U(A)) byLU(A) etc.

Underlying Fuzzy Order

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Definition 4. For an L-ordered set hhX,≈i,i andA∈L^X we define the L-setsinf(A)andsup(A)inX by

(inf(A))(x) = (L(A))(x)∧(U L(A))(x) for all x∈X, (sup(A))(x) = (U(A))(x)∧(LU(A))(x) for all x∈X.

inf(A) andsup(A) are called the infimum and supremum ofA, respectively.

Lemma 1. LethhX,≈i,ibe anL-ordered set,A∈L^X. If(inf(A))(x) = 1and (inf(A))(y) = 1thenx=y(and similarly forsup(A)).

Proof. The proof is almost verbatim repetition of the proof of Lemma 9 in [3].

Lemma 2. For an L-ordered set hhX,≈i,i and A ∈ L^X, the L-sets inf(A) andsup(A)are compatible with respect to≈.

Proof. The proof can be found in [2], namely in more general proof of Lemma 5.39

with regard to Remark 5.40.

Definition 5. For a set X with an L-equality ≈, an L-set A ∈ L^X is called an S-singleton if it is compatible with respect to ≈ and there is some x₀ ∈X such thatA(x₀) = 1 andA(x)<1for x6=x₀.

Remark 1. There are various definitions of fuzzy singletons (see e.g. [8] or [2]).

Our definition represents the simplest one, that is why we call it S-singleton.

Demanding more conditions than stated would lead to serious troubles in proof of Theorem 2. Note that in case ofLequal to the Boolean algebra2of classical logic with the support{0,1},S-singletons represent classical one-element sets.

Lemma 3. For anL-ordered sethhX,≈i,i and A∈L^X, if (inf(A))(x₀) = 1 for some x₀∈X theninf(A) is anS-singleton. The same is true for suprema.

Proof. The assertion immediately follows from Lemmata 1 and 2.

Definition 6. An L-ordered set hhX,≈i,i is said to be completely lattice L-ordered if for anyA∈L^X bothinf(A) andsup(A) areS-singletons.

Theorem 1. For anL-ordered setX=hhX,≈i,i, the relation¹is an order on X. Moreover, if Xis completely lattice L-ordered then ¹is a lattice order onX.

Proof. The proof is analogous to the proof of Theorem 13 in [3].

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4 Fuzzy concept lattices

We remind some basic facts about concept lattices in fuzzy setting. A formal L-contextis a tripplehX, Y, IiwhereIis anL-relation between the setsXandY (with elements called objects and attributes, respectively). For anyL-context we can generalize notions introduced in Definition 3 as follows. LetX, Y be sets with L-equalities≈X,≈Y, respectively;I ∈L^X×Y be an L-relation compatible with respect to≈_X and≈_Y. For anyA∈L^X,B∈L^Y, we defineA^↑∈L^Y,B^↓∈L^X (see e.g. [1]) by

A^↑(y) = ^

x∈X

(A(x)→I(x, y)) for all y∈Y,

B^↓(x) = ^

y∈Y

(B(y)→I(x, y)) for all x∈X.

Clearly,A^↑(y) describes the truth degree, to which “for eachxfromA,xandy are inI”, and similarlyB^↓(x). We will abbreviate (A^↑)^↓ byA^↑↓, (B^↓)^↑byB^↓↑

etc. The equationA^↑=A^↑↓↑ holds true for allA∈L^X (see e.g. [1]). Note that if X = Y and I = is an L-order on X, then A^↑ coincides with U(A) and B^↓coincides withL(B). AnL-conceptin a givenL-contexthX, Y, Iiis any pair hA, BiofA∈L^X andB∈L^Y such that A^↑=BandB^↓=A(see [2]).

We denote by B(X, Y, I) the set of all L-concepts given by an L-context hX, Y, Ii, i.e.

B(X, Y, I) ={hA, Bi ∈L^X×L^Y;A^↑=B, B^↓=A}.

For anyB(X, Y, I), we put (see [3])

(hA1, B₁i ShA2, B₂i) = S(A1, A₂) for allhA1, B₁i,hA2, B₂i ∈ B(X, Y, I).

TheL-relation_Sobviously satisfies the conditions of reflexivity and transitivity. As to the antisymmetry, we need anL-equality. Therefore, consider an arbi- traryL-equality≈on the setB(X, Y, I) such that_Sis compatible with respect to≈and the inequality

(hA₁, B₁i _ShA₂, B₂i)⊗(hA₂, B₂i _ShA₁, B₁i)≤(hA₁, B₁i ≈ hA₂, B₂i) holds true for every hAi, B_ii ∈ B(X, Y, I), i∈ {1,2}. (Existence of such an L- equality is demonstrated e.g. by (hA₁, B₁i ≈ hA₂, B₂i) = S(A₁, A₂)∧S(A₂, A₁) in [3].) Consequently,_S is anL-order onhB(X, Y, I),≈iand we get anL-ordered set hhB(X, Y, I),≈i,_Si which will act in further two theorems. Note that the L-ordered set is more general thanL-concept latticehhB(X, Y, I),≈i,_Siof [3]

because of more generalL-equality.

The next theorem characterizingL-concept lattices needs further denotation.

As usual, for anL-setAinUanda∈L, we denote bya⊗AtheL-set such that Underlying Fuzzy Order

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(a⊗A)(u) =a⊗A(u) for all u∈U. If Mis anL-set inY and eachy∈Y is anL-set inX, we define theL-setS

MinX(see [3]) by ([

M)(x) = _

A∈Y

M(A)⊗A(x) for allx∈X.

Clearly,S

Mrepresents a generalization of a union of a system of sets. For an L-setMinB(X, Y, I), we putS

XM=S

pr_X(M),S

YM=S

pr_Y(M) where pr_X(M) is anL-set in the set{A∈ L^X;A=A^↑↓} of all extents ofB(X, Y, I) defined by (pr_XM)(A) = M(A, A^↑) and, similarly, pr_Y(M) is an L-set in the set {B ∈ L^Y;B =B^↓↑} of all intents of B(X, Y, I) defined by (pr_YM)(B) = M(B^↓, B). Hence, S

XM is the “union of all extents from M” andS

YM is the “union of all intents fromM” (see [3]).

Theorem 2. Let hX, Y, Iibe an L-context. AnL-ordered set

hhB(X, Y, I),≈i,_Si is completely lattice L-ordered set in which infima and suprema can be described as follows: for an L-setMinB(X, Y, I) we have

1inf(M) = (*

([

Y

M)^↓,([

Y

M)^↓↑

+)

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1sup(M) = (*

([

X

M)^↑↓,([

X

M)^↑ +)

(3) Proof. The proof of (2) and (3) is analogous to the proof of part (i) of Theorem 14 in [3] where differently defined antisymmetry is not used anywhere. Further- more by Lemma 3, eachL-ordered sethhB(X, Y, I),≈i,_Siis completely lattice

L-ordered.

For any completely lattice L-ordered set X=hhX,≈i,i, a subsetK ⊆X is called{0,1}-infimally dense({0,1}-supremally dense)inX(cf. [3]) if for each x∈Xthere is someK⁰⊆Ksuch thatx=V

K⁰(x=W

K⁰). HereV (W

) means infimum (supremum) with respect to the 1-cut of.

Theorem 3. Let hX, Y, Ii be an L-context. A completely latticeL-ordered set V=hhV,≈_Vi,iis isomorphic to anL-ordered sethhB(X, Y, I),≈i,_Siiff there are mappings γ:X×L→V, µ:Y ×L→V, such that

(i) γ(X×L) is{0,1}-supremally dense inV, (ii) µ(Y ×L) is{0,1}-infimally dense inV,

(iii) ((a⊗b)→I(x, y)) = (γ(x, a)µ(y, b))for allx∈X, y∈Y, a, b∈L.

(iv) (hA₁, B₁i ≈ hA₂, B₂i) =

W

x∈X

γ(x, A₁(x))≈_V W

x∈X

γ(x, A₂(x))

for allhA₁, B₁i,hA₂, B₂i ∈ B(X, Y, I).

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Proof. Let γ and µ with the above properties exist. If we define the mapping ϕ:B(X, Y, I)→V byϕ(A, B) = W

x∈X

γ(x, A(x)) for allhA, Bi ∈ B(X, Y, I), then by the proof of Part (ii) of Theorem 14 in [3] (where differently defined antisymmetry is not used anywhere) the mapping ϕis bijective and preserves fuzzy order. Thus, we have to prove that it preserves also fuzzy equality. However this is immediate:

(ϕ(A₁, B₁)≈_V ϕ(A₂, B₂)) = ( W

x∈X

γ(x, A₁(x))≈_V W

x∈X

γ(x, A₂(x))) =

= (hA₁, B₁i ≈ hA₂, B₂i).

Conversely, let V andhhB(X, Y, I),≈i,_Si be isomorphic. Similarly to [3], it suffices to prove existence of mappingsγ, µwith desired properties for V= hhB(X, Y, I),≈i,Siand for identity onhhB(X, Y, I),≈i,Siwhich is obviously an isomorphism. The reason for this simplification lies in the fact that for the general caseV∼=hhB(X, Y, I),≈i,_Sione can takeγ◦ϕ:X×L→V, µ◦ϕ: Y ×L→V, whereϕ is the isomorphism ofhhB(X, Y, I),≈i,_Si ontoV. If we defineγ:X×L→ B(X, Y, I), µ:Y ×L→ B(X, Y, I) by

γ(x, a) = D

{a/x}^↑↓,{a/x}^↑E ,

µ(y, b) = D

b/y ^↓, b/y ^↓↑E

for all x ∈ X, y ∈ Y, a, b ∈ L, then by the proof of Part (ii) of Theorem 14 in [3] (where differently defined antisymmetry is not used anywhere) these map- pingsγ, µsatisfy conditions (i–iii) of our theorem. So, it remains to prove condition (iv), i.e. the equality

(hA₁, B₁i ≈ hA₂, B₂i) =

W

x∈X

γ(x, A₁(x))≈_V W

x∈X

γ(x, A₂(x))

.

Since we consider identity onhhB(X, Y, I),≈i,_Si, we have≈=≈_V and it suffices to prove that W

x∈X

γ(x, A(x)) =hA, Bifor allhA, Bi ∈ B(X, Y, I).

We start with proof of the equationA=S

x∈X

nA(x)/xo↑↓

for anyA=A^↑↓. On the one hand we get for eachx⁰∈X:

S

x∈X

nA(x)/xo↑↓

(x⁰) = W

x∈X

nA(x)/xo↑↓

(x⁰) =

= W

x∈X

"

V

y∈Y

nA(x)/xo↑

(y)→I(x⁰, y)

#

=

= W

x∈X

"

V

y∈Y

V

x∈X˜

nA(x)/xo

(˜x)→I(˜x, y)

→I(x⁰, y)

#

= Underlying Fuzzy Order

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= W

x∈X

"

V

y∈Y

(A(x)→I(x, y))→I(x⁰, y)

#

≥

≥ V

y∈Y

[(A(x⁰)→I(x⁰, y))→I(x⁰, y)]≥

≥ V

y∈Y

A(x⁰) =A(x⁰).

On the other hand we have:

S

x∈X

nA(x)/x o↑↓

(x⁰) = W

x∈X

"

V

y∈Y

(A(x)→I(x, y))→I(x⁰, y)

#

≤

≤ W

x∈X

"

V

y∈Y

V

˜x∈X

A(˜x)→I(˜x, y)

→I(x⁰, y)

#

=

= W

x∈X

"

V

y∈Y

A^↑(y)→I(x⁰, y)

#

=

= W

x∈X

A^↑↓(x⁰) =A^↑↓(x⁰) =A(x⁰).

Using also the definition ofγand Theorem 2, we obtain W

x∈X

γ(x, A(x)) = W

x∈X

n A(x)/x

o↑↓

,

nA(x)/x o↑

=

* S

x∈X

nA(x)/x o↑↓↑↓

,

S

x∈X

nA(x)/x o↑↓↑+

=

=hA^↑↓, A^↑i=hA, Bi.

Remark 2. Note that the essential difference between Theorem 3 in this paper and Theorem 14, part (ii) in [3] lies in differently defined L-ordered sets (see the notes at the end of Section 2). Therefore in comparison to Theorem 14 of [3], Theorem 3 must contain “additional” condition (iv) which is necessary for isomorphism between (more general)VandhhB(X, Y, I),≈i,_Si.

5 Work in progress

There is an interesting proposition which deals with a completely lattice L-ordered sethhB(V, V,),≈Si,Si such that

• is anL-order onV,

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• ≈_S denotes an L-equality defined by (hA₁, B₁i ≈_S hA₂, B₂i) = (v₁ ≈ v₂) wherev_i(i∈ {1,2}) is the (unique) element ofV such that (sup(A_i))(v_i) = 1.

(Thus fuzzy equality betweenL-conceptshA₁, B₁i,hA₂, B₂i ∈ B(V, V,) is expressed by fuzzy equality between suprema of their extents.)

Proposition 1. A completely latticeL-ordered setV=hhV,≈Vi,iis isomorphic tohhB(V, V,),≈_Si,_Si.

The proposition represents a corollary of Theorem 3, but an elegant proof of this fact is a matter of further investigations.

6 Acknowledgements

The author wishes to thank the anonymous reviewers for their criticisms and useful comments.

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Underlying Fuzzy Order