Fuzzy sets as texture spaces, I. Representation theorems

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www.elsevier.com/locate/fss

Fuzzy sets as texture spaces, I. Representation theorems

Lawrence M. Brown

^a;

∗ , Riza Erturk

^b;1

aHacettepe University, Faculty of Science, Mathematics Department, Beytepe, Ankara, Turkey

b Abant ˙Izzet Baysal University, Faculty of Science and Letters, Mathematics Department, ˙Izzet Baysal Campus, Bolu, Turkey Received August 1997; received in revised form April 1998

Abstract

The authors continue the development of a theory of texture spaces, introducing complemented products and sums, and applying these in a series of representation theorems for fuzzy lattices and the lattices ofL-fuzzy sets, generalized fuzzy sets and intuitionistic sets. The second paper in this series will extend this enquiry by introducing subtextures and quotient textures and a second series of papers is under preparation which consider topological aspects of this correspondence.

Keywords:Fuzzy lattices;L-fuzzy sets; Generalized fuzzy sets; Intuitionistic sets; Textures; Complemented textures;

Product textures; Sums of textures (VARF)

1. Introduction

The notion of a ditopological texture space, under the name ditopological fuzzy structure, was introduced by the rst author at the 2nd BUFSA Conference on Fuzzy Systems and Articial Intelligence held at Trabzon University in 1992. It is a natural sequel to the work of the second author on the representation of fuzzy topologies by bitopological spaces [8 – 10].

However, in place of the full lattice of subsets of some base setS, attention is focused on a suitable sublattice of subsets, called a texturing ofS. In this way, as will be conrmed in this paper, an exact point-set setting for the study of fuzzy sets is obtained.

∗Corresponding author.

E-mail address:[email protected] (L.M. Brown)

1Current address: Mersin University, Faculty of Science and Letters, Mathematics Department, Mersin, Turkey.

In the context of textures, bitopologies are replaced by dichotomous topologies, or ditopologies for short.

Some basic notions concerning these were presented in [2, 3]. In particular product textures were dened and the ditopological counterpart of joint compactness was introduced, and shown to be productive. Further results on ditopological compactness may be found in [4], and forms of ditopological paracompactness and connectedness are discussed in [5, 7], respectively.

The relation between textures and fuzzy sets was mentioned briey in [2], and it is the aim of this paper to investigate this relation in detail. This will involve a discussion of complementation on product textures, and the notion of a sum of textures will be introduced to facilitate the representation of generalized fuzzy sets in the sense of [13], and to provide an alternative formulation of the representation ofL-fuzzy sets.

Our present concern is to develop the theory of texture spaces within a rigorous mathematical setting,

PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 1 5 7 - 2

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but the potential for future applications is clear. Tex- tures enable fuzzy sets to be studied as crisp subsets of some basic set and the operations of product, sum, sub-texture and quotient studied in this paper and its sequel place fuzzy sets in a new framework. More- over, as we will see below, textures are strictly more general than fuzzy sets. This raises the exciting possi- bility of generalizing the notion of fuzzy set in ways that are mathematically advantageous and at the same time, of practical signicance. Some ideas along these lines will be developed in a future paper.

For the benet of the reader we will recall all nec- essary denitions and results from [2, 3]. [11] is an invaluable source of information about lattice theory.

Denition 1.1. Let S be a set. Then S⊆P(S) is called a texturing ofS, and S is said to be textured byS. If

(1) (S;⊆) is a complete lattice containingS and

∅, and the meet and join operations in (S;⊆) are related with the intersection and union operations in (P(S); ⊆) by the equalities

^

i∈I

Ai=\

i∈I

Ai; Ai∈S; i∈I;

for all index setsI;while _

i∈I

Ai=[

i∈I

Ai; Ai∈S; i∈I;

for all nite index setsI.

(2)Sis completely distributive.

(3) S separates the points of S. That is, given s16=s2 in S we have A∈S with s1∈A; s2∈=A, or A∈Swiths2∈A; s1∈=A.

IfSis textured bySwe call (S;S) atexture space, or simply atexturefor short.

The mapping s→P_s=T

{A∈S|s∈A} is a natural embedding of S in S. Recall that an element M6=∅ofSis a moleculeifM⊆A₁∪A₂⇒M⊆A₁ orM⊆A2 for allA1; A2∈S. Clearly{Ps|s∈S} is a set of molecules inSwhich is a base forSin the sense that

A= _

s∈A

Ps= [

s∈A

Ps

for allA∈S.

We callSsimpleif all the molecules ofSbelong to{P_s|s∈S}.

A texturingSofS induces a partial ordering onS given by

s6t⇔P_s⊆P_t:

Clearly this is equivalent tos6t⇔s∈Pt.

A mapping:S→Sis called acomplementation if ²(A) =A for all A∈S and A⊆B in S implies (B)⊆(A). A complementation is necessarily bijec- tive. A texture with a complementation is said to be complemented.

Examples 1.2. (1) Clearly (X;P(X)) is a texture, which represents the full set structure of X. Here Px={x}and these are the only molecules so the texture is simple. The induced partial order is trivial.

Clearly,=X:Y⊆X7→X\Y is a complementation on (X;P(X)).

(2) LetL= (0;1] andL={(0; r]|r∈[0;1]}. Then (L;L) is a texture for which the join is not always the same as the union. The only molecules arePs= (0; s], s∈S, so the texture is simple. The induced partial order is the usual order on (0;1]. Clearly (0; r] = (0;1−r] is a natural complementation on (L;L).

(3) Now let L= [0;1], and take L={[0; r]|r∈ [0;1]} ∪ {[0; r)|r∈[0;1]}. Then (L;L) is a texture for which the join and union coincide. In addition to P_r= [0; r], r∈L, the sets [0; r) are also molecules, so this is not a simple texture. Again the induced order is the usual order on [0;1]. A complementation may be dened by setting [0; r] = [0;1−r) and [0; r) = [0;1−r],r∈ L.

(4)S={∅;{a; b};{b};{b; c}; S}is a simple texturing ofS={a; b; c}. ClearlyPa={a; b},Pb={b}and Pc={b; c} so the induced partial order is the reex- ive closure ofb6a,b6c. It is not possible to dene a complementation on (S;S).

We shall nd the following properties of complementation useful in the sequel.

Lemma 1.3. Let (S;S; ) be a complemented texture. Then for allP∈Swe have

P= \

t∈(P)

(Pt):

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229

Proof. Take the image under of both sides of the identity(P) =W

t∈(P)P_t. Corollary 1. P_s=T

t∈(Ps)(P_t)for alls∈S.

Corollary 2. t∈(Ps)⇔s∈(Pt)∀s; t∈S.

If (Si;Si), i= 1;2 are textures, a mapping :S1→S2 is an isomorphism [4] if it is a bijection and the mapping:S1→S2dened by:A7→(A) is a bijection also. Clearly an isomorphism of textures preserves arbitrary meets and joins. If in addition, i is a complementation on (Si;Si), i= 1;2, and satises the additional condition 2◦=◦1 then is an isomorphism of (S1;S1; 1) with (S2;S2; 2).

2. Representation of fuzzy lattices andL-fuzzy sets Let L be a fuzzy lattice, i.e. a completely distributive lattice with order reversing involution ⁰. Let L denote the set of molecules in L and set

’(a) ={m∈L|m6a} and L={’(a)|a∈L} for a∈L. Then:

Theorem 2.1. (L;L) is a simple texture with complement(’(a)) =’(a⁰); a∈L;and’:L→Lis a lattice isomorphism which preserves complementation.

Conversely; every complemented simple texture may be obtained in this way from a suitable fuzzy lattice.

Proof. The fact that L is a complete lattice and

’:L→L is an isomorphism of complete lattices follows trivially from

a6b⇔’(a)⊆’(b) ∀a; b∈L

which holds by virtue of the fact that the molecules of Lform a base (see [11], where this result is given in terms of the dual notion of prime ideal). Likewise

\

∈A

’(a) = ^

∈A

’(a)

fora∈Land all index setsAis a consequence of m6^

a⇔m6a ∀∈A

form∈L, and [

∈A

’(a) = _

∈A

’(a)

for niteAfollows from m6_

a⇔m6a for some∈A

whenever m∈L. Since L=’(1)∈L and ∅=’(0)

∈L we see that (L;L) satises Denition 1.1(1).

L is completely distributive since it is isomorphic to L and it separates points of L since if m; n∈L satisfy m6=n then m66 n or n66 m so m =∈’(n) or n =∈’(m). Hence (L;L) is a texture. Finally, if’(a) is a molecule in L it is easy to see that a is a molecule in L, so all the molecules in L have the form’(m) form∈L. However, form∈Lwe clearly haveP_m=’(m) and so (L;L) is simple. It is trivial that (’(a)) =’(a⁰) denes a complementation and thatis preserved under’.

Conversely, suppose that (S;S; ) is a simple complemented texture. Then S is a fuzzy lattice with involution ⁰ dened by A⁰=(A), A∈S. For A∈S let ’(A) ={Ps|Ps⊆A}={Ps|s∈A}, S^?=’(S) ={Ps|s∈S}, ^?(’(A)) =’((A)) and S^?={’(A)|A∈S}, so that (S^?;S^?; ^?) is the complemented texture corresponding to S. Clearly :s7→Ps is an isomorphism between (S;S; ) and (S^?;S^?; ^?).

If (L;L; ) is dened as above for the fuzzy lattice Lwe will call (L;L) ((L;L; )) the (complemented) fuzzy texture ofL.

By Theorem 2.1, all fuzzy textures are simple. This means that many potentially important textures, such as the texture of Examples 1.2(3), do not correspond to fuzzy lattices, i.e. textures are strictly more general than fuzzy lattices. Moreover, the property of being simple does not seem to play an essential role in the theory of textures, at least not for those results presented in the present paper and its sequel.

This suggests that non-simple textures may provide a useful model on which to base denitions of generalized fuzzy sets with desirable properties. We will not pursue this line of inquiry further here, but as mentioned in the introduction some preliminary studies are underway which will be reported later.

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Now let L be a fuzzy lattice and X a non-empty set. Then W=L^X, the set of L-fuzzy sets [12] on X, is also a fuzzy lattice under the point-wise ordering f6g⇔f(x)6g(x) ∀x∈X, and the involu- tionf⁰(x) =f(x)⁰. We will refer to the fuzzy texture (W;W; !) ofWas thecomplementedL-fuzzy texture onX. The elements ofW are just the “fuzzy points”

ofW, i.e. the functions xm(z) =

m ifz=x;

0 otherwise,

for x∈X andm∈L, where as beforeL is the set of molecules of L. Representing xm by the pair (x; m) we obtain a canonical form for (W;W; !) by setting W=X ×L, and then

W={’(f)|f∈W};

’(f) ={(x; m)∈W|x_m6f}

={(x; m)∈W|m6f(x)} while

!(’(f)) =’(f⁰) ={(x; m)|m6f(x)⁰} gives the complement.

We wish to show that (W;W) is actually the product of the textures (X;P(X)) and (L;L). First let us recall from [3] the denition of the product of texture spaces.

Let (Si;Si),i∈Ibe textures, setS=Q

i∈ISiand for k∈I andA⊆Sk let

E(k; A) =Y

i∈I

Yi;

where Y_i=

A ifi=k;

S_i otherwise.

Then theproductof the texturingsSi; i∈I, is the tex- turingSofSwhich consists of arbitrary intersections of elements of the set

E=



 [

j∈J

E(j; A_j)

J⊆I; A_j∈Sj



:

It is shown in [3] that this does indeed dene a texturing in the sense of Denition 1.1. The following results from [3] will prove useful later on.

Lemma 2.2. 1.IfA_i∈Si; i∈I;thenQ

i∈IA_i∈S.

2. For s = (s_i)∈S we have P_s = Q

i∈IP_s_i = T

i∈IE(i; P_s_i).

Lemma 2.3. For ∈D suppose that J⊆I and for j∈J letA_j∈Sj. Then setting

E= [

j∈J

E(j; A_j)∈E for∈Dwe have

_

∈D

E= [

j∈∪J

E

j;_

{A_j|j∈J} :

We may now state:

Lemma 2.4. TheL-fuzzy texture(W;W)onXis the product of the set structure(X;P(X))of X and the fuzzy texture(L;L)ofL.

Proof. By denition, each element of the product texturing ofX ×L may be written as an intersection of sets of the form

(Y ×L)∪(X×’(a)) forY⊆X anda∈L.

To show the product texturing is coarser thanWit will be sucient to show (Y×L)∪(X×’(a))∈W. However, this is trivial for if we denef:X→Lby f(x) =

1 forx∈Y;

a forx∈X\Y;

then clearly’(f) = (Y ×L)∪(X ×’(a)).

Conversely, to showWis coarser than the product texturing, it suces to takef∈Wand note that

’(f) = \

z∈X

[((X\{z})×L)∪(X ×’(f(z)))];

while this set belongs to the product texturing.

Complementation on products was not considered in [2, 3]. It will be appropriate to do so now.

Theorem 2.5. Let(Si;Si; i); i∈I;be complemented textures and(S;S)their product. ForA∈Sdene (A) = \

s∈A

[

i∈I

E(i; _i(P_s_i)):

Thenis a complementation on(S;L).

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231

Proof. That is a mapping on S satisfying A⊆B⇒(B)⊆(A) for allA; B∈Sis trivial, so it remains to show((A)) =AforA∈S.

Take s∈A. Then for t∈(A) we have t∈ S

i∈IE(i; _i(P_s_i)) and hence there exists i=i(t)∈I with ti∈i(Ps_i). By Corollary 2 of Lemma 1.3 we deduce si∈i(Pt_i) for this i, whence s∈((A))

=T

t∈(A)

S

i∈IE(i; i(Pt_i)). This establishes A⊆ ((A)).

Now taker∈((A)) and suppose thatr =∈A. We may write

A= _

s∈A

P_s= _

s∈A

\

i∈I

E(i; P_s_i) = \

∈I^A

_

s∈A

E((s); P_s_(s))

by the complete distributivity ofS. For each∈I^Awe may now apply Lemma 2.3 withD=A, Js={(s)} fors∈AandA^s_j=Ps_j forj∈Jsto give

A= \

∈I^A

[

j∈(A)

E

j;_

{P_s_j|s∈A; (s) =j} :

Since r= (ri)∈=A there exists ∈I^A so that for all j∈(A) we have rj∈= W

{Ps_j|s∈A; (s) =j}. If for each j∈(A) we apply Lemma 1.2 to P= W{Ps_j|s∈A; (s) =j} we see that there exists tj∈ j(W

{Ps_j|s∈A; (s) =j}) =T

{j(Ps_j)|s∈A; (s)

=j} satisfying rj∈=j(Pt_j). For i =∈(A) we may chooseti∈Si\i(Pr_i) sinceri∈Pr_i6=∅ ⇒i(Ps_i)6=Si, whence ri∈=i(Pti) by Lemma 1.3, Corollary 2. Let t= (t_i). Thent∈(A). To see this we need only note that by denition (A) =T

s∈A

S

i∈IE(i; _i(P_s_i)) so by the complete distributivity of the latticeP(S) (A) = [

∈I^A

\

s∈A

E((s); (s)(Ps_(s)))

and for thefound abovet∈E((s); _(s)(P_s_(s))) for eachs∈A. Now

r∈((A)) = \

u∈(A)

[

i∈I

E(i; _i(P_u_i)) sor∈S

i∈IE(i; i(Pt_i)) sincet∈(A). However this is a contradiction since by the denition oftwe have ri∈=i(Pti) for alli∈I. Hencer∈Aand we have es- tablished ((A))⊆A. Thus is indeed a complementation on the product texture (S;S).

Denition 2.6. The complementation dened on the product (S;S) of the complemented textures

(S_i;Si; _i), i∈I by (A) =T

s∈A

S

i∈IE(i; _i(P_s_i)) for allA∈Sis called theproductof the complemen- tations_i. (S;S; ) will be called thecomplemented productof (S_i;Si; _i),i∈I.

We may note the following. The proof follows much the same lines as the proof of Theorem 2.5, and is omitted.

Lemma 2.7. If(S;S; )is the complemented product of(Si;Si; i); i∈I;then fori∈I andAi∈Si we have

(E(i; Ai)) =E(i; i(Ai)):

Corollary. Fors∈S=Q

i∈ISi we have (P_s) =[

i∈I

E(i; _i(P_s_i)):

Proof. (Ps) = (Q

i∈IPs_i) = (T

i∈IE(i; Ps_i)) = W

i∈IE(i; i(Ps_i)) by Lemma 2.7. Since S

i∈IE(i;

i(Ps_i))∈Swe obtain(Ps) =S

i∈IE(i; i(Ps_i)).

We are now in a position to compare the complementation! on (W;W) with the complementations on (X;P(X)) and (L;L).

Lemma 2.8. The complementation !on (W;W) is the product of the complementationson(X;P(X)) andon(L;L).

Proof. It will be sucient to show that the eect of the two complementations on sets of the form (Y×L)

∪(X×’(a)),Y⊆X,a∈Lis the same. By Lemma 2.7 the complement of this set for the product complemen- tationis((Y×L)∪(X×’(a))) =(Y×L)∩(X×

’(a)) = ((Y)×L)∩(X ×(’(a))) = ((X\Y)×L)∩ (X ×’(a⁰)) = (X\Y) ×’(a⁰). On the other hand (Y×L)∪(X×’(a)) =’(f), wheref:X→Lis dened by

f(x) =

1 forx∈Y;

a forx∈X\Y;

so!(’(f)) =’(f⁰) where f⁰(x) =

0 forx∈Y;

a⁰ forx∈X\Y:

It is clear that’(f⁰) = (X\Y)×’(a⁰) as required.

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Combining Lemmas 2.4 and 2.8 gives:

Theorem 2.9. The complementedL-fuzzy texture on X is the complemented product of(X;P(X); )and (L;L; ).

Examples 2.10. (1) For classic fuzzy sets onX we have L= [0;1]. Every non-zero element of L is a molecule, so L= (0;1] and L={(0; r]|06r61}. HenceW=X×(0;1] and the sets ofWare intersections of sets of the form (Y ×(0;1])∪(X ×(0; r]), Y⊆X and 06r61. Moreover!((Y×(0;1])∪(X× (0; r])) = (X\Y)×(0;1−r].

(2) Take L={0;¹₂;1}. Then L={¹₂;1} and L={∅;{¹₂}; L}. Hence in this caseW=X × {¹₂;1} and the elements ofWare intersections of sets of the formX×L,Y× {¹₂;1}and (Y× {¹₂;1})∪(X× {¹₂}), Y⊆X, whose complements are respectively ∅, (X\Y)× {¹₂;1}and (X\Y)× {¹₂}.

We end this section with a further characterization of intuitionistic textures. In [1] Atanassov introduced the notion of intuitionistic fuzzy set, and the crisp version of these sets, intuitionistic sets, was given by

Coker in [6]. In [4] intuitionistic textures were dened as those textures which correspond to intuitionistic sets, and several characterizations were given. We now show that intuitionistic textures are precisely the textures described in the last example, so showing that intuitionistic sets are essentiallyL-fuzzy sets, whereL is the fuzzy lattice{0;¹₂;1}.

Theorem 2.11. (S;S; ) is an intuitionistic texture if and only if it is isomorphic to the texture(W;W; !) corresponding to theL-fuzzy sets on some setX for L={0;¹₂;1}.

Proof. We prove suciency, leaving the proof of ne- cessity to the interested reader. Let (W;W; !) be a texture as in Examples 2.10(2). We verify the conditions of [4, Theorem 2.5(iv)]. W=X ×L=X × {¹₂;1}= (X × {¹₂})∪(X × {1}). Let T=X × {¹₂} and denef:T→W\T=X×{1}by (x;¹₂)7→(x;1), x∈X. Clearlyfis injective. ForA∈WletA1=A∩T andA₂=A∩(W\T). Then it is easy to verify that

(1)A=A₁∪A₂,A₁⊆T andA₂⊆f(A₁), (2)!(A) =W\(f(A₁)∪f⁻¹(A₂)).

Conversely, any subset ofW satisfying (1) and (2) belongs toW, whence the result follows by [4, The- orem 2.5 (iv)].

3. Sums of texture spaces – the representation of generalized fuzzy sets

In order to represent generalized fuzzy sets in the sense of Nakajima [13] we now introduce the notion of the sum of texture spaces. This will, in particular, lead to an alternative interpretation of the representation of L-fuzzy sets given in Theorem 2.9.

Lemma 3.1. Let(Si;Si); i∈I;be textures withSi∩ Sj=∅fori6=j. Let S=S

i∈ISi and S={A|A⊆S;

A∩Si∈Si∀i∈I}. Then(S;S)is a texture. Further;

if i is a complementation on(Si;Si); i∈I; then dened onSby

(A)∩Si=i(A∩Si); i∈I;

makes(S;S; )a complemented texture.

Proof. S;∅ ∈S follow trivially from the denition.

Moreover, forA∈Swe have (T

A)∩Si=T

(A

∩Si)∈Si for eachi∈I, so T

A∈S. ThusSis a complete lattice andV

A=T

A. Now let us verify that

_

A

!

∩S_i=_

(A∩S_i) ∀i∈I:

Clearly, _

A

!

∩S_i=\( A

A∈S; [

A⊆A )

∩S_i

=\( A∩Si

A∈S; [

A⊆A )

:

Now A∈S, S

A⊆A⇒A∩Si∈Si; S

(A ∩Si)

⊆A∩Siso _

(A∩S_i) =\( A_i

A_i∈Si; [

(A∩S_i)⊆A_i )

⊆ _

A

!

∩Si:

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233

On the other hand if A_i∈Si and S

(A ∩S_i)⊆A_i, let A=S

j6=iS_j∪A_i. Then A∈S, S

A⊆A and A∩Si =Ai, whence (W

A)∩Si⊆W

(A∩Si) as required.

Now for A1; A2; : : : ; An∈S and i∈I we have (Wn

k= 1Ak)∩Si=Wn

k= 1(Ak∩Si) =Sn

k= 1(Ak∩Si) = (Sn

k= 1Ak)∩SisinceSi is a texturing. Hence _n

k=1

Ak= [n k=1

Ak

so (S;S) satises Denition 1.1 (1). To show (S;S) is completely distributive, for ∈A, ∈B, take A∈S. Then for anyi∈I, (W

∈A

T

∈BA)∩Si= W

∈A((T

∈BA)∩S_i) = W

∈A

T

∈B(A∩S_i) = T

∈B

W

∈A(A₍₎∩Si) =T

∈B((W

∈AA₍₎)∩Si)

= (T

∈B

W

∈AA₍₎)∩Si using the above and the complete distributivity of (Si;Si). Thus

_

∈A

\

∈B

A = \

∈B

_

∈A

A₍₎

as required. Moreover, sinceSi⊆Sfor eachi∈I it is easy to see thatSseparates the points ofS, whence (S;S) is a texture.

Now supposei is a complementation on (Si;Si) and forA∈Sdene(A) by

(A)∩Si=i(A∩Si); i∈I:

Clearly (A)∈S and for A; B∈S with A⊆B we have(B)⊆(A). Finally

((A))∩Si=i((A)∩Si)

=_i(_i(A∩S_i)) =A∩S_i

for alli∈I, whence((A)) =A. This completes the proof thatis a complementation on (S;S).

Denition 3.2. The texture (S;S) dened as in Lemma 3.1 is called thesumof the disjoint textures (Si;Si),i∈I. Ifi is a complementation on (Si;Si) andis dened as in Lemma 3.1 then (S;S; ) is the complemented sumof the (Si;Si; i),i∈I.

Now let X be a set and for each x∈X let Lx be a fuzzy lattice. Then functions f on X satisfying f(x)∈Lxfor eachx∈X are called generalized fuzzy

sets in [13]. Clearly these form the family L= Y

x∈X

Lx= (

f

f:X→ [

x∈X

Lx; f(x)∈Lx

)

;

which is a fuzzy lattice under the point-wise order f6g⇔f(x)6g(x) ∀x∈X and involution f⁰(x) = (f(x))⁰,x∈X.

Denote by (L;L; ) the complemented fuzzy texture ofL. We wish to show that (L;L; ) is isomorphic in a natural way to a complemented sum of textures.

Let (Lx;Lx; x) be the complemented fuzzy texture of Lx; x∈X, and note rst that the molecules in L have the formf(x; m)forx∈X andm∈Lx, where f(x; m)(y) =

m ify=x 0y otherwise;

0y being the smallest element of Ly. Now let (Nx;Nx; x) be the complemented product of ({x};P({x}); _{_x_}) and (Lx;Lx; x) and dene N= [

x∈X

Nx= [

x∈X

({x} ×Lx):

SinceN_x∩N_y=∅forx6=ywe may consider the complemented sum (N;N; ) of the textures (N_x;Nx; _x).

Then

Theorem 3.3. The complemented texture (L;L; ) corresponding to a lattice of generalized fuzzy sets on X is isomorphic to the complemented sum (N;N; )of the textures(Nx;Nx; x)dened above.

Proof. Dene:N→Lby (x; m)7→f(x; m). Clearly is a bijection. We show it is an isomorphism between (N;N) and (L;L). TakeA∈N. For eachx∈X we haveA∩Nx∈Nxso there existsax∈Lx, necessarily unique, so thatA∩Nx={x} ×’(ax). Denef∈Lby f(x) =ax ∀x∈X. Then (A) =’(f)∈L, and it is now trivial to verify that, regarded as a mapping from NtoL, is one to one and onto. Hence is indeed an isomorphism of textures. It remains to show that it preserves complementation. ForA∈Nanda_x,fde- ned as above we have(A)∩N_x=_x(A∩N_x) ={x}

×_x(’(a_x)) ={x} ×’(a⁰_x), using Lemma 2.7. Since f⁰(x) =a⁰_x; ∀x∈X, we have

((A)) =’(f⁰) =(’(f)) =((A));

which is the required equality.

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Note. Nakajima only requires that the lattices Lx

be complete Heyting Algebras. We could weaken the requirement that textures be completely distributive in order to represent Nakajima’s generalized fuzzy sets in the general case. However, complete distributivity will prove invaluable when we come to discuss topological properties of textures, and we prefer not to make this change at the present time.

In the case where Lx=L ∀x∈X, the generalized fuzzy sets on X reduce to L-fuzzy sets on X and then the sum (N;N; ) of (Nx;Nx; x),Nx={x} ×L, provides an alternative representation of the fuzzy texture of L. Indeed it is trivial to verify directly that (W;W; !) is isomorphic to (N;N; ) under the mapping

(x; m)∈W=X ×L7→(x; m)∈N_x⊆N:

Now let us briey consider the converse situation.

Given a complemented simple texture (S;S; ), under what conditions will it correspond to a lattice of (generalized) fuzzy sets on some setX? As it stands the answer is trivial – by Theorem 2.1 any complemented simple texture will correspond to the lattice of L-fuzzy sets on a single point set X. To obtain a more useful result we restrict our attention to generalized fuzzy sets where the fuzzy latticesLxare product irreducible, i.e. cannot be expressed in a non-trivial way as a product of fuzzy lattices. We refer to these as irreduciblegeneralized fuzzy sets for short. We shall require the following:

Denition 3.4. Let (S;S; ) be a complemented texture. Then A∈Sis called-irreducibleif whenever A⊆A₁ ∪A₂, A₁; A₂∈S, B∩A_k=C ∩A_k for B; C∈Simplies(B)∩A_k=(C)∩A_k,k= 1;2 and A∩A₁6=∅ 6=A∩A₂we haveA∩A₁∩A₂6=∅.A∈Sis -reducibleif it is not-irreducible. The-component ofs∈S is the set

K(s) =[

{A|s∈A∈S; Ais-irreducible}: Lemma 3.5. For eachs∈S; s∈K(s)∈S;while any two-components are either equal or disjoint.

Proof. Since Ps is clearly -irreducible we have s∈P_s⊆K(s)⊆W

{A|s∈A∈S; Ais-irreducible}

∈S. It is easy to show that this last set is-irreducible, whenceK(s) = W

{A|s∈A∈S; Ais-irreducible}

∈S as required. The remaining properties are immediate.

Lemma 3.6. Let L be a fuzzy lattice and (L;L; ) the complemented fuzzy texture ofL. ThenLis isomorphic to a product of non-trivial fuzzy latticesL1; L2if and only ifLis-reducible in(L;L; ).

Proof. ⇒. IfLis isomorphic toL1×L2={f:{1;2}

→L1∪L2; f(k)∈Lk; k= 1;2}then by Theorem 3.3, (L;L; ) is the complemented sum of (N1;N1; 1) and (N2;N2; 2), where Nk={k} ×Lk, k= 1;2. Clearly the existence of these sets implies thatL=N1∪N2is -reducible in (L;L; ).

⇐. LetLbe-reducible in (L;L; ). Then we have L1; L2∈LwithL=L1∪L2, forA; B∈L,A∩Lk=B∩ L_k⇒(A)∩L_k=(B)∩L_k,k= 1;2,L₁6=∅ 6=L₂but L₁∩L₂=∅.Lk={L_k∩A|A∈L}is a fuzzy lattice on L_k,k= 1;2. Moreover a complement_k onL_k is well dened by_k(L_k∩A) =(A)∩L_k in view of the conditions onL_k,k= 1;2. It is clear that (L;L; ) is the complemented sum of (L1;L1; 1) and (L2;L2; 2), whence ifL1,L2 are the corresponding fuzzy lattices in the sense of Theorem 2.1, (L;L; ) corresponds to L1×L2. HenceLis isomorphic to a non-trivial product of fuzzy lattices, as required.

We may now state

Theorem 3.7. The complemented texture (S;S; ) corresponds to a lattice of irreducible generalized fuzzy sets if and only if

(1)EachK(s); s∈S;satises A; B∈S; A∩K(s) =B∩K(s)

⇒(A)∩K(s) =(B)∩K(s);

and:

(2) Arbitrary unions of the components K(s);

s∈S;belong toS.

Proof. ⇒. Suppose (S;S; ) corresponds to the lat- ticeQ

x∈XLxof irreducible generalized fuzzy sets. By Theorem 3.3, (S;S; ) may be expressed as a sum of the structures (Nx;Nx; x). By Lemma 3.6 eachNxis x-irreducible in (Nx;Nx; x), and hence-irreducible in (S;S; ). Since these sets form a partition we see that they are the -components, and conditions (1) and (2) are now easily veried.

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235

⇐. ChooseX⊆S with the following properties:

(a)S

x∈XK(x) =S, and

(b)x₁; x₂∈X,x₁6=x₂⇒K(x₁)∩K(x₂) =∅. For each x ∈ X let S_x =K(x),Sx = {S_x∩A| A∈S}and_x(A∩S_x) =(A)∩S_x,A∈S, which is well dened by (1). It is easy to verify that (S;S; ) is the complemented sum of the structures (Sx;Sx; x), x∈X. Moreover, by Lemma 3.6 and Theorem 2.1 each (Sx;Sx; x) corresponds to a product irreducible fuzzy latticeLx, whence (S;S; ) corresponds toQ

x∈XLx, as required.

Corollary. (S;S; )is isomorphic to a lattice of L- fuzzy sets with L product irreducible if and only if it satises the conditions of Theorem 3.7 and the textures onK(s); s∈S;are isomorphic.

In particular instances it may be possible to choose a distinguished representative in each component, so obtaining a canonical copy ofX inS.

Example 3.8. LetSbe the punctured unit disc andS the set of all subsetsAofS satisfying

(i) (r; )∈A; 0¡s6r⇒(s; )∈S, and (ii) (r_i; )∈A,i∈I⇒(supr_i; )∈A.

Clearly (S;S) is a texture space, and a complemen- tationmay be dened by sup{r|(r; )∈(A)}= 1− sup{r|(r; )∈A}. The reader may easily verify that for (s; )∈S we have K(s; ) ={(r; )|0¡r61}.

Indeed, (S;S; ) is isomorphic to the complemented fuzzy texture of the lattice of fuzzy subsets of X= [0;2).X may be embedded canonically inSas, for example, the circumference of the disc.

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Fuzzy sets as texture spaces, I. Representation theorems