T-eigen fuzzy sets

INFORMATION

SCIENCES

75,63-80 (1993)

63

T-E&en Fuzzy Sets M. J. FERNANDEZ F. SUAREZ and P. GIL Departamento de Matemciticas, Facultad de Ciencias, 2” Planta, c) Calvo Sotelo, s/n, Universidad de Oviedo, 33007 Oviedo, Spain

ABSTRACT This paper concerns the study of the solutions of sup-T equations such that ATR =A, where T is a triangular norm and R is a fuzzy relation. By generalizing a result from Sanchez [12], we determine the greatest T-eigen fuzzy set of a fuzzy relation (a T-eigen fuzzy set of R being a fuzzy set A which is a solution of ATR =A). We also study some algebraic properties for the class of the T-eigen sets of a fuzzy relation, and for the set of fuzzy relations leaving invariant a given fuzzy set. Finally, properties related with the transitive closure are established.

In [121 and [13], Sanchez dealt with the problem of determining a fuzzy set being invariant under the max-min composition (denoted by 01 with a binary fuzzy relation R; that is, he looks for the eigen solutions of fuzzy relations, namely, he obtains the greatest solution, in the sense of fuzzy set inclusion, of the fuzzy equation

(1) This theory was originally proposed for medical diagnosis applications. Amagaza and Tazaki presented in [l] an application of eigen fuzzy sets in connection with heuristic structure synthesis. Also, Cao Zhi Qiang [2] proposed an algorithm to obtain the eigen fuzzy sets of a fuzzy matrix. On the other hand, Goetschel and Voxman [6] extended the results of Sanchez regarding fuzzy set equations to fuzzy numbers, that is, they dealt with type-2 fuzzy sets whose membership functions are fuzzy sets on R. The problem of existence and determination of eigen fuzzy solutions in 0 Elsevier Science Publishing Co., Inc. 1993 655 Avenue of the Americas, New York, NY 10010

0020-0255/93/$6.00

64

M. J. FERNiiNDEZ ET AL.

a priori given regions (fuzzy intervals) has been recently investigated by Wagenknecht and Hartmann in 1171. In the above papers, except in [17], the max-min composition is employed, and the referential X is assumed to be a finite set. In this paper, we are going to extend some results to the case of sup-T and inf-T’ compositions and infinite X. The scheme in the paper is the following. In Section 1, we give basic definitions and some results needed in the sequel. In Section 2, we find the greatest T-eigen fuzzy set and the smallest T’-eigen fuzzy set associated with a fuzzy relation. In Section 3, we discuss some algebraic properties of the class of T-eigen fuzzy sets of a fuzzy relation and the set of fuzzy relations leaving invariant a given fuzzy set by generalizing some results from Sanchez [13]. Finally, in Section 4, we examine the T-eigen fuzzy sets associated with the transitive closure of a fuzzy relation. 1. PRELIMINARIES Let I be the unit interval [O,11 of the real line, and let x r\y and x respectively, denote the minimum and maximum of x and y.

Vy,

DEFINITION 1.1 [18]. A fuzzy set A of a nonempty set X is a function A: X-tl. The class of all fuzzy sets on X is denoted by fix). The referential set X itself can be seen as the “greatest” fuzzy set of 9(X>, with membership function X(t) = 1 for any t EX, and the empty set 0 is the “smallest” fuzzy set of 9’(X), with membership function 0(t) = 0 for any t EX. DEFINITION 1.2 1181. Let X,, X,, . . . , X,, be nonempty sets. A fuzzy n-ary relation R on X1,X,, . . . , X, is defined as a fuzzy set of the Cartesian product X, XX, X ... XX,, that is an element of 9(X,X X*X .** XX,). In particular, a fuzzy binary relation on X and Y is an element of *xx Y). In this paper, we are only interested in the case in which X= Y. Accordingly, by a fuzzy relation, we herein mean a fuzzy binary relation on X. DEFINITION1.3 [14]. A triungulur nom (t-norm) is a real-valued function of two variables T: ZXZ -+Z such that T(x, 1) =x, T(x, y) = T(y, n), T[T(x,y),zl= T[x,T(y,z>], and TCx,y)g T(x,z) if y GZ, for all x,y, z E I.

65

T-EIGEN FUZZY SETS

DEFINITION1.4 [14]. A triangular cononn (t-conorm) is a real-valued function of two variables T’: IXl+l such that T’(x,O)=x, T’(x,y)= T’(y,x), T’[T’(x,y),z]=T’[x,T’(y,z)l, and T’(x,y)~T’(x,z) if y
It is easy to prove that T(X,Y)
(max(x,Y)

GT’(x,Y)

for all x,yEZ.

Due to monotonicity of t-norms and t-conorms, the lower semicontinuity (respectively, upper semicontinuity) of a t-norm T (respectively, t-conorm T’) is equivalent to the left continuity (respectively, right continuity) of T (respectively T’) with respect to both arguments, x and y (cf. [4, 71). This can be shown in the equivalent equality

which holds for any family {yJi E J of real numbers in Z if T is assumed to be a lower semicontinuous t-norm, or in T’( x, infy, ) = inf T’(x,yi) iEJ

if T’ is an upper semicontinuous 2.

T-EIGEN

FUZZY

iEJ

t-conorm.

SETS

In this section, we extend certain results of the eigen fuzzy set theory to the context of sup-T and inf-T’ compositions and to the case of infinite X, that is, we want to obtain the greatest T-eigen fuzzy set (or the greatest solution of (2.0, stated below) and the smallest T’-eigen fuzzy set [or the smallest solution of (2.2)] of a fuzzy relation. DEFINITION2.1. Let X be a finite or infinite set and RESXXX). If T is a t-norm, a fuzzy set A ES~(X) is a T-eigen fuzzy set associated with R if ATR =A

(2.1)

66

M. J. FERNtiDEZ

ET AL.

where the sup-T above composition is defined by (Am)(Y)

for any y EX.

=;p(Aw(%Y))l

The problem to solve is to obtain the greatest T-eigen fuzzy set of R. In an analogous way, we have the following. DEFINITION 2.2. Let X be a set and RES(XXX). If T’ is a t-conorm, a fuzzy set A EY(X) is a T’-eigen fuzzy set associated with R if AT’R =A

(2.2)

where the inf-T’ above composition is defined by (AT’R)(y)=xitnf:[T’(A(x),R(x~~))l

for any y EX.

The T-eigen and T’-eigen fuzzy sets are closely related, so from now on, we may study only one of them since results regarding one of the concepts can immediately be deduced from those for the other one by means of the referring relation. EXAMPLE 2.3. a> If supx E x R(x, y) = 1 for any y EX, then the greatest solution of the equation (2.1) is X. In the case of lower semicontinuous triangular norms, there are also solutions A = k for any k E 10,11.b) When T = minimum, let g(y) = supx E x R(x, y) for each y EX and let (Y= inf, g(y). Then A(x) = a is a solution of the fuzzy equation (1). As a particular case, if X= [a, b] a, b E [wand R(x, y> =e-I’-Yl, then A(x)

= ;eYb

for any x EX is a solution of (1).

Our purpose is to find the greatest T-eigen fuzzy set of X, for any fuzzy relation R ES(XXX>. If we denote JdT,R={A~9-(X)/ATR=A}

this set is nonempty since 0 E_&~,~ for any t-norm T and for any fuzzy relation R. We can obtain the greatest element of s?~,~ by means of the next theorem requiring the following lemma.

67

T-EIGEN FUZZY SETS LEMMA 2.4. Zf A cB,

then ATR cBTR for any R G+‘(XXX).

The proof of the lemma is straightforward

from the monotonicity of any

t-norm. THEOREM 2.5. Let R E&XXX), and let T be an upper semicontinuous t-norm. Zf we define the following sequence of fuzzy subsets of X: A,=XTR,

A,=A,TR,

A,=A,TR=A,TR2,...,A,,,

=A,TR=A,TR”,... then the fuzzy subset A defined by A(x)

for each x EX

=lim{A,(x)}(n+m)

is the greatest element of MT, R. Proof. Let us prove by induction that

Since A, cX, then by the previous lemma, A, =XTR xA,TR =A,. On the other hand, if A,_ 1 xA,, we have that A, =A,_,

TRIA,TR=A,+,

from which there exists A(x) = lim(A,(x)) We will now see that A E&~, R. Indeed,

= inf(A,(x))

= ~~px[inf{T(A,(x),R(x,y))ll

=am[sup{T(A,(x),R(x,~))}] XEX

=lim{(A,TR)(y)}=lim(A,+,(y))=A(~).

for each x EX.

68

M. J. FERN6LNDEZ ET AL.

To see that A is the greatest element of Mr,R, let us suppose that there is a fuzzy set A’ E%X) and A’~,~, and a y, EX such that A’(y,) >A(y,). Then, we have that

since A’ =A’TR cXTR =A, and, if A’ CA,, A ,,+, =A,TR zA’TR =A’. In particular, AYy,) ,then A is the greatesteigen fuw set associatedwithR.

The proof is immediate since BoR =A,. In an analogous way to the preceding theorem, we can calculate the smallest T’-eigen fuzzy set associated with R. If we denote by d

T,,R=(~~9(~)/AT’R=A}

we wish to obtain the smallest element of &r-S,R(since the greatest is X). It is easy to see that the sup-T and inf-T’ compositions are connected as follows: (AT’R)‘=A’TR”

(2.3)

69

T-EIGEN FUZZY SETS

where T’ is the t-conorm dual of T, that is, T’(x, y) = 1 - TO -x, 1 -y> for all x,y EX and A’ is the complement fuzzy set of A, that is, A’(x) = 1 -A(x) for each x EX. By means of the preceding equality, we prove the following. THEOREM 2.7.Let R ES~(XXX) and let T’ be a lower semicontinuous t-conovn. The fuzzy set A, meaning the limit of the following sequence of j&y sets, A,=0T’R,

A,=A,T’R,

A3=A,T’R=A,T’R2

,...,

A,,+,

=A,T’R=A,T’R*,... is the smallest T’-eigen fuzzy set associated with R. Proof. By applying Theorem the sequence A; =XTR’,

2.5 to R’ and T = 1 - T’, the limit H of

A; =A;TR’,.

.., A”,,, =A;TR”,.

..

is the greatest T-eigen fuzzy set related to R”, that is, H = HTR’. H(x)=;$a{A’,(x)}=l-&nm(A,(x)}

Consequently,

for any x EX.

A = H”. By (2.31, we have that A = H’ = (HTR”)’

= H”T’R =AT’R

and hence A E&~, R. We will see that it is the smallest element of Mr,,R. Indeed, if B E$~., R, it is easy to see that B’ E&~,~~. Then, B” cH and B IH” =A. 3.

ALGEBRAIC

n

ASPECTS

In this section, we present some theoretical results related to the class of the T-eigen fuzzy sets of a given fuzzy relation, and the class of fuzzy relations leaving invariant a given fuzzy set. First, we shall study the set dT,

R*

If T = minimum and X is a finite set, Sanchez [13] pointed out that &r,R is an upper semilattice since the max-min composition is distributive for fuzzy unions, but not for fuzzy intersections.

M. J. FERNtiDEZ

70

ET AL.

We generalize this result for any t-norm and X not necessarily being finite in the following. PROPOSITION3.1. Let T be a t-norm and RE~IXXX). then AUBWr

IfA, BEJZ’~,~,

R.

Proof.

=;$T(A(x),R(x,y))]

= (ATRHY) =

[VW

v;$T(B(X))R(v))]

v (BTWY)

uW-R)l(y)

=(AuB)(y)

for every y EX.

n

The above proposition has an extensive application since it has been proven whatever the fuzzy relation R, the T-eigen sets of R, A, and B, and the t-norm T in the sup-T composition may be. It has been shown in practice [15, 191 that in some situations, there are operators more appropriate than maximum and minimum to define the union and intersection of fuzzy sets. So, if we define the union of two fuzzy sets A, B ESZ(X) by means of any t-conorm T’ as

(Au,,B)(x)=T’[A(x),B(x)]

for every x EX

we can be wondering if the above proposition remains valid irrespectively of the t-conorm T’ used to define the union. The answer is negative, as we can see in the following counterexample. COUNTEREXAMPLE memberships: A( xi) =

3.2. Let X={x,,x,}

0.5

1

ifi=l ifi=

B( xi) =

and let A, B Ed’

0.5

0.4

ifi=l if i=2.

with

71

T-EIGEN FUZZY SETS Let R czs(XxX)

be defined by the matrix R=[;::

“;“I.

Then, if T(x, y) =x /\y for all x,y EZ, we have that (ATR)(x,) (

=T(0.5,0.5)

ATR)( x2) = T(0.50.3)

vT(1,0.4)

=0.5=A(x,)

v T( 1,1) = 1 =A( x2)

(BTR)(x,)

=T(0.5,0.5)

v

(BTR)(x,)

=T(0.5,0.3)

vT(0.4,1)

Z-(0.4,0.4) =0.5=B(x,) =0.4=B(x,)

from which A, B EzZ~, R. On the other hand, if

T’(XvY)

=

i

.:,

(AU,,B)(x,)=l

if xAy>O if xAy=O for i= 1,2

[(A~,~B)TR](x,)=T(1,0.5)~T(1,0.4)=0.5+(Au,.R)(x,) and this implies that A U,. B Ecs?~,~. n The next theorem characterizes the maximum as being the unique t-conorm transforming two T-eigen fuzzy sets associated with a fuzzy relation in another T-eigen fuzzy set associated with the same fuzzy relation. THEOREM 3.3. A U,, B is a T-eigen fuzzy set of R for any t-norm T, for anyfizzy relation R E~IX XX) whatever X may be, and for all T-eigens A, B ofR, iff T’(x,y)=xVyforallx,yEZ. Proof. The sufficiency of the condition is proved in Proposition 3.1. We now see that it is also necessary. If AU,, B is a T-eigen of R, then (A’JTBHY)

= [W+B)TR](Y) =x~~~x[T(T’(A(x),B(x)),R(x,y))l

for any yEX (3.1)

72

M. J. FERNhLNDEZ ET AL,. On the other hand,

(A

UT'B)(Y)

= EWW W~TWl(Y)

=7’[sup qA(x),qx,y))Y X6X

sup T(B(x),R(x,y))]

XEX

for any y EX.

(3.2)

We can always consider T(x, y) =x r\y and X= {x0} such that A(x,) =B(x,)=R(x,,x,)=a for any a EZ. Then, if (3.1) and (3.2) coincide, we have that T’(a, a) A a = T’(a, a>, from which T’(a, a) G a for any a EZ, and hence T’(x, y) =x Vy since x Vy Q T’(x, y) G T’(x Vy, x Vy> G x vy.

n

REMARK. It should be pointed out that from the preceding theorem, we cannot conclude that for a particular case, a t-conorm different from maximum is not able to transform (by means of the fuzzy union) two T-eigen fuzzy sets of a fuzzy relation into another T-eigen fuzzy set of the same fuzzy relation, but rather, that result can only be always guaranteed when the t-conorm used to define the union of two fuzzy sets is the maximum. Now we shall study some properties of the set 9 A,T={z?E.F(XXX)/ATR=A},

that is, the set of the fuzzy relations leaving invariant the fuzzy set A by the sup-T composition. A,T is nonempty for any A lflX) and for any t-norm T since the LG? fuzzy relation 0 defined by qx,Y)

=o

if x#y,

O(x,y) = 1

if x=y

is always in zA, T. Moreover, when T is a lower semicontinuous t-norm, as a straightforward consequence of a theorem from Pedrycz in 191,we have that sA,r has a maximum given by R(x, y) = q[A(x), A(y)], 40 being the operator associated with T. When T = minimum and X finite, Sanchez [13] proved that gA, r has minimal elements R, defined by: for all y EX, the only possible element R,(x, y) f 0 is just Z?,(x,, y) =A(y) for some x0 EX such that Ax,) aA(

73

T-EIGEN FUZZY SETS

Analogously, we can prove for the set _~‘r,~ that if we establish an order relation on 9 A,T by means of the inclusion of fuzzy sets, then SA,r is an upper semilattice, but not a lattice. The ordering defined by the inclusion of fuzzy sets in L%?~, T and &r, R is a dense order, as we can see in the following. PROPOSITION 3.4. a) Let R,, R, be in 9A,T with R, CR,; then there also exists an R, in zZ~?~,~ such that

R,cR,cR,. b) Let A,, A, be in Sal,,, with A, CA,; then there also exists an A, in JZ’~R such that A, CA, CA,. Proof a) Let defined by

R, = AR, +(l-

A)R,

for any hel

that is pointwise

then R, CR, cRZ, and the monotonicity of t-norms implies the monotonicity of the sup-T compositions: A =ATR, cATR, cATR, =A; then A =ATR, and R, ES?~,~. We can prove b) in a similar way. H On the other hand, ~%‘~,ris a semigroup, 0 being the identity element. To prove it, we need the following lemma. LEMMA

3.5. Let T be a lower semicontinuous t-norm. Then

a> QT( RTS) = (QTR1T.S b) AT(QTR) = (ATQITR whatever A E~@X) and Q, R, S E~IXXX)

may be.

Proof. We shall prove a) because the proof of b) should be analogous. [QT(RTs)l(x,y)

= SUP [T(Q(~,~>~
74

M. J. FERNhLNDEZ ET AL.

= sup tex

supT[T(Q(x,~>,R(z,t)),S(t,y)] ZEX

= [(QTR)TS](n,y)

forall X,Y

PROPOSITION3.6. Let T be a lower semicontinuous

Q,R-&;

EX.

n

t-norm, and let

then QTR ES@~,~ and RTQ ES~,~.

Proof. In effect, by section (ATQ)TR =ATR =A.

b) of the above lemma,

ATCQTR) =

Analogously, we should prove that RTQ ES’~, T.

n

PROPOSITION3.7. OTR= R for any R 6%XxX). Proof. The result can immediately be derived.

n

Then %,T with the sup-T composition is a semigroup, identity element. We conclude this section with the next theorem.

0 being an

THEOREM3.8. Let R, Q E~~XXX) be R zQ, and let T be an upper and lower semicontinuous t-norm. Then, if A is the greatest T-eigen fuq set associated with R and Q, A is the greatest T-eigen fuzzy set associated with RTQ and with QTR. Proof. Let A,=XTR,

Az=A,TR,...,A,=A,_,TR=AITR”-’

B, =XTQ,

B,=B,TQ

,...,

B, =B,_,TQ=B,TQ”-?

Under the assumed hypothesis, the greatest T-eigen fuzzy set associated with R and with Q is the same, and hence by Theorem 2.5, we have that

)ya PnWl= ,‘-nm Pn(x)l=A(x)

for any x EX.

(3.3)

75

T-EIGEN FUZZY SETS Let us now define the sequence {CJ of fuzzy sets given by C, =XT( RTQ) = (XTR)TQ =A,TQ, C, = C,T( RTQ) ,...,C,=C,,_,T(RTQ) =CIT(RTQ)“-I,...,

for all integer n > 0,

But Q CR, from which A, I&, C, =A,TQcA,TR

=A,

C, = C,T( RTQ) cA,TR2 =A,

and then

and

C, 3B,TQ=B,

and

C,=C,T(RTQ)xB2TQ2=B,.

Let us prove by induction that B,,

CC,

=A2n

c,cA,TR~(“-“=A,TR’“-~

=A,,;

c,~B,TQ~n-“=B,TQ2”-’

=B

2Il’

By (3.31,

Then

and on the basis of Theorem 2.5, A is the greatest T-eigen fuzzy set associated with RTQ. Analogously, we shall prove that A is the greatest T-eigen fuzzy set associated with QTR. m It is obvious that analogous results can be stated using upper semicontinuous t-conorms, and similar considerations can be derived for T’-eigen fuzzy subsets.

76

M. J. FERNANDEZ

4. T-EIGEN TRANSITIVE

FUZZY SETS ASSOCIATED CLOSURE

ET AL.

WITH THE

Let T be a lower semicontinuous t-norm, and let R be a fuzzy relation defined on X. Due to the associativity of the sup-T composition for the lower semicontinuous t-norms (Lemma 3.9, we can define Rk recursively by Rk ,Rk-

‘TR =RTRk-

1

Now we can give the following. DEFINITION4.1. Let T be a lower semicontinuous t-norm, and let R E~(XXX). We define the T-transitive closure of R, C,(R) by

C,(R)

=RuR2uR3u

.a. uRku

..a

If T = minimum, we note that the above definition coincides with that proposed by Kaufmann in [8]. If T = minimum and card(X) = n, then (cf.

[81) C(R)

=RuR2uR3u

... uR”

and, in this case, Sanchez proved [131 that the greatest eigen fuzzy set associated with C(R) is the same one associated with R. Our purpose is now to prove this result in the general case when X is an infinite set and T is an upper and lower semicontinuous t-norm. The next proposition generalizes the result from Sanchez [13]. PROPOSITION 4.2. Let R E~(X x X), and let T be a lower semicontinuous t-norm. If we define, as in Theorem 2.5, A, =XTR,

A, =A,TR,.

. ., A,, =A,_,TR

then, for any n > 0 and for any y EX, XTR” =A,.

=A,TR”-

1,...

77

T-EIGEN FUZZY SETS Proof.

= sup(RTR”-‘)(x,Y) XEX

= sup sup T( R( X,Z),RqZ,Y))

xex ZEX

= sup sup T(R( X,z),Rn-l(z,Y))

26X xex

= (A,TR”_‘)(y)

for any y E Y.

=4(Y)

n

We will see that the greatest T-eigen fuzzy set preserves invariance if we make some transformations to the fuzzy relations. THEOREM 4.3.Let R E~IXXX), and let T be a lower and upper semicontinuous t-norm. Then, the greatest T-e&en fuzzy set associated with R coincides with that associated with Rk and with that associated with C,(R).

Proof. In accordance with Proposition 4.2, if we denote A’,k’ =mRk

=A,

A(2k)=:A’,k,TRk =AkTRk =Azk A(3k) =A$

k ‘TR k

=A,,TRk

=A,,

Aik’ =Aik? lTRk =A,,_ l,kTRk =A,,

M. J. FERNhLNDEZ ET AL.

78 we have that

for any x EX. In this way, if we denote

AfT=x7&(R), A?‘(Y)

then = SUP x

=: sqp[C,(R)](x,Y)

=

lJ

WGY)

fc=1,2,...

1

sup S~p{Rk(x,Y)} x

=

SUP 4(Y)

=4(Y)

k= 1,2,...

= SUP [‘( x

4w

SU*pR*(XTYl)]

=sup[supT(A,(x),~~(x,Y))]=suP[suP~(A,~~~,R”~~~Y~)] x k x k = s;p

(

A,TRk)(Y)

=

k=~~,,,Ak+l(Y) 9 I

=A2(y)’

By induction, we then obtain MY)

=(&l~GuwY)

=(~sG(~))(Y)

=suP[~(A,-,(x),s~pR*(x,Y))] x = SUP[ SUPT(A,_,(X),RX(X,Y))]

x

k

=s;p(&TRk)(y)

=

SUP 4f+n-1(Y)

k=1,2,...

=4t(Y)

T-EIGEN FUZZY SETS

79

and by Theorem 2.5, A = lim A, (n -+ ml is the greatest T-eigen fuzzy set n associated with C,(R). 5.

CONCLUDING

REMARKS

We are in agreement with Prof. E. Sanchez, who has claimed that an eigen fuzzy set theory can be developed in areas such as belief systems, transportation problems, fuzzy clusterings, human decision processes, pattern recognition, medical diagnosis, etc. In some of these fields, such as medical diagnosis, Sanchez has already proposed some applications of this theory, and it could be reasonable to consider the referential as a finite set and the composition as the max-min one. Nevertheless, in other areas, it can be useful to extend this theory to infinite referentials and sup-T compositions in the sense exposed in this paper. In this respect, Pedrycz in [ll] points out that the notion of an eigen fuzzy set can be directly used in questions of stability control in fuzzy models that are essential in the development of fuzzy control procedures. Thus, it can be employed to determine the fuzzy states stabilized by a given fuzzy control, and we suggest defining the fuzzy relation of the controller in an infinite universe of discourse, and analyzing different triangular norms to fit the structure of data (see [lo]). REFERENCES 1. M. Amagaza and E. Tazaki, Heuristic structure synthesis in a class of systems using a fuzzy algorithm, in Summary of Papers on General Fuzzy Problems, No. 5, The Working Group on Fuzzy Sets of Japan, Dec. 1979. 2. Z. Q. Cao, The eigen fuzzy sets of a fuzzy matrix, in Approximate Reasoning in Decision Analysis (M. M. Gupta and E. Sanchez, Eds.), North-Holland, Amsterdam, 1982, pp. 61-63. 3. M. K. Chakraborty and M. Das, Studies in fuzzy relations over fuzzy subsets, Fuzzy Sets and Syst. 9:79-89 (1983). 4. A. Di Nola, S. Sessa, W. Pedrycz, and E. Sanchez, Fuzzy Relation Equations and Their Applications to Knowledge Engineering, Kluwer Academic, 1989. 5. M. J. Femandez, F. Suarez, and P. Gil, Equations of fuzzy relations defined on fuzzy subsets, Fuzzy Sets and Syst., 52:319-336 (1992). 6. R. Goetschel and W. Voxman, Eigen fuzzy number sets, Fuzzy Sets and Syst. 16:75-85 (1985). 7. S. Gottwald, Fuzzy set theory with T-norms and cpoperators, in Topics in fhe Mathematics of Fuzzy Systems (A. Di Nola and A. Ventre, Eds.), Verlag TUV Rheinland, Koln, 1986, pp. 143-196. 8. A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, Vol. I, Academic, New York, 1975.

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M. J. FERNhDEZ

ET AL.

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