Fuzzy Sets and Systems 47 (1992) 105-108 105 North-Holland

Generalisation of the Goetschel-Voxman embedding* L a j o s G e r g 6 Computer Centre of EOtv6s Lordnd University, Bogddnfy u. l O/b, H-1117 Budapest, Hungary

Received February 1990 Revised November 1990

Abstract: Goetschel and Voxman [3] gave a mapping from the space of t%crisp fuzzy numbers of the real line into 22. Their embedding is a homeomorphism and its range is convex. Using their construction and introducing a new metric we can extend their embedding to a subspace of the metric space of normal, fuzzy convex, upper semicontinuous and compactly supported fuzzy numbers of ~ which have their supports contained in a fixed compact cube of R ~. If we consider the case of Goetschel and Voxman (i.e. the fuzzy numbers of the real line) the extra properties of our mapping are: we do not need the ~-crispness and both the embedding and its inverse are Lipschitz continuous with respect to different metrics.

Keywords: Fuzzy numbers; metric space; embedding.

1. Notations

Denote by E n the set of all normal (i.e. there exists to • R n such that X(to) = 1), fuzzy convex, upper semicontinuous and compactly supported fuzzy numbers in ~", where fuzzy convex means that for the function x : ~" --0 I,

x(oa + (1 - a0s ) ~> min(x(t), x(s)}

holds for each t, s • supp(x). Define the metric D. by the equation

D.(x, y) = sup d.([x] ~, [y]") a ' ~ l

where I denotes the closed interval [0, 1]; d. is the Hausdorff metric in PK(ff~"), the set of nonempty, compact, convex subsets of ~":

d. (K, L ) = max/su p p(x, L), sup p(K, y)~, I .X~K y ~ L J

where p(x, L) denotes the p-distance of the point x and the subset L in R~; Ix] ~ = {t e ~ Ix(t) >I ~) for 0 < tr ~< 1 (the tr-level set of x); [x] ° denotes the support of x.

It is known that the ~t-level sets of x are nonempty convex compact subsets of R ~ and the space (E n, D~) is a complete metric space.

First let us consider a definition:

Definition 1.1. The product x = x l × x 2 x - - - x x n of the following element of En:

x(t) = min{xl(4), x2(t2) . . . . . xn(t , ) ) for each t • ~n.

fuzzy numbers xl , x2, • . . , xn e E 1 is the

* Research supported by the Hungarian Young Scholars' Fund No. 400-0113.

0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved

106 L. Gerg6 / Goetschel-Voxman embedding

We set

~ n :-~ { X I X X2 X " " . × x, • E': xi • El for each l <~i <~n}.

Now we introduce two subspaces of E n which will be denoted by E~(J) and ~ ( J ) . Let J be a fixed compact cube of R n. Then

E"(J) := {x • E ~ [ supp(x) c J}, ~n(J) := {x • ~n [ supp(x) c J}.

2. A new metric in En(J) and its properties

First we order the set I fq ~ of rational numbers that lie in the interval I into a sequence by a bijection (p : [~--->IN (2 where t~ is the set of positive integers. Let us define the function D,* by the equation

D*~ (x, y ) = (k~__l (d~([X]~(~k[Y]~'(k)))Z)l/2

where d~ is the Hausdorff metric and x, y • En(J).

Lemma 2.1. The function D* defines a metric on En(J).

Proof. First we can see that Dn* is bounded. All the ~b(k)-level sets are in J so we have dn([x]~(k), [y]~(k)) ~< [j[ for all k • ~ where [J[ denotes the diameter of J. Consequently,

(k~__l (]/])2x 1/2 i j I D*. (x, y) <~ _ --4~- / - V ~ .

The symmetry is evident, and the triangular inequality can be shown after straightforward calculation using the fact that d, is a metric and the Minkowski inequality in 22. The only thing we have to show is that if D*(x, y ) = 0 then x =y . Let D*(x, y) be equal to zero. Then

d,([x]~(k), [y]~(k)) = 0 for each k • N.

This means that Ix] ~= [y]O~ for all tr • I A (2. Let tr be an irrational element of L Then there exists a sequence tr k of rational numbers of I such that irk/Z a:. Using the well known properties of the it-level sets of fuzzy numbers,

[x] ~= f~ [x] ~k= ~ [y]"k= [y]" k=l k=l

which means that [x] ~ = [y] ~ for all tr • I that is x = y. It is clear from the definition that

1 D*(x, y) <- ~ D,(x, y) for each x, y • ~ ( J ) ,

but there exists no positive constant c such that

cD~(x, y) <~ D*~(x, y) for each x, y • ~n(J)

since levelwise convergence does not imply convergence in the metric D~.

Why did we choose the constant 4 in the definition of metric D*? It will be clear in the next section.

L. Gerg6 / Goetschel-Voxman embedding 107

3. Construction of the embedding

Let us consider the subspace gn(j) of En(J). There exists a continuous linear operator A : ~n--+ ~n such that the following holds:

HA(t)II2<~ 1 for each t e J ,

where II 112 denotes the Euclidean norm in ~n. For x e gn(j),

[xl = x [xd " 'k) x . . . x [xnl

This is equal to the product of intervals in R, that is

Ix] ' 'k) = [a,,k)(xa), b,(k)(X,)] × ' ' " × [a.(k)(Xn), b.,k)(X.)].

Let us introduce a notation:

A(a,,k)(x)) := A(a.,k)(Xl), a,(k)(X2), a.,k)(X3) . . . . . a.(k)(X.)),

A(b.,~)(x)) := A(b.,k)(Xl), b.(k)(X2), b.(k)(X3) . . . . . b.,k)(Xn)).

Now we define the mapping G : gn(j)_+ e2(Rn) as in [3], where ~2(~ n) is a Hilbert space with the n o r m

Hall = ~/k~__ ' Ila~[I 2 for a e #2(an), a = (a,, a2 . . . . . an , . . . ) , a, e ~n.

For x e gn (j),

(A(a.21)(x)) A(b .o) (x) ) A(a.(2)(x)) A(b. ,e,(x)) A(a.(k)(x)) A(b.,k)(X)) ) G(x) := , 2 2 , 2 3 , 24 . . . . . 2 2 k _ 1 , 2 2 k . . . . .

From the choice of A we can see that G(x) e ~e2(Nn), since

IIG(x)II 2 = ~ I[(G(x))kll2< 1. k = l

Theorem 3.1. I f we choose the operator A in the form A(t) = ct with appropriate positive constant c. the following will hold:

cO*. (x, y) <- IIa(x) - a (y ) l l ~ X/~ cOn(x, y),

that is the embedding G has an inverse and both G and G-1 are Lipschitz continuous with respect to different metrics above.

Proof. We have

{. A(a,fk)(x)) -- a(a , ,k) (y)) A(b.fk)(X)) - A(b, ,k)(y)) G(x) G(y ) \ • " ' 2 2 k - 1 ' 2Zk ' . . . . / /

Taking into account that

Ila.,k)(x) - a.(k)(Y)ll 2 ----- ~ la.,k)(Xi) -- a.,k)(Y~)l 2, l~j~n

it follows that

I IG(x ) -G(Y) ' I2=c2 t=I~L" \\{{lla'~'k)(X)--aee'k)(Y)H2~222k_, ] + ([ 'b"k)(X)~b'~k'(Y)l '2) (*)

~-~ {n maxl~,.n la.,k)(X,) - a,,k,(y,), 2 n maxl . j~ , l b , , k ) ( . ) - b.,k)(y,)[2\ ~ c 2 2., 4 2 k - l ~- 42k ) "

k = l

108 L. Gerg6 / Goetschel-Voxman embedding

From the definition of the Hausdorff metric it is clear that

max{la~(k)(xj) - a~,(k)(Yj)l, Ib ~,(k)(Xj) -- b~(k)(yj)[} = dl([Xj] ~'(k), [yj]~(k)).

Hence

[ IG(k ) - G(y)ll2<~nc 2 ~,°" \(maxl~</~n (dl([xj] ep(k),~ [yj]*(k)))24. maxl<~j~n (dl([xj] *tk),42k [y/]~)tk)))2). k=l

1 <~ nc2(O,(x, y))2 ~ ~ = ~nc2(O~(x, y))2, k=l

where we used the formula (see [2]) l¢(k)

max dl([Xjl , [yj]~fk)) = dn([x]~fk), [y]~Ck)) ~< Dn(x, y) for each k e ~. l~j~n

On the other hand, starting again with the formula ( * ),

(,)I> C 2 ~ (.~l~j~n [a~(k)(4X2J ) --aq~(k)(Yj)]2.t ~l<<-j<<-n ]b~(k)(xj)--b~,(k)(Yj)]2~

~l~j~n (dl[Xj] dp(k), [yj]ga(k)))2 C 2

k=l~'d 42k

~, maxl , j ,~ (dl([xj] *~k), [yj],<~)))2 C 2 zL~ 4zk k=l

= c 2 ~ (d~([x],fk), [yl,fk)))2 = c2(O*( x, y))2. 42k k=l

Remarks

Goetschel and Voxman proved the continuity of G with respect to the topology of the pointwise convergence. Our result is that the embedding of a more general space of fuzzy numbers and its inverse, G and G -1, are Lipschitz continuous with respect to the topology generated by the given metrics.

Diamond and Kloeden in [1] gave an embedding into a Banach space but our goal was to give a mapping into a Hilbert space because of its good properties.

References

[1] P. Diamond and P. Kloeden, Characterization of compact subsets of fuzzy sets, Fuzzy Sets and Systems 29 (1989) 341-348. [2] L. Gerg6, Some remarks on the topological properties of fuzzy numbers in R n, Fuzzy Sets and Systems 48 (1992), to appear. [3] R. Goetschel, Jr. and W. Voxman, Topological properties of fuzzy numbers, Fuzzy Sets and Systems 10 (1983) 87-99.