smooth preuniform and preproximity spaces

Fuzzy Sets and Systems 59 (1993) 95-107 95 North-Holland

Smooth preuniform and preproximity spaces

R. B a d a r d , A . A . R a m a d a n a n d A . S . M a s h h o u r CARSOP, PrO Marzy, Frontenas, Le Bois d'Oingt, France

Received March 19!)2 Revised March 19921

Abstract." Fuzzy sets and fuzzy logic are very suited to interpret every day life sentences. Fuzzy interpretations are defined with the use of membership functions whose role is simply to give a kind of graphical, visual support to help us to capture the meaning. We think that generally membership values are not very important, but only some characteristics of membership functions have to be kept. We think that monotonic relations between degrees with which some properties are satisfied is an important characteristic. We translate in this sense axioms corresponding to preuniform structures and preproximity structures. Contrary to fuzzy extensions, like fuzzy topology for instance, which generally keep the original axioms but adapt few of them to deal with fuzzy sets, our approach induce much more transformations. Finally we study how classic constructions relating these concepts can be translated in this framework.

Keywords: Fuzzy sets; topology; logic.

Fuzzy sets and fuzzy logic have opened new horizons and new ways of thinking. But these theories are very semantical in the sense they lie on the definition of membership functions. In fact many statements are not very dependent on such membership functions. Membership functions are there to help us to reason, they recall us the underlying interpretation, like figures help us to prove theorems in geometry. But like in geometry, even if these interpretations are not exact we must be able to build correct proofs and results. For instance, an assertion like 'when winter is cold the fuel consumption is high' can be interpreted in many ways. Particularly in fuzzy set theory we have to interpret the meaning 'winter is cold' and the meaning of 'fuel consumption is high'. But in fact, such an assertion simply says that everything else being equal, the more the winter is cold the more the consumption of fuel is high. It does not matter to define that an average temperature of say - 1 0 centigrade degrees agrees 0.97 to 'cold winter' and that an average consumption of 1500 fuel litres per person and per month agrees 0.83 to 'high consumption' . The only interesting thing is that when we are in winter in a particular geographical area and the average temperature decreases then generally the fuel consumption increases. Such assertions can be translated as systems of inequalities between truth values. On the other hand we feel that many mathematical structures have been extended to fuzzy sets, but in fact the structures themselves are almost the same, only their definitions have been adapted to deal with fuzzy sets. For instance the way Chang [4] defined fuzzy topologies is exactly the classic definition for a family of open sets, only the fact that some of these sets can be fuzzy makes the difference. We think that it could be more interesting to reformulate the defining axioms themselves in term of multiple valued logics, in this case the corresponding axioms become systems of inequalities. Such an approach is very aesthetic and brings us to precisely analyse all the concepts we need and the way they may be smoothed. This approach is not only aesthetic in a pure mathematical point of view, but it could be fruitful for applications. It is well known that the classic topological approach is of little interest for application to digital image processing. Such digital images must be described on different levels of

Correspondence u~: Dr. R. Badard, CARSOP, Pr6 Marzy, Frontenas, 69620 Le Bois d'Oingt, France.

0165-0114/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved

96 R. Badard et al. / Smooth preuniform and preproximity spaces

abstraction and particularly at a topological one. But topology does not work well for this goal and certainly pre topology and preuniformity could be of practical use (recall that if we want to endow a finite set with a topology for which it becomes a separated space then we get the discrete topology). In a paper [2], we gave some rules and developed a rough methodology to deal with these problems. Here we will show how the concepts of preuniformity and preproximity [1, 6] can be smoothed in this way and how we still get theories which are well articulated and have nice properties.

In this paper we will note L and L ' two lattices which will be copies of [0, 1] or {0, 1}. In fact these sets could be more general lattices but for simplicity we will restrict the study to these sets. We will use the same notations for fuzzy or crisp sets, when confusion is possible we eventually underlined symbols referring to non-fuzzy concepts. We will note [A]~ for the a cut of a fuzzy set A, [A]~ = {x • E ]A(x) /> a}, and [A]~ + for the strong cut [A]~ + = {x • E ]A(x) > ce}.

1 . S m o o t h p r e n n i f o r m s p a c e s

1.1. Definitions and general properties

A fuzzy preuniform space, as defined in [1] is:

D e f i n i t i o n 1.1.1. A fuzzy preuniform structure OR on a set E is a family of L-fuzzy subsets of E x E, called entourages, which satisfies - Vu E OR, u ~ D, where D is the diagonal: D = {(x, x) I x e E} - Vu e OR, v ~ u implies v • OR. The pair (E, OR) is said to be a preuniform space.

The preuniform structure OR is said to be - symmetrical, when for every u • OR we can assert that u 1 • OR, where u l(x, y) = u(y, x). - type D, when for every u, v • OR we can assert that u N v • OR. - t y p e S, when for every u • O R there is v • O R such that v ® v c u , where ( v®v) (x , y )= sup(v(x, z ) ^v (z , y) l z • E}.

Obviously a fuzzy preuniform space which is symmetrical, type D and S is a fuzzy uniform space.

Now, a smooth preuniform space will be defined by smoothing the preceding axioms.

D e f i n i t i o n 1.1.2. A smooth preuniform structure OR on E is a L ' - fuzzy set on the L-fuzzy sets of E x E. OR is an element of L,L ExE which satisfies - I f u ~ D then OR(u) = 0L,, -- VU, V in L E×E, if v ~ u then °R(v) >/OR(u), in words it says that when v contains u then v belongs at least as u to the family OR, - OR(E × E) = 1L,.

The pair (E, OR) is said to be a smooth preuniform space. The smooth preuniform structure is said to be - symmetrical when OR(u -1) = OR(u), - type D when OR(u n v) >! OR(u)^ OR(v), but with the second axiom OR(u) and OR(v) t> OR(u n v), so we can write OR(u n v) = oR(u) ^ OR(v). - type S when sup{oR(v) [ v ® v c u} ~> OR(u).

Many constructions f rom a preuniformity use the concept of ball. For every A E L F" and v e OR, we will denote ~3(A, u) the ball ~3(A, u ) (x )=sup{u(x ,y )^A(y ) lY E E}. When we deal with a smooth preuniformity the same construction can be used, but now we must be aware that this fuzzy ball is more or less interesting, depending on how u belongs to OR. We remark that L can be the Boolean lattice {0, 1}. In this case we build a smooth preuniform structure with crisp entourages. We have in fact different possibilities, for this reason we have distinguished the lattices L and L ' which play different

R. Badard et al. / Smooth preuniform and preproximity spaces 97

roles. As for the classical case we can c o m p a r e s m o o t h p r e u n i f o r m s t ructures by way of the inclusion of L-fuzzy sets.

Proposition 1.1.1. The set o f smooth preuni form structures on E & a lattice when endowed with the thinness relation >, where U1 > U2¢:~ U1 ~ U2 (in the sense o f inclusion o f L - fu z zy sets).

Proof . T h e p r o o f is s t ra ight forward , inher i ted f rom the inclusion re la t ion of L- fuzzy sets. It is also clear that the sup, inf, def ined f rom inclusion, give smoo th p r e u n i f o r m s t ructures when appl ied to smoo th p r e u n i f o r m structures.

Proposition 1.1.2. We note OR~ v °~2, O~ 1 ̂ O~ 2 the supremum, preuni form structures O~ 1 and OR2.

When OR1, OR2 are symmetrical then so are OR1 ̂ ore, OR1 v ~12. When OR1, OR2 are type D then so is OR1 ̂ ORe. When OR~, OR2 are type S then so is all1 v OR2.

respectively inf imum, o f the smooth

Proof. If OR1, ~12 are symmet r i ca l then for every L-fuzzy set u on E × E we have ORffu) = OR~(u ~) and OR2(u) = OR2(u-1). So we conclude tha t

(ORI V 0~2)(U ) = 0~I(U ) V 0~2(U ) = 0~I(U-1 ) V 0~2(U 1) ~_ (0~1 V 0~2)(U 1),

(0~,~ 1 A 0~2)(U ) ~- 0~I(U ) A 0~2(U ) = 0~I(U 1)A 0~2(U 1) = (0~1 A @/2)(u- l ) .

If OR~, ORe are type D then for every u, v in L e×F, we have

(oR, ^ OR2)(u N v) = OR,(u n v) ^ % ( u c~ v) = ORl(u) ^ ORffv) ^ % ( u ) ^ OR2(v)

= ( % ^ % ) ( u ) ^ ( % ^ % ) ( v ) .

If OR1, 0~2 a r e type S then for every u in L e×L" we can write

sup{(oR1 v ORz)(V) I v ® v = u} = sup{oRl(V) v °~2(v ) ] V @ V = U}

= sup{ORl(V) [ V @ V = U} V sup{OR2(V) I V @ v = U}

/> %(u) v %(u) = (% ,1%)(u).

A n o t h e r opera t ion , which will be used subsequent ly , will be no ted Q and is def ined as:

Definition 1.1.3. Let 0~1, 0~ 2 be s m o o t h p r e u n i f o r m s t ructures on E. We deno te OR~ C)°//2 the s t ructure def ined as ORj Q OR2(u) = sup{oRl(v) ^ ~//2(w) [ v fq w c u}.

Proposition 1.1.3. OR1 (3 °//2 is a smooth preuni form structure. We have OR~ (3 °R2 > OR~ v OR2. When OR1 and OR2 are symmetrical, type D, type S then so is O~ 1 (~ 0~2, respectively.

Proof. Let u be in L E×E, such tha t u :p D. T h e n for any v, w such that v N w c u at least one of t hem does not conta in D. H e n c e ORI(v) ̂ OR2(w) = 0L,, which p roves tha t OR1 Q OR2(u) = 0L,.


Le t u, v be in L E×E and u ~ v. T h e n

9/1 (3 9/2(u) = sup{9/~ (x) ^ 9/~2(y) [ x n y = u} and 9/1 (3 9/2(v) = sup{9/l(x) ^ 9/2(Y) [ x n y = v}.

But if x n y ~ v then x N y = u which p roves that 9/1 (3 9/2(u)/> 9/~ (3 °Re(v). F r o m

9/1 (3 0~2(U) ~--- sup{9/l(v) ^ 9/2(w) [ v n w ~ u}

i> sup{9/l(v) ^ 9/2(E X E ) [ v = u} = sup{9/l(V) [ V ~ U) = 9/1(U),

we can write 9/1 (3 9/2 ~ 9/1, 9/2, and 9/a (3 9/2 > 9/1 v 9/2. W h e n 9/1 and °?/2 are symmet r i ca l it is c lear that

9/1 (3 9 / 2 ( u - l ) = sup{9/l(X) A 9/2(Y) [ x n y ~ u 1}

= sup{9/l(x 1) ̂ 9/2(y 1) [ x -1 n y-1 ~ u} = 9/1 (3 9/2(U) •

Le t us suppose 9/a and 9/2 are type D. W e consider a = 9/~ (3 9/2(u) a n d / 3 = 9/~ (3 9/2(v), for any u and v. For any e > 0 , a rb i t ra ry small, which verifies e < - a ^ / 3 we can find x~, y~ such tha t 9/1(Xl) A 9/2(Yl) ~> of -- E, X 1 N ya = u, and x2, Y2 such that 9/1(X2) A 9/2(Y2) ~ / 3 -- 6, X 2 n 22 ~ P. But (x~ n x2) n (yl n Y2) ~ u n v, implies 9/1 (3 9/2(u n v ) /> 9/1(x~ n x2) ^ 9/2(Yl n Y2). 9/1 and 0~ being type D we have

9/1 Q) 9/2( u n v ) ~ 9/1(Xl) A 9/1(X2)/x. 9 /2(Yl) A 9/2(Y2) ~ (Of -- e)/x. (/3 -- e ) .

This p roves tha t 9/1 (3 9/2(u n v ) /> 9/1 (3 9/2(u) ^ 9/1 (3 ~ ( v ) .

1.2. The cuts and the representation o f smooth preuniform structures

W e now s tudy the cuts of a s m o o t h p r e u n i f o r m s t ructure 9/. Le t us no te

9/~ = [9/]~ = {u • L e×E ] 9/(u) 1> of} and 9/~+ = [9/]~ = {u E LE×E I 9/(u) > of}.

As we will see the second k ind of cut is m o r e interest ing because it enables the m a n y p roper t i e s of U to be conve r t ed in co r respond ing p roper t i e s for the cuts.

Proposition 1.2.1. The cuts 9/~ and °71+ of a smooth preuniform structure are L- fuzzy preuniform structures. When 9/ is respectively symmetrical, type D then so are the 9/~ for any of > O. When 9/ is respectively symmetrical, type D, type S then so are the °'11+ for any a. For every a >!/3, we have Us < U s and U+~ < U~.

Proof . W h e n a = 0 9/~ is the trivial s t ructure which is a fuzzy preuni formi ty . W h e n a > 0, if u • 9/~ then 9 / (u) /> a > 0 and so u ~ D (if u :~ D then 9/(u) = 0L'). Fo r any of, if u • 9/+ then 9/(u) > a / > 0 and so u ~ D. W h e n v • 9/~, resp. 9/+ and u ~ v, we have 9/(u) i> 9/(v) and so u • 9/~, resp. 9/~+. W h e n U is symmet r i ca l u ~ 9/~, resp. 9/~+, implies 9/(u) I> a , resp. 9/(u) > a, but 9/(u -1) = 9/(u)

implies that u -1 • 9/~, resp. 9/~+. W h e n 9/ is type D, then for every u, v • 9/~, resp. 9/+, we have 9 / ( u ) ~ >of , 9 / ( v ) ~ >of , and so

9 / (unv )>~9/ (u )^9 / (v )>~of , thus u n v • 9 / ~ , resp. 9 / ( u ) > o f , 9 / ( v ) > o f and 9 / ( u N v ) > ~ 9 / ( u ) ^ 9/(v) > of, which implies u n v • 9/+.

W h e n 91 is type S it is poss ible tha t 9/~ would not ver i fy the co r respond ing p roper ty . Le t us show that


it holds for ~+ . If u • ~/~+ then ~ ( u ) > a , bu t sup{°//(v) I v @ v = u}/> °//(u) > a and so there is a v such that v ® v ~ u and °//(v) > a , which p roves that o?/+ is type S. The last pa r t is trivial to verify.

O n the cont ra ry , for a r ep re sen ta t i on t heo rem, we would like to see if the ax ioms for s m o o t h p r e u n i f o r m s t ructures are effect ively satisfied when we star t f rom families of fuzzy p r e u n i f o r m s t ructures associa ted with every a in L ' . M o r e precisely:

Proposition 1.2.2. Let °V(a)l~a,\~o~, be a family o f L- fuzzy preuniform structures on E, such that if al ~ a2 then °U(al ) < °F(a2). Let all be the L' fuzzy set built by ~//(u) = sup{a I u • °V(a)}, when this set is non empty, and °ll(u) = Ot' else. °ll is a smooth preuniform structure and i f the °F(a) are symmetrical, type D, type S, then so is all (respectively).

Proof. I f u does not conta in D, then for every a u ~ 7/ ' (a) and so °?/(u) = 0L,. Le t be u ~ v. T h e n for every a, v E °U(a) implies u E 7r(a) , so sup{a ] u • ?/ ' (a)}/> sup{a [ v • 7/'(a)} implies ~ / (u) /> a//(v).

If the °U(a) are symmet r i ca l then f rom u E ~ ( a ) it fol lows that u ~ E ~ ( a ) and sup{a I u - I E c~(O~)} = 0~(U 1) ) ~/(U).

I f the °V(a) are type D then for every u, v and a we have that if u, v e °V(a) then u f9 v • °V(a). But {a [u n v E ~ V ( a ) } = { a ]u • ~ ( ~ ) and v e °V(a)} implies that ~ ( u N v ) ~ ° ~ ( u ) ^ ° l l ( v ) and so 0// is type D.

If the °V(a) are type S then ~ ( u ) = sup{~ [ u e °V(a)}, but u • °V(a) implies that there is v such that v • °V(a) and v Q v c u , so {a 13v, v Q v c u and v • ° V ( a ) } = { a [u e °V(a)}. But

sup{a ] 3v, v ® v = u and v • °V(a)} = sup{sup{a [ v • °V(a)} [ v ® v = u}

= sup{~(v) l v ® v =u}>~ U(u),

which concludes the proof .

U being a L ' fuzzy set, with L ' = [0, 1], we can s tate a r ep re sen ta t i on theorem:

Proposition 1.2.3. Let oR be a smooth preuniform structure and let U, and U + be the a cuts as defined precedingly. From the families o f fuzzy preuniform structures ~ and °ll + we build 7~ and °W2 by 74/'1(u) = sup{a 1 u E 0//,~} and °Wz(u ) = sup{a [ u E 0//+}. We can assert °W 1 = 7~U 2 = %

Proof. T h e p r o o f is trivial f rom the p reced ing results and the well known fact that

sup{a ] u ~ ~ } = sup{a I ~ ( u ) I> a} = °~(u), and sup{a ] u ~ U +} = sup{a I ~ ( u ) > a} = ~//(u).

1.3. Constructions o f smooth preuniform structures, the use o f basis

Le t us now exhibi t some s imple const ruct ions of s m o o t h p r e u n i f o r m structures . A t radi t ional cons t ruc t ion lies in the use of basis, such basis can also be def ined here .

Definition 1.3.1. Le t ~3 be a L ' fuzzy set on L E×E such tha t u :p D implies ~ ( u ) = 0L,, where D is still the d iagonal of E × E. The L ' fuzzy set ~ on L e×E def ined as ~//(u) = s u p { ~ ( v ) I u ~ v} is a s m o o t h p r e u n i f o r m s t ructure , and Y3 is said to be a basis for 0//.

W e mus t p rove that this cons t ruc t ion gives a s m o o t h p r e u n i f o r m s t ructure on E. Obv ious ly when u ~ b D we have u ~ v ~ v : p D ~ 3 ( v ) = 0 L , and so °?/(u)=0L,. Le t be u ~ v . T h e n °//(u) = sup{N(w) [ u = w}, ~ ( v ) = sup{N(w) ] v = w}, but {w [ u ~ w} ~ {w ] v ~ w} so ~ ( u ) / > ~ ( v ) .

Like in the t radi t ional case we can define the countabi l i ty p roper ty .


Definition 1.3.2. A smooth preuniform structure OR on E is said to be countable if it is generated by a countable basis 99 ( ~ being a fuzzy set, it is said countable if the set {v E LE×e[99(v)>O} is countable).

Proposition 1.3.1. Let ~ be a bas& for the smooth preuniform structure °R, which verifies sup{~(v) l v ® v = u} ~ 99(u), for any u, We can assert that OR is type S.

Proof.

sup{OR(v) [ v Q v c u}/> sup{~(v) ] v ® v = u}/> 99(u)

and

OR(u) = sup{~(x) Ix = u} ~< sup{sup{oR(x) ] v ® v c x } [ x = u} = sup{oR(v) [ v ®v c u},

which concludes the proof.

Proposition 1.3.2. Let 99 be a basis for OR which verifies for every u, v 99(u n v) >! ~3(u) ^ ~(v) , then OR is type D. Let ~3 be a basis for OR and let ~3" be defined from 99 by 99"(u)= sup{A~z 5~(vi) lOiEivi = u}, where I is any finite set. Then the smooth preuniform structure OR* generated from 5~* is type D. I f 5~ is countable then OR* is countable.

Proof. For every u and v we have

{99(w) I w = u n v} = {99(x n y) I x c u, y c v},

but 99(x D y)/> 99(x) A ~ ( y ) , SO take the supremum

OR(u n v) = s u p ( 9 9 ( w ) [ w = u n v} i> sup{99(x) A 99(y) [ X c u, y = v} ---- OR(U) A OR(V).

For the second part we only have to prove that 99* verifies the property ~*(u n v)/> ~* (u ) ^ ~*(v). Let be a=99*(u)=sup{ / \ i~199(v i ) ] (h iE lv i=u} and /3=99*@). Then for every e such that O < e ~ < a A / 3 , we can find finite families I0 and 11 such that Ai~toVi=U, f ~ v ~ = v , A i ~ l o ~ ( V i ) ~ Ol -- E, A i ~ l 2 ~ ( O i ) ~ /3 -- E and so Oi~t0ul, vi = u n v, AiEloUl199(vi) ~ ( ol A / 3 ) - - E. But this can be done for arbitrary small e, which proves that 99"(u rq v)/> a ^/3. If ~ is countable then 99* is also countable and so is OR*.

Another very practicable construction is to start from the set S of fuzzy preuniform structures on a set E. Let us consider a L ' fuzzy set F on S. We compute OR(u) = sup{F(oR) ] OR • S, u • OR}. When this set is non empty and we take OR(u) = 0L, else, we can assert:

Proposition 1.3.3. OR as defined precedingly & a smooth preuniform structure on E.

Proofi Let us consider u • L e×e such that u zp D. Then for every fuzzy preuniform structure OR we have u ~ OR and so OR(u)= 0L,.

Let us consider u ~ v in L exe. For every OR in S we have that if v•OR, then u•OR, so {F(OR) [ OR e S, v • OR} c {F(OR) ] OR • S, u • OR}, which shows that OR(u) t> OR(v).

1.4. Maps between smooth preumform spaces, initial and final structures

Let us now study maps from a smooth preuniform space into another.

Definition 1.4.1. Let (E, °R) and (F, ~ ) be smooth preuniform spaces and f a map from E into F. The map f i s said to be a smooth preuniform map if for every v in L F×F the inverse image ( f x f ) l(v) is such that OR((f x f ) l(v)) t> ~(v) .

We have the traditional results about the product of maps:

R. Badard et al. / Smooth preuniform and preproximity spaces t01

Proposition 1.4.1. Let (E, OR), (F, or) (G, °W) be smooth preuniform spaces and f, g be smooth preuniform maps f : E --> F and g : F ---> G. The map g of is a smooth preuniform map from E into G.

Proof. Let w be in L c × c and v = ( g x g ) l(w) and u = ( f x f ) ~(v). From the assumptions we have or(v) 1> °M#(w) and OR(u)/> or(v) which shows that OR(u) ~> °#2(w), where u is in fact (g of × g o f ) - t(w).

A natural way to endow a set with a smooth preuni form structure is to build it such that some maps are smooth preuniform maps. In fact we choose extremal preuniform structures, in particular the initial structure is defined as:

Definition 1.4.2. Let E, F be sets and °R a smooth preuniform structure on F. Let f be a map from E into F. We consider the set of smooth preuniform structures on E and select the coarsest for which f i s a smooth preuniform map. This structure is said to be the initial one.

This structure verifies:

Proposition 1.4.2. The initial structure or on E is generated by the basis N: N ( ( f × f ) ~(u)) = OR(u), where u is any element o f L Fxe, and we have or((f × f ) l(u)) = OR(u).

When OR is symmetrical, type D, type S, then so is the initial structure.

Proof. We have °V(v) = sup{~(w) I w c v} = sup{oR(u) I ( f x f ) - l ( u ) = v}. So, obviously f is a smooth preuniform map, because for every u in L F×F ~/~((f X f) I(U))/> 0~(U). In fact we have equality because °V((f x f ) l (u))/> ~ ( ( f x f ) - l ( u ) ) = OR(u). But let us suppose the inequality would be strict. Then we could find w = u ~ ( ( f x f ) - l ( w ) ) > U(u), but u = w implies OR(u) ~> OR(w) > OR(u), a contradiction.

Let us show that °7 is the coarsest structure for which f is a smooth preuniform map. We suppose on the contrary that there is °/4/" on E such that f is a smooth preuniform map and that ~ < °7 strictly. There would be vo with °W(v0)< °V(Vo), but °W(Vo)< °V(Vo) implies that there is uo such that ( f x f ) '(u0) c vo and ~V(Vo) < ~(Uo) = oR(Uo). But ~V((f X f ) - l (u0 ) ) ~< °/4/'(V0) because ( f X f ) I ( u o ) c v0, and °/4/'(v0)< oR(Uo) which shows that (°W((f xf ) - l (Uo))~< oR(u0), a contradiction.

When OR is symmetrical then obviously °7 is symmetrical. When OR is type D we can write Vu, v OR(u D v) = OR(u)^ OR(v). But let us consider ff = ( f x f ) l(u) and ~ = ( f x f ) l(v), ~ ( g ) = OR(u) and ~ ( ~ ) = OR(v). We have g D ~ ~ u D v = ( f x f ) - l ( u D v), so we conclude

or(a n ~) >/or(u n v) = oR(u n v) = oR(u) ^ oR(v) = ° r (a ) ^ o r ( J )

Let us consider the case where OR is type S, so for every u s u p { o R ( v ) [ v ® v c u } ~ o R ( u ) . Let us suppose that or is not type S. We could find w such that sup{or(v) Iv ® v c w} < or(w) and for every v such that v @ v c w we would have o r ( v ) < or(w). So we could find u such that g = ( f × f ) - ~ ( u ) c w and o r ( g ) > or(v) for every v such that v ® v c w . But {v [ v ® v c w } ~ { v [ v ® v c g } so

sup{or(v) [ v @ v c a} ~< sup{or(v) ] v @ v c w} and or(g > sup(or(v) [ v ® v c u).

Let us consider {x ] x @ x cu} . We then have

sup{oR(x) Ix ® x = u} i> OR(u) = or(g) > sup{or(v) I v ® v = w}

and x ® x c u implies Y@Y ~ u and so

sup{or(v) ] v ® v c u} ~> sup{oR(x) Ix ® x c u},

a contradiction.

Definition 1.4.3. Let (E, OR) be a smooth preuniform structure and F be a set. Let f be a map from E onto F. We define the final structure as the finest smooth preuniform structure on F for which the map f is a smooth preuniform map.


Proposition 1.4.3. The final structure 7# on F is generated by

7#(u) = sup{oR(v) I v = ( f x f ) - l ( u ) } = OR((f x f ) - l ( u ) ) .

When ~ is symmetrical type D, then so is 7#.

Proof. The proof of this proposition is very similar to the preceding one about the initial structure. Let us sketch the steps. With the structure 7# the map f is a smooth preuniform map. Let us consider

u in L FxF 7#(u) = sup{oR(v) I v c ( f / f ) - l ( u ) } . But v c ( f x f ) - l ( u ) implies OR(v) ~< OR(a), where t /= ( f x f ) - l ( u ) , and so 7#(u) = OR(a) and OR(a) I> 7#(u).

Let us now show that 7# is the finest smooth preuniform structure for which f is a smooth preuniform map. Let us suppose on the contrary that there is 74/" which was strictly finer than 7#. There would be u such that °W(u) > 7#(u) and OR(t/) i> °W(u) > 7#(u), a contradiction.

Let us now consider the case where OR is type D, and we consider u, v in L FxF. We have 7#(u) = OR(t/), 7#@) = OR(0), and 7#(u n v) = OR(u n v), where ~ is ( f Xf) - I (x ) . But u n v ~ ff n ~ and so OR(u n v)/> ~ ( t / n ~)/> OR(t/) ̂ OR(0), which shows that 7#(u Cl v)/> 7#(u)^ 7#@) and that 7# is also type D.

1.5. Products o f smooth preuniform spaces

We now deal with the construction of products of smooth preuniform spaces, as traditionally we would like the projections to be smooth preuniform maps. But the traditional approach which lies in taking the coarsest structure for which the projections are smooth preuniform maps, is not very interesting when the smooth preuniform spaces are not type D. In fact we will define:

Definition 1.5.1. Let (E, OR) (F, 7#) be smooth preuniform spaces. The product E x F can be endowed with the following smooth preuniform structures: - OR ® 7# generated from the basis: for every u in L ExE and v in L F x F we take

~ ( ( P l x p 0 - 1 ( u ) ) = OR(u) and ~ ( (P z Xp2)- l (v) ) = 7#(v),

where Pl and P2 are the projections on E and F. - OR x 7# generated from the basis: for every u in L exE and v in L F x F w e take

~ ( ( P l Xp1)-a(u) n (P2 × Pz)-I(v)) = OR(u) A 7#(V).

The first product a//® 7# is obviously a smooth preuniform structure for which the projections are smooth preuniform maps. But without the axiom for type D we do have not to consider the intersections of the inverse images of entourages from OR and 7#.

The second construction is closer to traditional definitions. Obviously projections are still smooth preuniform maps, but it is principally when OR and 7# are type D that this construction is justified.

In fact E x F can be endowed with a smooth preuniform structure 7#1 which is the coarsest such that the projection Pl is a smooth preuniform map. In the same way let 7/'2 be associated with the projection P2. 7#1 is the initial structure induced from Pl and 7#2 is the initial structure induced from P2- So we see that OR ® 7# is simply 7#~ v 7#2 and °R x 7# is 7#1 @ 7#2- So we can state:

Proposition 1.5.1. When all and 7# are respectively symmetrical type D, type S, then so is all X 7#. When °ll and 7# are smooth uniformities then so is OR x 7#.

Proof. The proof is clear from the preceding discussion. In fact we have just to use the preceding results about initial structures. Here , 7#1 and ~2 are initial structures on E x F relative to the maps Pl and P2, and so they are symmetrical, type D, type S, when OR and 7# have these properties. From preceding results on the operation E) and from the identity OR x 7#= 7/'1Q 7#2 we conclude the proposition.

R. Badard et aL / Smooth preuniform and preproximity spaces

2. S m o o t h preproximity spaces

103

Let us now recall what is a smooth preproximity space.

Def in i t ion 2.1. A relation 8 on L e (the fuzzy sets on E) is said to be a fuzzy preproximity on E if it satisfies (pl) 6(A, B ) ~ A ~0 and B ~0 , (p2) A NB ¢ 0 ~ 6 ( A , B).

The pair (E, 8) is said to be a fuzzy preproximity space. The preproximity may satisfy the following additional axioms and is said to be

- symmetrical when it verifies p3: 8(A, B) ~ 8(B, A), - type I when it verifies (p4): A ~ B and C ~ D and 8(B, D) ~ 8(A, C), - type D when it verifies (p5): 8(A U B, C)¢~ 6(A, C) or 8(B, C), - t y p e S when it verifies (p6): 6 ( A , B ) ~ 3 C such that 6(A, C) and 8(C, B), where C is the complementary of C in the sense of the decreasing involution r on L.

A fuzzy preproximity which verifies p3, p5 and p6 is a fuzzy proximity as defined by Katsaras [5]. In this case it satisfies also p4 because p3 and p5 implies p4. We can remark the very resemblance with the definition of classic proximity spaces.

A possible smoothing of this structure may be the following:

Def in i t ion 2.2. We will now consider 6 as a fuzzy relation on the set of fuzzy sets on E. 8 : L E × L E ~ L' , or 8 E L,L e X L E, which satisfies (P1) ~b(A)^ ~b(B) 1> 8(A, B), where ~b is a non-vacuity degree mapping L E into L'.

In fact 4~ yields a truth value which characterizes how a fuzzy set is far to be empty. Such a map must obviously verifies A~B~c~(A)>~qb(B), ~b(0)=0L,. We will only use ~b0 and oh1 defined by ~bo(A) = 1L, when A ~ 0 and 0L, else; and q~l(A) = sup{A(x) [ x ~ E}. (p2) 4~(A f) B) <~ 8(A, B).

The pair (E, 8) is said to be a smooth preproximity space. The preproximity will be said to be

- symmetrical when it verifies (P3): 8(A, B) = 8(B, A), - type I when it verifies (P4): C ~ A and D = B implies 8(C, D)>! 6(A, B), - type D when it verifies (P5): 8(A t3 B, C) = 8(A, C) v 8(B, C), - type S when it verifies (P6): for every A and B there is C such that 8(A, B) >! 8(A, C)v 6(C, B), where C is the complementary of C.

We compare preproximity by way of inclusion of the corresponding relations.

Def in i t ion 2.3. A preproximity (fuzzy or smooth) 81 is finer than 82, we will note 81 > 82, if and only if 81 ~82.

3. Links b e t w e e n s m o o t h preuni formity and s m o o t h preproximity

3.1. Construction from the cuts

Let E be a set and °R be a smooth preuniform structure on E. Let us consider for every a the set OR, = {u ~ LE×e[ OR(u)/> a}. When a > 0 ~//~ is a fuzzy preuniform structure, and the family OR~[~I0,~I verifies: if a />/3 then ORs c ORs (°Rs is finer than ORs). To every a we can associate with ORs a fuzzy preproximity 8, by taking 6,~(A, B) if and only if Vu E ORs : 5~(A, u) N ~(B, u) ~ 0, where ~(A, u) is the ball around A built from entourage u.


But a t>/3 implies OR~ = ORt~ and 6t~ is finer than ~ . This comes from

6tffA, B) ,~ Vu ~ ORt3 [~3(A, u) n ~3(B, u) # 01 ~ Vu E OR~ [~(A, u) n ~3(B, u) # 0] ¢~ 6~(A, B)

and so 6~ > 6~. For every A, B in L E we consider the sets DAB = {O~ ~ ]0, 1] I 6~(A, B)} and /)AB the complementary

of DAB. We have a ~ / 3 ~ ( / 3 ~ D A B ~ ~ DAB) and we will take $~(A, B)=sup£)AB (0 when /)aB = 0) and 61(A, B ) = v(6~(A, B)), r is the decreasing involution V(x)= 1--x . We first consider 6 ~ we define as

6'(A, B ) = T(~I(A, B)) = 1 -sup/SAB = 1 --infDAB,

where inf DAB is taken as 1 when DAn = O.

Proposition 3.1.1. The fuzzy relation 61 on L E × L E & a smooth preproximity (with the non-vacuity degree qbo, dpo(A ) = 1 when A # 0 and 0 when A = 0), which is symmetrical and type I. I f the smooth preuniform structure °R is type D then so is 6J.

Proof. Let us consider A = 0. We have Va > 0 6~(0, B) and so bob = ]0, 1] implies 61(0, B) = 1 and 61(0, B) = 0. A = 0 or B = 0 implies 61(A, B) = 0 and 4~o(A) ̂ 00(B) i> 61(A, B).

When A n B # 0 we have for every a > 0 6~(A, B), so /)AB = 0 implies 61(A, B) = 0 which implies 61(A, B) = 1 and we conclude that Cko(A N B) <- 6~(A, B).

Let us consider C ~ A and D ~ B. For every a > 0 if 6~(A, B) then 6~(C, D) (the fuzzy preproximity generated from a fuzzy preuniform structure are type I, see Theorem 5.1 of [6]), so if DAB ~ DCD then DAB ~ DCD SO 61(A, B) >i 6~(C, D) and 61(A, B) <~ 6~( C, D ).

Symmetry is obvious by construction because N(A, u) N N(B, u) = Y3(B, u) n N(A, u). When OR is type D the cuts OR~ are type D, because OR(uNv)>~OR(u)^OR(v) and u e OR~, v e OR~ implies OR(u n v)/> a so u n v E OR~. The associated 6~ are also type D (Theorem 5.1 of [6]), so for every a > 0 we have 6~(A U B, C) iff 6~(A, C) or 6~(B, C), and so DAUB, C = DAC U DBC, from which we conclude that 61(A U B, C)= 61(A, C ) v 6~(B, C).

3.2. Construction with the use of a non-vacuity degree

We will now generalize the construction of a smooth preproximity from a smooth preuniformity by considering a 'smoothing', a soft version, of the predicate relative to non-vacuity. In fact we will consider the non-vacuity degree Ckl:LE---~L ', which assign to a fuzzy set a measure of how it is non-empty. We will have to replace ~(A, u) n ~(B, u) # 0 by the degree 4h(~(A, u) n ~ (B , u)), (with 4h defined as ~bl(A)= sup{A(x) lx E E}). When we use a fuzzy preuniform structure we can do a construction like in Theorem 4.1 of [3], but here the entourages belong more or less to OR. The classic construction of a fuzzy preproximity from a fuzzy preuniformity starts from

6(A,B) ¢::> Vu [uEOR ~ ~ ( A , u ) n ~ ( B , u ) # O ] ,

or equivalently

6 ( A , B ) ¢:~ 3u [u e oR and ~ (A, u) n ~ 3 ( B , u ) = 0 ] .

This last formula can be generalized to a smooth structure as

62(A, B) = sup[oR(u)^ [1 - ~bl(O3(A, u) n ~(B, u))]], and 62(A, B) = 1 - 62(A, B), t l

in which we have associated inf with the conjunction and v(x) = 1 - x with the negation and the sup with the existential quantifier (which is a form of disjunction). With these choices, if we want to keep the equivalence between A ~ B , not (A) or B, not (A and not (B)), we must take 1 - ( a ^ ( 1 - b ) ) = (1 - a ) v b as the truth value of A ~ B, where a, b are the truth values of A and B.

R. Badard et al. / Smooth preuniform and preproximity spaces

So starting from the first formula to generalize 6 (A, B) 63(A, B) = inf,[(1 - ~ ( u ) ) v ~ b f f ~ ( A , u) n ~ (B , u))]. We now verify that these identical.

105

we would get" two smoothings are

Propos i t ion 3.2.1. 6 2 is a smooth preproximity which is symmetrical and type I. When ~)l is type D then so is 6 ~. 6 2 and 6 ~ are identical.

Proof . Our interpretation of connectives keeps the De Morgan laws. We can write

1 - 62(A, B) = 1 - sup [ag(u)/x (1 - ~bl(~(A, u) n ~ (B , u))] tg

= inf[1 - (°g(u) ^ ( l - &,(~(A, u) N ~ (B , u))))] 11

= inf [(1 - ~ (u ) ) v (1 - (1 - 4,1(~(A, u) n Yd(B, u))))] t l

= inf [(1 - °g(u)) v &I(~(A, u) n Yd(B, u))] = 63(A, B). I I

Let us now show that ~2 is a smooth preproximity.

62(A, B ) = inf [(1 - ~ / ( u ) ) v & , ( ~ ( A , u ) n ~ ( B , u))] , I t

but ~(A, u) ~ A and ~(B, u) ~ B and so ~b,(~(A, u) n ~(B, u)) ~ &I(A N B). This enables us to write

62(A, B) >t inf [(1 - ~ (u ) ) v &I(A N B)] > 4,l(A n B). i t

From

~(A, u) _-v ~(A, u) n @(B, u) and ~ (B , u) ~ ~(A, u) n ~ (B , u)

we have

&I(~(A, u) n ~(B, u)) ~< 61(~(A, u))A 6 , ( ~ ( B , u)).

62(A, B) <~ (1 - ~//(u)) v 61(~(A, u) n B(B, u)) <~ (1 - °//(u)) v &,(~(A, u)) A 6 , ( ~ ( B , u)),

this for any u. Let us consider uo = E × E. Then ~(Uo) = 1 and 32(A, B) ~< ¢h~(~(A, u)) A &I(~(B, u)), but from the construction of ~ 4 ,1 (~(A ,u ) )=(a~(A) , and so 62(A,B)<~dpl(A)AckI(B). These properties show that 62 is a smooth preproximity.

Symmetry is obvious from commutativity of N. Let us now consider 62(A, B) and C ~ A and D ~ B . Then

62(C, D ) = inf [(1 - °//(u))v 6,(~3(C, u ) n ~ (D , u))], u

but ~(C, u) ~ B(A, u) and ~ ( D , u) ~ ~ (B , u), which shows that &I(~(C, u) n ~ (D , u)) > ~I(~(A, a) n ~(B, u)) and so 62(C, D) >~ 62(A, B).

We now suppose that ~ is type D. We know that 62 is type I, so we have

and

A O B ~ A ~ 6Z(A O B, C) >I 62(A, C)

A U B ~ B 0 62(A 0 B, C) >~ 82(B, C), ~2(A U B, C) >I 82(A, C) v 32(B, C).

We note 32(A O B, C) = a, so for every u (1 - °R(u)) v &I(N(A, u) N N(B, u)) >1 a. Let us suppose 62(A, C) V 62(B, C) < oz. There would be u and v such that

(1 - ~ (u ) ) v 4~,(Y3(A, u) N N(B, u)) < oz and (1 - ~/(v)) v ~b,(N(A, v) N ~(B, v)) < oz.


We consider w = u n v. We have

~ I ( ~ (A , w) n ~ (B , w)) ~< (~I(~(A, u) n ~(B, u)), ~bl(~(A, v) n ~ (B , v)).

But ~ (w) /> °//(u) ^ °//(v) implies 1 - °//(w) ~< 1 - ~//(u), 1 - ~(v) , so

(1 - °//(w)) v (hl(~(A, w) N ~ ( B , w)) ~< (1 - ~//(u) v (~I(~(A, u) n ~ (B , u))

and the same inequality holds for v, which shows that

62(A U B, C) ~< (1 - ~U(w)) v 61(~(A, w) n ~(B, w)) < c~,

a contradiction.

It is interesting to relate this construction with the first one, in which we started from the cuts ~ .

Proposit ion 3.2.2. The smooth preproximity 6 2 c a n be constructed from the cuts ~ , a > O, o f the preuniformity by use o f the equality

62(A, B) = sup [6(.)(A, B ) ^ a], oL:>0

where 6(.)(A, B) is defined f rom ~ll. by

6(.)(A, B) = inf{¢(g3(A, u) n ~ ( S , u)) l u E ~.} ,

ch is ehl and 6(.)(A, B) = 1 - 6(.)(A, B ).

Proof.

62(A, B) = sup [°//(u) ^ (1 - (~I(~(A, u) n ~ (B , u)))]. u

But let A(x) , B (x ) be maps from a set X into [0, 1] and let us consider s u p { A ( x ) ^ B ( x ) l x ~ X}. We have

s u p { A ( x ) ^ B ( x ) I x E X} = sup s u p { A ( x ) ^ B ( x ) [A(x)/> a, x E X}. a c [ 0 , 1 ]

But if for some y in X we have A(y )> i a, then we can write A ( y ) ^ B ( y ) > 1 a ^ B ( y ) and

sup{A(x) ^ B(x) [ A (x ) >t a}/> sup{B(x) ^ a [ A(x ) >1 a}.

But for every x there is a such that A(x ) >t a, which proves that taking the global supremum we have

sup sup{A(x) ^ B ( x ) [A(x)/> a} = sup sup{B(x) ^ a [A(x) i> a}. ~ c[o,1] ~[0,11

Applying this equality to the formula giving ~2(A, B) we obtain

62(A, B) = sup[~(u) ^ (1 - ~bl(~(A , u) n 93(B, u)))] u

= sup[sup{(1 - ch~(O3(A, u) n ~ (B , u))) ^ a I U(u) >1 a}] a ~ > 0

= sup[(1 - inf{¢l(~(A, u) n 9¢(B, u)) I u E ~ .}) ^ a] = sup[(~(.)(A, B) ^ ol]. a > O a ~ ' O

Corollary 3.2.1. When we take ¢ = 4>o, with C~o(A ) = 1 i r A # 0 and 0 if A = O, in the construction o f 6 2, we get the structure 61 we have defined f rom the cuts.


Proof. The proof of the preceding proposition shows that in this case we still have

~ 2 ( a , B) = sup[6(~)(a, B) ^ a],

but

6(~)(A, B) = inf{~b0(~(A, u) n ~(B, u)) [u • 0//~}

is equal to 1 when ~(A, u) n ~(B, u) ~ t3 for every u in ~ , and 0 else. So $(~)(A, B) is 1 when there is u in ~/~ such that ~(A, u) n ~(B, u) = 13, and 0 else. g(~)(A,B)^o~ is ~ when $~(A, B) and 0 else. In this case we have

sup[g(~)(A, B) A a] = sup{ol I ol > 0 and g~(A, B)}. c~>[)

It is just sup/SAB we used in the definition of 61.

4. Conclusion

In this paper we have presented smooth extensions of preuniform and preproximity structures. These structures are obtained by weakening axioms of the corresponding classic mathematical structures. This weakening consists in the embedding of the structure in a multiple valued logic context. In this way axioms are converted in system of inequalities between truth values.

These structures, we qualify with smooth, are deeper than classic fuzzy extensions which generally lie on slight adaptations in order to work with fuzzy sets. This approach has more fundamental basis: Human beings are unable to have absolute feeling but are very sensible to differences. Any person is very efficient to compare sounds, co lours , . . , but only few persons have an absolute hear (can recognize a note in an absolute manner). For this reason we think that our 'intern logic' is basically a logic of monotony. In this logic, assertions must be interpreted as monotonic relations between truth values. For instance A implies B must be interpreted as: the more property A is satisfied then the more B is.

References

[1] R. Badard, Fuzzy preuniform structures and the structures they induce Part 1: Main results, J. Math. Anal. Appl. 100 (2) (1984).

[2] R. Badard, Smooth axiomatics, First IFSA Congress, Palma de Mallorca, July 1986. [3] R. Badard, A.S. Mashhour and A.A. Ramadan, Smooth preproximity spaces, to appear in Fuzzy Sets and Systems. [4] C.L Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968). [5] A.K. Katsaras, On fuzzy proximity spaces, J. Math. Ana l Appl. 68 (1979). [6] A.S. Mashhour, R. Badard, A.A. Ramadan, Fuzzy preproximity, to appear in Fuzzy Sets and Systems.

smooth preuniform and preproximity spaces

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