AbstractIt is shown here how the Galois connections theory provides a canonical procedure to fuzzify a crisp closure space. Roughly speaking, if S is a space equipped with an ∩-closed set F of subsets (the closed subsets), the closed fuzzy subsets of S ⪡are⪢ the Galois connections between F and the valuation set. The closed crisp subsets are particular closed fuzzy subsets

Achache, A

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 130, 538-544 (1988) 
How to Fuzzify a Closure Space 
A. ACHACHE 
140, Rue Dedieu, 69100 Villeurhannr, France 
Submitted by L. A. Zadeh 
Received May 17, 1985 
It is shown here how the Galois connections theory provides a canonical 
procedure to fuzzify a crisp closure space. Roughly speaking, if S is a space equip- 
ped with an n-closed set 3 of subsets (the closed subsets), the closed fuzzy subsets 
of S ((are)) the Galois connections between .!F and the valuation set. The closed 
crisp subsets are particular closed fuzzy subsets. 7’ 1988 Academic Press, Inc. 
Let us recall what a Galois connection is. A Galois connection between 
the ordered sets P and Q is a couple of mappings P c’:Q (or 
(f, g) E Qp x PQ) such that f and g are decreasing, 1, <gof and 1, <fog. 
Let us denote by T(P, Q) the set of such Galois connections. This set can 
clearly be cartesianily ordered. 
When P and Q are complete lattices, it is known that r(P, Q) is itself a 
complete lattice. In this case, it is also known that, whenever 
(f, g)ET(P, Q) and A c P, f(VA) =AatA f(u). But we have better. Let us 
define 
A(P, Q) = h E Q’/t’A c P, h(VA)= A h(a) . 
i UEA I 
The mapping (f, g) t+ f from lJP, Q) toward d(P, Q) is an isomorphism 
(of complete lattices). (See [I or 31). 
RESULTS 
In the whole following text, S is a set, equipped with a closure structure: 
9 is an n-closed set of subsets of S. The associated closure F: 2’-+ 9 is 
defined by F(X) = n { YE P/Xc Y}( = x). 
Let L be a complete lattice (with 0 < 1). The fuzzy subsets of S are (by 
definition) the members of LS. 
538 
0022-247X/88 $3.00 
Copyright f.) 1988 by Academtc Press, Inc 
All rights of reproduction in any form reserved. 
HOW TO FUZZIFY A CLOSURE SPACE 539 
THEOREM 1. The complete lattice T(L, F) is isomorphic to: 
9={,ELs/vxCS, A x(x)= /j a(x)}. 
YEX reR 
More precisely, we have the following reciprocal isomorphisms (of complete 
lattices): 
with 
and 
0, = (6 j), i(t)=cr-‘( tt), j(X)= /j a(x). 
YEX 
Proof. Let us first prove that u and v are well defined. We shall denote 
by V the supremum in the complete lattice 9. Assume first that (A g) is in 
r( L, F). We have to prove that u~,( = cp) is in 3. For each subset X of S we 
have 
I=,( u {xI)=F(U~({x}))=VF({x~). 
XGX 
We deduce g(X)=g(V F( {x})) = Ag(F({x})) = A q(x). But then also 
g(X) = g(F(X)) = l\rsfi-CxI q(x). Hence uf,n is in 9. Assume now that c1 is in 
8. We have to prove that i(t) is a member of 9 and that (i, j) is Galois. 
For the first point, let z be a member of i(t): since 
i(t)= (xEE/cI(x)> t} (with t E L), 
we get 
and z is a member of i(t). The second point is easily checked. 
Since u and v are obviously increasing, we just have to prove now that u 
and v are reciprocal bijections. For each member x of S, 
409'130%16 
540 A.ACHACHE 
In order to finish, let us prove that, for each member (f, g) of T(L, F), 
V u/,n = (f; g). Let us put for the moment u,,,* = (i,j). We get (by using 
successively the definitions of u and u): 
Then .f = i. Now, for XE 9, 
Note that, whenever CI is a member of 93, LX -‘(tt) is a member of 9 for 
every t in L. Thus, each level of a (i.e., each a-‘(tt)) is closed. We now 
extend the notion of closed subset by the following definition. 
DEFINITION 2. A fuzzy subset a of S is said F-closed (or closed) if and 
only if 
vxcs /j CL(x)= /j r(x). 
TEX rtR 
So, the fuzzy subsets of S are the members of Y. In order to justify the 
terminology of Definition 2, we have to prove that Y is /l-closed. Then, we 
shall make the link precise between the closed subsets and the fuzzy closed 
ones and we shall prove that the closure associated to 3 is a prolongation 
of F. 
THEOREM 3. (1) The set 9 is an A-closed subset of Ls. Let us denote by 
G: Ls + Y the associated closure. 
(2) 9n2’=9. 
(3) For every member CI qf 2’, F(a) = G(U). 
HOWTOFUZZIFYA CLOSURE SPACE 541 
ProoJ (1) We have to prove that, for each m c 8, the fuzzy subset 
y = Am is a member of 9. For a given subset X of S, 
(2) We identify a member M of 2’ to the crisp A = a ~ ‘( 1). Such a 
crisp subset a belongs to 9 if and only if, for each XC S, 
,?, a(x) = 1 * /j a(x) = 1. 
.Y E R 
In other words, a is in 9 if and only if XC A *XC A. And this is 
equivalent to A E F. 
(3) We still identify a to A. Let us identify A = B to the associated 
characteristic function /3 (>a). By (2), we know that b is in $9. By the 
definition of G, 
Hence 
G(a)=/\ {yE%/a<y). 
a < G(a) (=cp)G.. 
For each UCA, 1 = a(u) < q(a), then q(a) = 1. For each XE S\A, 
cp(x)bfl(x)=O, then cp(x)=O. Finally 1 =/jUtA cp(a)=&.,-q(a), and 
C&X) = 1 for each x E 2. Then rp = /I. 1 
EXAMPLE 1. Initial subsets of an ordered set. A subset X of the ordered 
set S is said to be initial if and only if 
The set 9 of the initial subsets of S is obviously n-closed. Which are the 
fuzzy closed subsets of S? Assume a is such a subset. Then 
Then y d x implies a(x) < a(y) and a is decreasing. 
542 A.ACHACHE 
Conversely, assume that u is a decreasing mapping from S toward L. 
Whenever y<z~X, we can deduce A,.Xa(x)<a(z)<a(y) and then 
A rEX~(~)<</\),ER a(y). Finally a is in Y. 
So the fuzzy closed subsets are exactly the decreasing mappings. 
EXAMPLE 2. Subgroupoids of a groupoid. A groupoid is a set uniquely 
equipped with a binary operation. A subgroupoid of a given groupoid is, 
by definition, a subset which is closed by that operation. 
THEOREM 4. Let (S, 9) be the closure space constituted by the complete 
lattice of the subgroupoids of a groupoid S. A .fuzzy subset (p(&L’) is closed if 
and only if, whenever x and y are in S: 
Proof: Assume that cp is closed. Let P be the subset (of S) the only 
elements of which are x and y. By Definition 2, we can write 
(W)) * (cp(v))= A dz). 
:EP 
Since xy is in P, we get (q(x)) A (tp( y)) < cp(xy). 
Conversely, assume that cp satisfies (for all x, y) the inequality of the 
theorem. Let X be an arbitrary (crisp) subset of S. If z is in X, it must be 
(at least in one manner) composed of a finite (nonnull) number of elements 
of X: let A( cX) be the considered finite set. By using finitely many of the 
inequality of the text of the theorem we get 
CP(Z)~ A da) 3 A v(x). 
USA x E x 
We deduce AZER q(z) >/jXEX q(x). Hence 40 is closed. 4 
These fuzzy closed subsets can be named the fuzzy subgroupoids of S. By 
the same procedure, we can define, for instance, the fuzzy subgroups of a 
group, and we find again the definitions of [4]. 
EXAMPLE 3. 9=2T It is clear that 59 = L’: F and G are the “identity” 
closures. Then, we have the reciprocal isomorphisms p and q and the 
reciprocal isomorphisms u and v: 
A(L, 2’) ; T(L, 2’) : Ls, 
Y I, 
HOWTOFUZZIFY A CLOSURESPACE 543 
where u and u are given in Theorem 1 and where p and q are deduced (for 
instance) of [ 1 ] : 
P(f) = (A 8) with g(X)= V t 
XCf(l) 
4t.L 8) =f: 
Let us compute the reciprocal isomorphisms up and qu: 
qx)=!F({x))= v t= v t F(lxl)cf(r) .XE/(O 
q”@(t) = i(r) = a-‘(p). 
So, we find again the representation theorem of Negoita and Ralescu [2]. 
EXAMPLE 4. 9 = 2’ and L = 2’ (T is a set). By defining again u and v 
as in Theorem 1 and by defining the reciprocal isomorphisms a and b by 
a(y) = {(x, y) E S x T/y E y(x)} we get the compositions: 
u 0 
I-(2T, 2s) 2 (27s 2 2sx7. 
” b 
Then, whenever p is a member of 2’” ‘, ub, = (i, j) is the Galois connection 
2’ 2 i 2’ defined as in Theorem 1. More precisely, we get 
J 
i(Y)=b;‘(fY)= {x~S/Ycb,(x)} 
= {XEWYE KY~~,W} 
= {-=WYE y, (x,Y)Ep}; 
AN= n b,(x)= (Y~~x~~,Y~~x)} 
x.5X 
= {YE TPxEX, (x,y)q+ 
Then ub, is the classical Galois connection of the binary relation p. We find 
again the fact that each binary relation can be identified with a Galois con- 
nection (see [3]). 
OPEN PROBLEMS 
(1) For a given 9, what are the m ( c LE) verifying the conditions (l), 
(2), and (3) of Theorem 3 (where Y would be replaced by m)? 
544 
(2) Let us define: 
A.ACHACHE 
F = { 9 c 2’/9 (“j-closed ) 
C = (27 c L’jW A-closed}. 
Study the mapping 
F-+C defined by 9 F-+ $9’. 
CONCLUSION 
It seems that the proposed procedure is the best one to fuzzify a 
mathematical structure. It can easily be used to “soften” such things as 
convexity structures or topological spaces. 
REFERENCES 
1. A. ACHACHE, Galois connexions of a fuzzy subset, Fuzzy Sers and Sysrems 8 (1982), 
215-218. 
2. C. V. NEGOITA AND D. A. RALESCU, “Applications of Fuzzy Sets to Systems Analysis,” 
Birkhaiiser, Basel/Stutgart, 1975. 
3. Z. SHMUELY, The structure of Galois connexions, Pacific J. Math. 54 (1974), 209-225. 
4. A. ROSENFELD, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512-517. 


How to fuzzify a closure space 

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