Fuzzy Sets and Systems 38 (1990) 365373 NorthHolland
FUZZY
LINEAR
AND
365
AFFINE
SPACES
Godfrey C. M U G A N D A Department of MathematicalSciences, Memphis State University, Memphis, TN 38152, U.S.A. Received September 1988 Revised February 1989
Abstract: Elementary properties of fuzzy linear and atiine spaces are studied. The concept of a basis for a fuzzy linear space is defined, and an algorithm for computing a basis when the underlying space is finite dimensional is given. Keywords: Fuzzy linear space; fuzzy attine space; basis.
1. Introduction The study of fuzzy sets has proceeded in many different directions, since its inception in [10]. Fuzzy linear spaces have been studied by Nanda [7], Katsaras [1, 2], Katsaras and Liu [3], and Lowen [6]. This paper is a 'fuzzy point' approach to the study of fuzzy linear spaces, and is in part, a response to Nanda's appeal in [7] for a development of a theory of fuzzy linear algebra.
2. Preliminaries and notation denotes the set of positive integers, R the set of real numbers, and I c R is the unit interval, i.e. I  [0, 1]. Let X be a set. A fuzzy set s in X is a map s : X* I. When s is a fuzzy set in X, we will refer to X as the host space of s. T h r o u g h o u t this paper, X will be the host of all fuzzy sets considered, unless otherwise specified. In general, host spaces of fuzzy sets will not be mentioned if they are clear from the context. W e also adopt the convention that for any function g:X~l, sup{g(x) l x ~ O } = O and inf{g(x) Ix • 0} = 1. If Y is a set, and f is a function mapping X into Y, then for any fuzzy set s in X, we define the image f ( s ) of s under f to be the fuzzy set in Y defined by
f ( s ) ( y ) = sup{s(x) Ix e f  l ( y ) }
(2.1)
for each y ~ Y. The intersection and union of a family of fuzzy sets {s}~, are defined to be the infimum, inf~ s,~, and the supremum, sup~ s~ respectively. For two fuzzy sets s~ and s2, we say s~ is a subset of s2, and write Sl<~S:, if s~(x)<s2(x) for all x e X . If X1, X2 are sets, and sl, s2 are fuzzy sets in X~, X2 respectively, then the Cartesian product of s~, s2, denoted by Sl x s2, is the fuzzy 01650114/90/$03.50 © 1990ElsevierScience Publishers B.V. (NorthHolland)
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set in XI x X2 defined by
sl x SE(Xl, x2) = min{sl(x0, s2(x2)},
(2.2)
where xi • Xi, (We omit the extra pair of parentheses to enhance readability.) A fuzzy point in X is a fuzzy set s, for which there exists a point x • X such that s(y) = 0 for all y ~ x . We denote such a fuzzy point by xx, where ~. = s(x). Thus we have:
Xz(Z)
j2 [0
if x = z, ifx=/=z,
for all z ~ X. An ordinary subset of a set X will sometimes be called a crisp set for emphasis. When it suits our needs, we will regard crisp subsetf, of X as fuzzy sets in X by identifying them with their characteristic functions. For a fuzzy point xx and a fuzzy set s, we write xx • s instead of xx ~ s. In the sequel, X denotes a vector space over R.
3. Vector space operations Let A d d : X x X   > X , M u l t : R x X   ~ X denote vector addition, and scalar multiplication, respectively. Let s, Sl, s2 be fuzzy sets, and let p • R. The sum Sl + s2 and the scalar multiple ps are defined by Sl + s2 = Add(s1 x s2),
(3.1)
ps = Mult({p} x s).
(3.2)
Using (2.1) and (2.2), it can be shown that for any z • X,
f s ( z / p ) P s ( z ) = ~ o U p { s ( x ) [x • X }
if p :/: 0, ififp=0andz=0'p=0andz:/=0.
Further elementary properties of these definitions may be found in [3].
Proposition 3.1. Let x~, ya be f u z z y points, s a f u z z y set, and let p • R. Then (a) x~ + y/~ = (x + Y)min{~,/J), (b) px~ = (px)~, and (c) for all z ~ X, (x~ + s)(z) = min{tr, s(z  x)}. Proof. Elementary, using (3.1) and (3.2).
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4. Fuzzy linear and aline spaces Definition 4.1. A fuzzy set s is called a linear space if for all x, y • X, and for all p, Q • R, s(px + Qy) >I min{s(x), s(y)}.
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Definition 4.2. A fuzzy set s is called an afJine space if for all x, y e X, and for all P, 0 E R satisfying p + p = 1, s(px + Oy) >1min{s(x), s(y)}.
Proposition 4.3. Let s be a f u z z y set. Then (a) s is a linear space if and only if for all p, Q ~ R, and for all f u z z y points xo~ e s, Yt~ E s, px~ + OY~ ~ s, (b) s is an aflfine space if and only if for all p e R, and for all f u z z y points x~ e s, yt~ e s, px~ + ( 1  p )yo e s. Proof. Elementary.
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We now give some simple, but useful, properties of fuzzy linear spaces. Proposition 4.4. Let s be a f u z z y linear space. Then (a) p E R , p 4=O~ s(px) = s(x) for all x E X, (b) for all x, y E X, s(x) 4=s(y) ~ s(x + y) = min{s(x), s(y)}, (c) s(O) >/s(x) for all x ~ X. Proof. To prove (c), note that for any x e X , s(0)=s(0x+0.x)/> min{s(x), s(x)} = s(x). To prove (a), let/9 e R, p 4: 0, and x e X. Then s(px) = s(px + 0)/> min{s(x), s(0)} = s(x), hence s(px) >i s(x) for every p ~ 0. But then, s(x) = s ( ( p x ) / p ) >i s(px). This proves (a). To prove (b), assume s ( y ) > s(x). Then s(x) = s(x + y  y) /> min{s(x + y), s (  y ) } = min{s(x + y), s(y)} /> min{s(x), s(y)} = s(x), from which it follows that min{s(x + y ) , s(y)}. Since s ( y ) > s ( x ) ,
(b)
this proves
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Delinitioa 4.5. The linear hull of a fuzzy set s (denoted by *~s) is the intersection of all fuzzy linear spaces l such that 1/ s. The attine hull of s (denoted by ~ts) is the intersection of all fuzzy atiine spaces a such that a/> s. The linear and attine hulls of a fuzzy set s can easily be characterized in terms of fuzzy points: ~, = sup
O,(Xi),(~,) I xj e X, p j e B~, j = 1. . . . .
n~l~ t.i= 1
}
n ,
(4. I)
which leads to the characterization •(x) = supt min s(xi) lx, c X , x = ~ Pig, p i e R 1 n~~Ll<i~n
by an easy application of Proposition 3.1.
i=1
9
(4.2)
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Similarly, it can be shown that the atiine hull of s can be defined by ~t~ = sup pi(xi),(x,)I Xi e X , P i e N, n~Nti=l
Pi = 1
(4.3)
i=1
or by Ms(x) = supt min s(xi) IxieX, x = ~ pixi, p i e R , ~ pl = 1}. n~l~tl<~i<<n i=1 i=1
(4.4)
For any fuzzy point xx, and fuzzy set s in X, the translate of s by xx is the fuzzy set xx + s. When X is finite dimensional, Lowen [6] has shown that every fuzzy affine space is the translation of some fuzzy linear space, by some fuzzy point. In infinite dimensions, there exist fuzzy atiine spaces that are not the translate of any fuzzy linear space.
Example 4.6. Let X = R ~, the vector space of all real sequences. For each i e [~, let ei e X be the sequence ei(]) = {10 if i = j , if i :/:], and let s : X~ I be defined by
s(x)={lo1/i
i=eiotherwise. fx for some i,
Then ~t~, the affine hull of s, does not attain a maximum on X, since each fuzzy linear space attains its maximum (Proposition 4.4). ~s can not be the translate of any fuzzy linear space.
5. Linear independence and bases Let Xp denote the (crisp) collection of all fuzzy points in X, i.e. Xp = {xx I x • X, ~, e I). We use 2 x to denote the power set of Xp. If S ~_Xp, by the foot of S, denoted foot(S), we mean the set {x ] xx e S}. Definition 5.1. Let s be a fuzzy linear space. A map B:I~2 xp is called Gadmissible for s if (i) 0,(0~ • B(s(O)), (ii) xx • B ( # ) ~ Z = s(x) = I~, (iii) foot({xx I x :/: 0, xx • U~,~t B(/z)}) is a linearly independent set, and (iv) if x is a linear combination of points in foot(Ux>0 B(A)), then it is a linear combination of points in foot(Ux_>_>s~x~B(A)).
Example 5.2. Let X
= ~ 2 , and s be the fuzzy linear space
{;
s(xl, x2) 
ifxl=x2, otherwise.
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If B is defined by B(3.)
J'{(1, 1)1, (0,
[0
0)1 }
if 3.= 1, otherwise,
Then B is Gadmissible for s. For a fuzzy linear space s, the class qgs of setvalued maps that are Gadmissible for s is clearly nonempty. We define a partial order _< on ~¢s as follows. For any 81, BE, E c~s, B 1  2xp that is Gadmissible for a f u z z y linear space s is maximal in qgs if and only if for each x* • X, x* lies in the linear hull of the foot of
B ( 3.). Proof. ( ~ ) By way of contradiction, suppose there exists x* • X, with x* lying outside the linear hull of the foot of Ux~,(x*)B(3.). It follows that (X*)o, [._Jx~>~(x*)B(3.) for any oc, and, by the admissibility of B, (x*)~ ~ [,.Jx~oB(3.) for any a~. Now define B* : I> 2x by ~B(3.) B*(A) = [B(s(x*)) U {x*(x.)}
if 3. #=s(x*), if 3. =s(x*).
Then B _< B*, B ~ B*. Since B* is clearly admissible, B can not be maximal, a contradiction. ( ~ ) We prove the contrapositive: let B be a map admissible for s, and suppose B is not maximal. Then there exists x* • X, and a map B* admissible for s, such that x* :/:0, B _< B*, and (x*)s(x.) • B*(s(x*))\B(s(x*)). We claim x* does not lie in the linear hull of the foot of [..Jx~,~.)B(Z): for if it does, then there exists n • N , such that x* = ~i=lP~i n x for some p i • l ~ , x i e X , and pi:/:O for i = 1 , . . . , n, and such that (xi)~(x,)• B(s(x~))~_ B*(s(x~)). It then follows that the foot of
{x lx o is not linearly independent, contradicting the admissibility of B*.
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By the linear hull of a map B : I> 2x,, denoted by ~B, we mean the linear hull of the fuzzy set sup{xx I xx • U~,~ B(/~)}. Theorem 5.4. Let s be a f u z z y linear space, and let B : I> 2x~ be admissible for s. Then ~B = s if B is maximal in ~ . Proof. By (4.1) or (4.2), SgB ~
(5.1)
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and
xiE foot(x~[7~<~) B(~.)).
(5.2)
By (5.2), and the admissibility of B, s(xl) >! s(x) for each i = 1 , . . . , n. Since s is a subspace, (5.1) implies that s(x)=minl<~<~ns(xl). We therefore have x~<~)= En=l pi(Xi)s(xi), therefore, s(x) <~~n(x). [] The preceding two theorems allow us to define the concept of a basis for a fuzzy linear space. Definition 5.5. Let s be a fuzzy set, and let B : I> 2xÈ be a ~admissible map for s. B is called a bas/s for s if it is maximal in qg~.
6. Computing a basis
In this section, X is a finite dimensional space with some norm [I'll For any e > 0, B, will denote the open ball of radius e about the origin of X a n d / ~ will denote the closed unit ball in X. Thus /~ = {x I IIxll< 1} and B, = {x e X I
IIx[I < ~}. We now outline an algorithm for computing a basis for a fuzzy linear space s :X>I. The input is any procedure for computing the value s(x) for any x e X, and the output is a basis B:I>2 x. for s. The algorithm is of theoretical importance in that it may be regarded as a constructive proof of the existence of a basis that avoids the use of Zorn's lemma. (1) (2) (3) (4) (5) (6) (7)
(8) (9) (10) (11) (12)
vat E: an ordinary subspace in X ; b: a subset of X ; begin E := {0}; B ( 0 ) : = {0s~0)}; b := 0; for each 6 e/k{0} do B ( 6 ) := 0 endfor; E := {x [ s(x) >I s(0)}; let b be a basis for E; for each x ~ b do B ( s ( x ) ) := B ( s ( x ) ) tO {xsex)} endfor; while E =/:X do find x* ~ X such that s(x*) = max{s(x) I x e X \ E } ;
B(s.(x*)):=B(s(x*))tO(x%)}; E : = (x Is(x)>~s(x*)}; extend b to a basis for E; for each x ~ b do B ( s ( x ) ) := B ( s ( x ) ) tO {xs¢x)} endfor; endwhile end
We will now justify the correctness of our basis finding algorithm. In using this algorithm, one would avoid trying to execute the loop of the line 3 by storing only those B(6)'s that are nonempty, and modifying the rest of the algorithm accordingly. Clearly, after lines 25 of the algorithm have been executed, B will be admissible for s. We show below that the x* of line 7 can be found. Assuming for the time being that line 7 causes no problems, each iteration of the loop of lines 611 will result in an increase in the dimension of E, while preserving the
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admissibility of B, hence, the loop must eventually terminate. When it does terminate, a basis for E, and therefore for X, will be included in the foot of the linear hull of [._J0~
max{s(x) I x
X\E}
which has to be solved in line 7. Clearly, each time we (re)enter the loop of lines 611, but before executing line 7, the condition min{s(e) I e e E} > max{s(x) Ix e X \ E } . holds. (It follows as a consequence of Theorem 6.1 below that this maximum actually exists.) Theorem 6.1 also shows that instead of solving (P), we can instead solve (P')
max{s(x)
Ix e [~\(E + B,/z)}.
By a theorem of Riesz (see, for example, [5, p. 16]), the feasible set for (P'), /~\(E + B,/2), is a compact set which is nonempty if E :/:X. Now, for each f i e R, the level set {y e X I s(y) I r } is a finite dimensional linear suhspace of X, and hence closed. This means that s is upper semicontinuous, and hence must attain its maximum on the feasible set of (P'). (P') is an optimization problem that is interesting in its own right. It has a quasiconcave objective function and reverse convex constraints (an optimization problem is said to have reverse convex constraints if its feasible set is the difference of two convex sets). Maximization problems with concave objective functions and reverse convex constraints arise in economic and management type applications [11], and have recently begun to receive attention in the literature [8, 9]. It may be that some of the techniques that have been developed in [8, 9] can be adapted to provide algorithms for solving problems such as (P'). The author hopes this paper will stimulate some research in that direction. Theorem 6.1. Let s be a f u z z y linear space in X , and let E be an ordinary proper linear subspace o f X such that inf{s(e) I e e E} > s ( x ) f o r all x e X \ E . Then f o r every e, 0 < e < 1, and f o r every y e X \ E , there exists y* ~ B \ ( E + B~) satisfying the condition s ( y * ) = s ( y ) .
and y e X \ E be given. By replacing y with y/llyll if necessary, we may assume y e / ~ \ E (Proposition 4.4). We may farther assume that y e E + B~; otherwise, there is nothing to prove. By [4, p. 36], there exists e* e E such that Proof. Let e, 0 < e < l ,
[lY  e*ll = min{lly  ell I e ~ E} > 0.
(6.1)
Now y  e* g E, thus s ( y  e*) < s ( e ) , and s(y)
= s ( e * + y  e*) =
min{s(e*), s ( y  e*)} = s ( y  e*)
by Proposition 4.4. Now put y* = e ( y  e*)/llY  e*ll.
(6.2)
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We then have s ( y * ) = s ( y  e * ) = m i n { s ( y ) , s(e*)} = s ( y ) . We now complete the proof by showing that y* ~ E + B,. Suppose on the contrary that y* ~ E + B,. Then y* = e + v for some e e E, and v e X, Iloll < e. From (6.2), it follows that IIY  (e* + (IIY  e*ll/e)ell = II(Y  e*)  (IlY  e*ll/e)ell = (IlY  e*ll/e) II(Y*  e)ll
e*ll Ilvll/e
Ily

Since e* + (IIY  e*ll/e) e ~ E, this contradicts (6.1).
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7. Dimension of a fuzzy linear space In this section, we give a definition of the dimension of a fuzzy linear space, which extends Lowen's definition to cover the case where the host space may have infinite dimension. Let B be a basis for a fuzzy linear space, and let o~ ~/. For convenience, linz(a;) will denote the ordinary linear space that is the linear hull of the foot of k_Jx~>~B(oO, and dimn(o;) will denote the dimension of the same (subspace).
Theorem 7.1. Let B, B' be bases for a fuzzy linear space s. For all tr~ I, (a) linn(a:) = linn,(a;), and (b) damn(a) = dimn,(te). Proof. It suffices to prove (a). Suppose x* e linn(o 0
(7.1)
but x* ¢ linn,(a 0. Since x* e linn,(s(x*)) by T h e o r e m 5.3, it must be that c~> s ( x * ) .
(7.2)
By (7.1), there exist xl . . . . . x, satisfying minl~a~, and such that x*=~i~=ipixi for some scalars p~. But then this means that s(x*)>~ m i n ~ n S ( X i ) >t ol, contradicting (7.2). It follows that linB(a)~_ linn,(c0 , and by symmetry, we have equality. [] Definition 7.2. Let s be a fuzzy linear space with basis B. The dimension of s is the set of ordered pairs {(oG dimn(c0) I c~ ~ s(X)}. The linear space s has finite dimension if sup0,~s
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8. Conclusion This p a p e r has taken a first step in the d e v e l o p m e n t of a fuzzy linear algebra. The concept of a basis for fuzzy linear spaces has b e e n defined, and an outline for an algorithm for computing a basis has been given. The algorithm is nondeterministic in a n u m b e r of its steps, allowing an actual ' i m p l e m e n t a t i o n ' to m a k e choices that will lead to a specific basis. T h e algorithm calls for the solution of a certain type of optimization p r o b l e m , for which it has been shown that an optimal point exists. Methods for actually solving this optimization p r o b l e m have not been explored here. Our definition of a basis is compatible with L o w e n ' s definition of the dimension of a fuzzy linear space, and the basis finding algorithm given here can be used as a basis for decomposing a fuzzy linear space into the c o m p o n e n t s specified in Lowen's decomposition principle [6]. The decomposition principle has proved useful in proving certain properties of fuzzy convex sets.
References [1] A.K. Katsaras, Fuzzy topological vector spaces I, Fuzzy Sets and Systems 6 (1981) 8595. [2] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984) 143154. [3] A.K. Katsaras and D.B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. AppL 58 (1977) 135146. [4] S. Lang, Real Analysis (Addison Wesley, Reading MA, 1983). [5] R. Larsen, Functional Analysis (Marcel Dekker, New York, 1973). [6] R. Lowen, Convex fuzzy sets, Fuzzy Sets and Systems 3 (1980) 291310. [7] S. Nanda, Fuzzy fields and fuzzy linear spaces, Fuzzy Sets and Systems 19 (1986) 8994. [8] I. Singer, Minimization of convex functionals on complements of convex sets, Math. Operationsforschung Statist. Ser. Optim. 11 (1980) 221234. [9] H. Tuy, Convex programs with an additional reverse convex constraint, J. Optim. Theory Appl. 52 (1987) 463486. [10] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 238353. [11] Zaleesky, Nonconvexity of feasible domains and optimization of management decisions (in Russian), Ekonom. i Mat. Metody 16 (1980) 10691081.