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Equations of fuzzy relations defined on fuzzy subsets
Fuzzy Sets and Systems 52 (1992) 319-336 North-Holland
319
Equations of fuzzy relations defined on fuzzy subsets M.J. Fern~indez, F. Su~irez and P. Gil Department of Mathematics, University of Oviedo, Spain Received August 1991 Revised November 1991
Abstract: In this paper some results of the theory of fuzzy relation equations are generalized, when we consider fuzzy relations defined on fuzzy sets instead of crisp sets. If the extension of classical definitions for composition of relations [12, 13] is not straighforward, we provide suitables counterexamples illustrating the necessity of change in definitions. We find the greatest (or the least) solutions of fuzzy relation equations with different types of compositions, like the sup-T, inf-T', inf-q0 or sup-fl compositions.
Keywords: Fuzzy relations; fuzzy relation equations; types of equations.
The theory of fuzzy relation equations was introduced by Sanchez [16], obtaining the author the greatest solution of the equation Q o R = S (for the sup-min composition), knowing Q and S or knowing R and S, where the 'greatest' is viewed like the maximum of the set of solutions of the equation (in the sense of set-theoretic inclusion). Since Sanchez, many papers dealing with such equations have appeared (see Chapter 15 of [3] for an extensive bibliography until 1989). But, in all of them, the fuzzy relations are defined on crisps sets. This paper deals with the study of fuzzy relational equations, when the fuzzy relations are defined on fuzzy sets instead of classical sets. The necessity of defining fuzzy relations over fuzzy subsets was already observed by various authors, see [2, 19, 20]. Obviously, since the crisp sets are a particular case of fuzzy sets, certain results of this paper are generalizations of some results in works previously refered. This paper is organized as follows: Basic definitions and notations are given in Section 1, introducing in this section the properties that we need in the next sections. In Section 2 we introduce the definitions of s u p - T and inf-T' compositions of two fuzzy relations defined on fuzzy sets as a generalization of the definitions given when the fuzzy relations are defined on crisp sets [13]. In Section 3 the fuzzy equations of the form Q T R = S, where Q T R is the s u p - T composition defined in the section before are proposed and solved. In an analogous way, in Section 4 we propose and solve under restrictive conditions the fuzzy equations of the form Q T ' R = S, with the inf-T' composition defined in Section 2. Finally, Section 5 studies the adjoint fuzzy relation equations.
1. Introduction
We will use the standard notation of fuzzy sets, employing capital letters to denote fuzzy sets and relations, and denoting the family of fuzzy sets defined in a set X (crisp or fuzzy) by F ( X ) . Correspondence to: Prof. Dr. P. Gil, Departamento de Matematicas, Facultad de Ciencias, 2a. Planta, Calvo Sotelo, s/n, 33007 Oviedo, Spain. 0165-0114/92/$05.00 (~) 1992--Elsevier Science Publishers B.V. All rights reserved
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Let I = [0, 1]. A triangular norm (t-norm) [10, 18] is a real-valued function of two variables such that T(x, 1)=x, T(x, y ) = T ( y , x ) , T[T(x, y ) , z ] = T[x, T(y,z)] and T(x, y)<~
T:I×I~I
T(x, z) if y <~z; for all x, y, z • l. A triangular conorm (t-conorm) is a real-valued function of two variables T ' : I x I--->I 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) i f y ~ < z ; for all x, y, z • L Given any t-norm T, the corresponding t-conorm T', dual of T, is defined by setting
T'(x, y) = 1 - T(1 - x, 1 - y)
for all x, y • 1.
Some examples of t-norms and related dual t-conorms are reported below: Tl(x, y) = min(x, y),
T'l(X, y) = max(x, y),
T2(x, y) =x . y,
T~(x, y) =x + y - x . y,
T3(x, y) = max(0, x + y - 1),
T~(x, y) = min(1, x + y).
It is immediate that, for any t-norm and t-conorm, the following inequalities hold for all x, y • I:
T(x, y) <~Tl(x, y), T~(x, y) <~T'(x, y). We now recall the definition of the pseudocomplement of a t-norm T (resp. t-conorm T'):
Definition 1.1 [14]. An operator tp : I × I---~ I (resp. /3) is associated with a t-norm T (resp. t-conorm T') if (a) (b) (c)
for all a, b, c • I: ~(a, max{b, c}) = max{qg(a, b), qg(a, c)} (resp. /3(a, min{b, c}) = min{/3(a, b),/3(a, c)}), T(a, qg(a, b)) <~b (resp. T'(a, /3(a, b)) >! b), cp(a, T(a, b)) >! b (resp. fl(a, T'(a, b)) <~b).
If T = T1 and T' = T'I these operators were introduced by Sanchez [16]. Gottwald has proved in [6] that: " A n operator q~ (resp. fl) is associated with a t-norm T (resp. t-conorm T') iff T (resp. T') is lower semicontinuous (resp. upper semicontinuous)". Then from now on, we consider lower semicontinuous triangular norms and upper semicontinuous triangular conorms. All t-norms (resp. t-conorms) mentioned in the above examples are lower (resp. upper) semicontinuous. The corresponding operators q0 and/3 are listed below:
{1,
a<-b, a >b,
tpl(a , b) = b,
{0, /31(a, b) = b,
~2(a, b)=
/32(a, b)= ,
cp3(a,b)={lb, -
a>b, a+ l,
a<-b, a>b,
/33(a,b)=
a >~b, a
b-a 1 a {~'_
a,
,
a
a>~b' a
The operator tp verifies a lot of properties that we need in this work. We recall some of them in:
Proposition 1.2 [3, 4, 11, 12]. Let T be a lower semicontinuous t-norm and cp the operator associated with T. Then: (a) q0(a, bl) ~< qg(a, bE) if b~ ~ b2, (b) cp(al, b) >~cp(a2, b) if al <~a2, (c) tp(qg(a, b), b)/> a, (d) ~(a, b) = sup{c • I I T(a, c) <~b}.
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In the same way, we have:
Proposition 1.3. Let T' the dual t-conorm of T and cp (resp. /3) be the operator associated with T (resp. T'). Then /3(a,b)=l-q~(1-a,l-b)
foralla, b • l .
Proof. We have /3(a, b) = inf{c • I I T'(a, c) >/b} = inf{c • I ] 1 - T(1 - a, 1 - c) ~> b}
=inf{ccIlT(1-a,
1-c)~< l-b}
=inf{1-c
•I I T(1-a,
c)~< l - b }
= 1 - sup{c • ! [ T ( 1 - a , c)~< 1 - b} = 1 - q0(1- a, 1 - b).
[]
The next is an immediate consequence of the two previous propositions.
Proposition 1.4. (a) /3(a, b 0 <~/3(a, b2) if bl <<-b2, (b) /3(al, b)~>/3(az, b) ifal<~a2, (c) /3(/3(a, b), b) ~< a. Let now Q • F ( X x Y), R • F ( Y × Z), T be any t-norm and T' be any t-conorm. Definition 1.5 [13]. The s u p - T (resp. i n f - T ' ) composition of Q and R, denoted by QTR (resp. QT'R) is defined as a fuzzy relation on X x Z whose membership function is given by:
(QTR)(x, z) = sup [T(Q(x, y), R(y, z))]
resp.
yEY
(QT'R)(x, z) = inf [T'(Q(x, y), n(y, z))]. yeY
for all x • X, z • Z .
Definition 1.6 [12]. Let q0 (resp. /3) be the operator associated with a t-norm T (resp. t-conorm T'). The inf-q9 composition (resp. sup-/3 composition) of Q and R, denoted by QcpR (resp. Q/3R) is defined as a fuzzy relation on X × Z whose membership function is given by: (QcpR)(x, z) = inf [qg(Q(x, y), R(y, z))] y~Y
resp.
(Q/3R)(x, z) = sup [/3(Q(x, y), R(y, z))] y~Y
for all x • X, z • Z. Let X be the initial set and A, B • F(X). We will use the next definition of Cartesian product proposed by Zadeh [20]: A × B is the fuzzy subset of X × X defined by
(A × B)(x, y) = min(A(x), B(y))
for all x, y • X
and hence the following: Definition 1.7 [1, 20]. Let R • F(A x B), i.e. R(x, y) <~min(A(x), B(y)) for all x, y • X. Then R is a fuzzy relation from A to B.
2. (sup-T) fuzzy relation equations The max-min (sup-min) composition of two fuzzy relations is closely linked with the extension principle [21], as it was stated by Sanchez [17] and Pedrycz [15].
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We propose a generalization of this composition, using t-norms, as in Dubois and Prade [5] to define the fuzzy operations between fuzzy numbers. Due to the associative property of the t-conorms, in [8] it is defined:
T'{xl, x2 . . . . , x,} = T'{T'{xl, x2 . . . . , x~-l}, Xn}
for n >i 2.
As well, by the monotonity property of t-conorms and by its definition on the closed interval [0, 1], we can define on a countable set, see [9]:
T'
{Xi}
=
i~N
lim T'{xl,
X 2 ....
, Xn}.
n--~
Let A, B and C be fuzzy subsets of X (finite or countable crisp sets), we can define the composition of fuzzy relations by means of: Definition 2.1. Let Q • F(A × B) and R • F(B × C). The T ' - T composition of Q and R, in symbols Qr,rR, is the fuzzy subset given by the membership function
(Qr,rR)(x, y) = T' [T(Q(x, z), R(z, y))]
for all x, y • X
z~X
where T and T' are any t-norm and t-conorm respectively, not necessarily duals. When T' -- max and T -- rain, we have the classical definition of max-min composition between two fuzzy relations. The next theorem proves that we cannot be very ambitious in the definition of the composition of fuzzy relations defined on fuzzy sets, provided that we wish that the result continues to be a fuzzy relation too. Theorem 2.2. For any t-norm T, Qr,rR will be a fuzzy relation on fuzzy subsets, for any Q and R fuzzy
relations on fuzzy subsets if and only if T'(x, y) = max(x, y)
for all x, y • [0, 1].
Proof. Let Q e F(A × B) and R e F(B x C). We have
T(Q(x, z), R(z, y)) <- min(Q(x, z), R(z, y)) ~
for all x, y, z • X
whatever X be. Consequently,
(Qr,rR)(x, y) = max [T(Q(x, z), R(z, y))] ~
for all x, y • X
zeX
and the sufficiency condition is proved; let us regard its necessity. By hypothesis,
T'z~xtT(Q(x, z), g(z, Y))I <~min(A(x), C(y))
for all x, y • X
(2.1)
for each t-norm T and for any fuzzy relations Q • F(A × B) and R • F(B x C), where A, B and C are fuzzy subsets of a referential X. Let us suppose that there exists a t-conorm T', different from the maximum, such that (2.1) is verified. Then there exists at least two elements a, b • [0, 1] such that T'(a, b) > max(a, b) and let a > b. Then
T'(a, a) >I T'(a, b) > a.
(2.2)
Let X = {x, y} and A, B and C be three fuzzy subsets of X such that
a(x)=a,
B(x)= B ( y ) - - 1 ,
C ( y ) = l.
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We define Q ~ F(A x B) and R ~ F(B x C) such that
Q(x, x) = min(A(x), B(x)) = a,
Q(x, y) = min(A(x), B(y)) = a,
R(x, y) = min(B(x), C(y)) = 1,
n(y, y) = min(B(y), C(y)) = 1.
Then, from (2.1), we have
T' [T(Q(x, z), R(z, y)) = T'[T(Q(x, x), R(x, y)), T(Q(x, y), R(y, y))]
z~S
= T'[T(a, 1), T(a,
1)1 =
T'(a, a) <~min(A(x), C(y)) = a,
which is in opposition to (2.2). The theorem is proved. Therefore, we will use the t-conorm T' = max to define the composition between two fuzzy relations that, furthermore, can be extended to any set X, not necessarily countable or finite. In this way, we give the following definition, that will be used for infinite X as well as in the finite case.
Definition 2.3. Let Q ~ F(A x B) and R ~ F(B x C). The s u p - T composition of Q and R is the fuzzy relation QTR ~ F(A × C) defined by
(QTR)(x, y) = sup [T(Q(x, z), R(z, y))] z~X
for all x, y • X
This definition coincides, when A, B and C are crisp sets, with that proposed by various authors, like [13]. When the fuzzy relations are defined on crisp sets, the s u p - T composition is associated with the i n f - T ' composition (in symbols QT'R, where T' is the dual t-conorm of T) according to the relationship
(QT'R)C= (QCTRC). In fact, this was one of the reasons for attempting the extension of the i n f - T ' definition to the case of fuzzy relations defined on fuzzy subsets. Unfortunately, the extension cannot be made so straightforward, because, in general, the i n f - T ' composition of two fuzzy relations defined on fuzzy sets is not a fuzzy relation on fuzzy subsets, as we can see in the next counterexample (where the asterisk is the min-max composition).
Counterexample. Let X = {x, y} and let A, B and C be defined by A(x) = 0.3,
A ( y ) = 0.5,
B(x)=0.1,
B(y)=0.7,
C(x)=0,
C(y) =0.1.
Then
AxB=
BxC=
1 0.5 '
Assuming that Q = [~.1
0.1 0.5]'
R=
O.
=AxC.
01]
0.1 '
we have
0.
'
and we see that Q • F(A x B), R • F(B x C), whereas Q * R ~ F(A x C).
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Then, we need to modify the definition of inf-T' composition in a manner such that the composition continues to be a fuzzy relation on fuzzy subsets and coincides with the classical definition when the subsets are crisp. Thus we propose: Definition 2.4. Let A, B and C • F(X). The inf-T' composition of Q • F(A x B) and R • F(B × C) is
the fuzzy relation, QT'R • F(A x C), given by the membership function
(QT'R)(x, y ) = min{~nfx [T'(Q(x, z), R(z, y))], (A × C)(x, y)}. The following proposition gives the main properties of the inf-T' composition. Proposition 2.5. (a) QT'(R f3 S) = (QT'R) fq (aT'S), O • F(A x B) R, S • F(B × C).
(b) If Q c R then QT'S c RT'S, Q, R • F(A × B), S • F(B x C).
(C)
R-1T'Q -1, Q • F(A × B), R • F(B x C). (d) Q , ( R * S ) = ( Q , R ) * S , Q , R , S • F ( A x A ) . (aT'R)
-1 =
Proof. (a) For all x, y • X, (
[QT'(R f3 S)](x, y ) = min[~nf[T'(Q(x,z), min{R(z, y), S(z, y)})], ( A x C)(x, Y)I = min{~nf [min{T'(Q(x, z), R(z, y)), T'(Q(x, z), S(z, y))}], ( Z x C)(x, y)} =
min/min/inf {T'(Q(x, z), R(z, y))}, inf {T'(Q(x, z), S(z, y))}], (A x C)(x, Y)I I
LzeS
)
z~X
= min/minlinf_ L... (T'(Q(x, z), R(z, y))}, (a× C)(x, y)], min/inf~.~..(r'(a(x, z), S(z, y))), (a× C)(x, y)]} = min[(QT'R)(x, y), (QT'S)(x, y)] = [(QT'R) A (QT'S)](x, y). (b) For all x, y • X,
(QT'S)(x, y)= min{infx [T'(Q(x, z), S(z, y))], ( Z x C)(x, y)} ~ min{~nf [T'(R(x, z), S(z, y))], (A x C)(x, y)} =(RT'S)(x, y). (c) For all x, y • X,
(QT'R)-I(x, y)= (QT'R)(y, x)= min{~nf [T'(Q(y, z), R(z, x))], (a × C)(y, x)} = min{~nfx [T'(Q-X(z, y), R-l(x, z))], ( C x Z)(x, y)} = (R-1T'Q-1)(x, y). (d) For all x, y • X, [Q * (R * S)](x, y) = inf [max(a(x, z), (R *S)(z, y))] zEX
= inf
lmax(a(x, z),
z~X L
\
inf teX
[max(R(z, t), S(t, Y))l)/ /-I
= inf inf max[Q(x, z), max(R(z, t), S(t, y))] z ~X t ~ S
= inf inf max[max(O(x, z), R(z, t)), S(t, y)] tEg z~S
= i n f / m a x ( i n f [max(Q(x, z), R(z, t))], S(t, t~X L
\z~iX
y))/ /J
=inf[max((Q*R)(x,t),S(t,y))]=[(Q*R)*S](x,y). t~X
[]
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3. Solving equations on fuzzy relations with the sup-T composition Let X be a set; A, B and C be three fuzzy subsets of X, and let T be a lower semicontinuous t-norm. Consider the fuzzy relation equations
QTR = S where Q • F(A x B), R • F(B x C) and S • F(A x C) and point out the following typical problems: (pl) Determine R if Q and S are given. (p2) Determine Q if R and S are given. Let cp be the operator associated with the t-norm T. To broaden these questions we extend the definition of inf-q~ composition [12], to fuzzy relations defined on fuzzy subsets, in the following way:
Definition 3.1. Let Q • F(A x B) and R • F(B × C). Define the fuzzy relation Qq~R • F(A x C) by (Qq~R)(x, y) = minlinf/z~X[qv(Q(x, z), R(z, y))], (Z x C)(x, y)}
for all x, y • X.
Note that we have modified the definition of [12] in order to get QcpR becoming a fuzzy relation on the fuzzy set A x C. The previous definition verifies the following properties:
Proposition 3.2. (a) Let Q • F(A x B) and R, S • F(A x C) be such that R c S. Then QqgR c QcpS. (b) Let Q, R • F(A x B) such that Q c R and S • F(B x C). Then QcpS D RqgS. The proof is a straightforward verification. In the next theorem we characterize the solutions of the problem (pl). It is based on:
Lemma 3.3. Let Q • F(A × B), R • F(B x C) and S • F(A × C). Then: (a) R c Q-'cp(QTR). (b) S = QT(Q-'cpS). Proof. (a) For all x, y • X,
IQ ¢¢Q .>ICx
z> CQ .>>l <" ×
=min
y,}
min z~xinf[q~(Q(z, x), T(Q(z, x), R(x, y)))], (B x C)(x, y)) min{R(x, y), (B x C)(x, y)} = R(x, y). (b) For all x, y • X,
[QT( Q-1cpS)I(x, y) = sup [T(Q(x, z ), ( Q-' cpS)(z, Y))I zEX
~ sup [T(Q(x, z), q~(Q(x, z), S(x, y)))l <~S(x, y). zEX
[]
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Equations of fuzzy relations defined on fuzzy subsets
Theorem 3.4. Let ~ = {R • F(B × C) QTR = S}. If ~ --/=~then the fuzzy relation Q-lqgS • F(B × C) is the maximum of ~. Proof. Let R • ~. By Lemma 3.3(a), R ~ Q-topS. Then S = QTR = QT(Q-IqgS) ~ S, from (b) of the lemma. [] When in (pl) we have A = B = C and Q = S is immediate to prove that the fuzzy relation 0, x4=y, R(x,y)=
supQ(t,x), x = y ,
is a minimal solution of the equation QTIR = Q (Tl=min). Moreover R • F ( A ×A), because R(x, x) = supt~xQ(t, x) <~A(x). Thus • :/:0 and we have: Corollary 3.5. Let A • F(X) and Q • F(A × A). Then the fuzzy relation Q-lCpl Q • F(A × A) defined
by (Q-~cp~Q)(x, y) = min~ inf [q91(Q-~(x, z), O(z, y))], (Z x A)(x, y)~ for all x, y • X )
tzeX
is the greatest solution of QT1R = Q.
In an analogous way, by means of the following lemma we will prove results that allow us to obtain the greatest solution of the problem (p2). Lemma 3.6. Let Q • F(A x B), R • F(B x C) and S ~ F(A x C). Then (a) Q c (Rcp(QTR)-')-L (b) S ~ (RcpS-')-ITR. Proof. (a) For all x, y • X,
(RcR(QTR)-')-I(x, y) = (Rcp(QTR)-I)(y, x)
= min(~nfx [(p(R(y, z), (QTR)-I(z,x))], (BX Z)(y,x)} =min/tz~xinf[(p(R(y, z), supz~x{T(Q(x, t), R(t, z)})], ( a x B ) ( x , y)} = min{}nf [qo(R(y, z), max{ T(Q(x, y), R(y, z)), sup T(Q(x, t), R(t, z))})], ( Z x B)(x, y)} ~>min{}nf [(i0(R(y, z), T(Q(x, y), R(y, z)))], (Ax B)(x, y)[ I> min{O(x, y), (A x B)(x, y)} = Q(x, y). (b) For all x, y • X, ((Rq~S-1)-ITR)(x, y) = sup [T((RqgS-1)-'(x, z), R(z, y))] z~g
=sup [T(min[inf [qg(R(z, t), S(x, t))], ( B x A)(z, x)}, R(z, y))] z~S
L
\
~.t~S
<~sup[T(inf[qg(R(z,t),S(x,t))],R(z, y))] z~X
L
\t~X
~
[]
z~S
Theorem 3.7. Let Q= {Q • F ( A x B) I QTR =S }. If Q4~O, then the fuzzy relation (RcpS-1)-I • F(A x B) is the maximum of Q.
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Proof. Let Q ~ ~. By L e m m a 3.6(a), Q c (Rq~S-L) -l. Then S = QTR c (RqgS-~)-ITR ~ S (from Lemma 3.6(b)). []
4. On inf-T' composition of fuzzy relations on fuzzy subsets Let X be a set; A, B and C be three fuzzy subsets of X and T' be an upper semicontinuous triangular conorm. Parallel to the results established in Section 3 for s u p - T composition, we will study in this section the solutions of equations:
QT'R = S, that is
S(x, y) = min/inf [T'(Q(x, z), R(z, y))], (Z x C)(x, Y)I (t~g
for all x, y ~ X,
J
where Q e F(A x B), R ~ F ( B x C) and S ~ F(A x C), and we will point out again the following two problems: ( p ' l ) Determine R if Q and S are given. (p'2) Determine Q if R and S are given. Let fl be the operator associated with the conorm T'. To broaden these questions we extend the sup-fl definition to fuzzy relations defined on fuzzy subsets. If we define the sup-fl composition of fuzzy relations on fuzzy subsets like when the fuzzy relations are defined on crisp sets, that is
( QflR )(x, y) = sup [fl(Q(x, z ), R(z, y))], z~X
then the result can be a fuzzy relation that is not defined on the fuzzy subsets, as we can see in: Example. Let X = {x, y} and A = (0.7, 0.3),
AxB=
0.3
B = (0.5, 0.9),
o7]
0.3 '
BxC=
C = (0.7, 0.4), 0.7
o4] 0.4 '
[o7 o4] AxC=
0.3
0.3 "
If we consider T' = max, fi = fl~ and
o.3
o.4
LU.I
then
We see that QfllR ~ F(A × C). Therefore we need to change the definition and we give the following
Definition 4.1. Let Q ~ F(A x B) and R ~ F(B x C). We define the fuzzy relation QflR e F(A × C) by means of
(Qflg)(x, y) = min/su p [fl(Q(x, z), g(z, y))], (A × C)(x, Y)I t z~X
The sup-fl composition have the following properties:
J
for all x, y e X.
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M.J. Ferndndezet al. / Equations of fuzzy relationsdefined on fuzzy subsets
Proposition 4.2. (a) Let Q • F(A x B), R, S • F(B x C) with R c S. Then Q~R c QflS.
(b) Let Q, R ~ F(A x B) with Q ~ R and S • F(B x C). Then QflS ~ RflS. The proof is straightforward. To obtain the smaller solution (p'l) we need analogous results to those of Subsection 3.3. Here we have: Lemma 4.3. Let Q • F(A x B) and R • F(B x C). Then R ~ Q-Ifl(QT'R). Proof. For all x, y • X,
[Q-Ifl(QT'R)](x, y ) : min{szup [fl(Q-l(x, z), (QT'R)(z, y))], (B X C)(x, y)}
= min{SzUp[fl(Q(z, x), min{inf [T'(Q(z, t), R(t, y))], ( Z x C)(z, y)})], (Bx C)(x, y)} ~
~
[]
Remark 1. Let S • F(A x C). The following counterexample shows that S c~ QT'(Q-lflS). Let A = (0.7, 0.9), B = (0.8, 0.3), C = (0.4, 0.6), [0.7 0.3] B × C = [ 0"4 0.6] AxC=[0.4 0.6] A x B = 0.8 0.3 ' 0.3 0 . 3 ' 0.4 0.6" If we take
[06
Q = 0.7 0.2 ' then Q-lfllS=
0.4 0.5 '
[00] 0.3 0.3
and QT'I(Q-1fl1S)=
[06 0 l ,r0 0] 0.7 0.2J 1[0.3 0.3 =.0.3
0.3 "
To solve this trouble, we will restrict ourselves to the case in which supA(z)<-min(B(x), C(y)) for all x, y •X. z~X
Proposition 4.4. Let Q • F(A x B) and S • F(A x C). If (4.1) holds, then
(Q-lflS)(x, y) = sup [fl(Q(z, x), S(z, y))] ~ (B x C)(x, y) for all x, y • X. zEX
Proof. For all x, y • X,
sup [fl(Q(z, x), S(z, y))] ~
z~S
~
= sup A(z) <~min(B(x), C(y)). zEX
z~X
[]
(4.1)
M.J. Fern~indez et al. / Equations of fuzzy relations defined on fuzzy subsets
4.5. Let Q ~ F(A x B) and S ~ F(A × C), such that (4.1) holds. Then S c QT'(Q-IflS).
Lelnma Proof.
329
For all x,
y ~ X,
[QT'(Q-t~S)](x, y)= min{infx [T(Q(x, z), (Q-I~S)(z, y))], (Ax C)(x, y)} =min / inf [T'(Q(x, z), sup [fl(Q(t, z), S(t, y))])], ( Z x C ) ( x , y)} kzeX L
t~X
~>min{~nf [T'(Q(x, z), fl(Q(x, z), S(x, y)))], ( Z x C)(x, y)} ~>min{inf S(x, y), (m× C)(x, y)} = min{S(x, y), (Z x C)(x, y)} = S(x, y).
[]
Now we can give: 4.6. Let Q s F ( A x B), S eF(A × C) such that (4.1) holds, and let ~ = {R e F(B × C) QT'R = S}. Then, if ~ 4:0, the fully relation Q-lfls e F(B × C) is the minimum of ~, where
Theorem
(Q-'flS)(x, y) = sup [fl(Q-'(x, z), S(z, y))] for all x, y ~ X. zeX
follows immediately from Lemma 4.3 that R ~ Q-~flS. Then S = QT'R ~ QT'(Q-IflS) ~ S (from Lemma 4.5), that is QT'(Q-~flS) = S. [] Proof. Let R e E. It
Now we solve the problem (p'2) in an analogous way. 4.7. Let Q c F ( A x B ) , Nevertheless S dF (RflS-1)-IT'R.
Lemma
Proof.
R6F(BxC)
and S e F ( A x C ) .
Then Q~[Rfl(QT'R)-I] -~
For all x, y • X,
[R#(QT'R)-']-'(x, y) = [R[3(QT'R)-q(y, x)
min/sup [/~(R(y, z), (QT'R)-~(z, x))], (B x A)(y, x)/ Lz~X
J
=min{sup[fl(R(y, <~min{:up[fl(R(y, z),
t~xinf [T'(Q(x,
t), R(t, z))])], ( Z × B)(x, y)}
=min{sup[fi(R(y, z), man{ T'(Q(x, y), R(y, z)), inf T'(Q(x, t), R(t, z))})], (A× B)(x, y)} t~:y
f min]sup [fl(R(y, z), T'(Q(x, y), R(y, z)))], (A x B)(x, Y) I ~
i07 0.8 0.3 o3] 0.2 0.2
AxB= With R=
'
BxC=
' S=
{o3 0.2 0.4
[04 06] 0.3 0.3 '
AxC=
i04 06] 0.4 0.6 "
330
M.J. Ferndndezet al. / Equationsof fuzzy relationsdefined on fuzzy subsets
we have
[050.3 0.3°4] and
0.31T,[0.1
0.3,
0.3
0.3 0.3
o.3]
[]
Let A, B and C be three fuzzy subsets of X such that sup C(z) <- min(B(x), A(y))
for all x, y • X.
(4.2)
zEX
Proposition 4.8. Let R • F(B × C) and S • F(A × C). If (4.2) holds, then
(RflS-1)(x, y) = sup [fl(R(x, z), S(y, z))] ~<(B × A)(x, y)
for all x, y • X.
zEX
Proof. For all x, y • X,
sup [fl(R(x, z), S(y, z))] ~
zeX
[]
~
zeX
Lemma 4.9. Let R • F(B × C) and S • F(A × C), such that (4.2) holds. Then S c (RflS-1)-IT'R. Proof. For all x, y • X,
[(RflS-1)-IT'R](x, y ) = min[inf [T'((RflS-1)(z, x), R(z, y))], ( A x C)(x, y)}
= min{~nf [T'(sup [fl(R(z, t), S(x, t))], R(z, y))], (A× C)(x, y)} ~>mintinf [T'(fl(R(z, y), S(x, y)), R(z, y))], (A x C)(x, y)} kzeX
t> min / inf S(x, y), (A x C)(x, y)} = min{S(x, y), (Z x C)(x, y)} = S(x, y).
[]
tzeX
The next theorem characterizes the minimum solution of the problem (p'2). Theorem 4.10. Let R • F(B × C) and S • F(A × C) such that (4.2) holds. Let Q = {Q • F(A × B) I Q T ' R = S}. Then, if Q ¢ fl, the fuzzy relation ( R f l S - 1 ) -1 • F(A × B) defined by
(RflS-1)(x, y) = sup [fl(R(x, z), S(y, z))] for all x, y • X zEX
is the minimum of Q. Proof. Let Q • Q. From Lemma 4.7, Q ~ (RflS-1) -t. Then S = Q T ' R D (RflS-1)-'T'R ~ S (from Lemma 4.9). [] 5. Adjoint fuzzy relation equations
Let q0 be the operator associated with a t-norm T which is lower semicontinuous. The adjoint fuzzy relation equations were studied by Miyakoshi and Shimbo [12] and Izami, Tanaka and Asai [7], when
M.J. Fermindezet al. / Equationsof fuzzy relationsdefinedon fuzzy subsets
331
the fuzzy relations are defined on crisp subsets, dealing with fuzzy relation equations of the form: QqoR = S, i.e. S(x, y) = inf [q0(Q(x, z), R(z, y))]
for all x, y • X.
z~X
Here we are interested in this type of fuzzy relation equations for which the fuzzy relations are defined on fuzzy subsets. In this way our equations are of the form Oq~R = S where Q • F(A x B), R • F(B x C) and S • F(A x C), and S(x, y) = min/inf [¢p(Q(x, z), R(z, y))], (A x C)(x, Y)I )
tz~X
for all x, y • X.
And we formulate, as usual, the two problems: (dl) Determine R if Q and S are given. (d2) Determine Q if R and S are given. The questions (dl) and (d2) are solved with the help of the following lemmas: Lemma 5.1. Let Q • F(A x B), R • F(B x C) and S • F(A × C). Then (a) R = Q-1T(QcpR). (b) S ~ Qqg(Q-'TS). Proof. (a) For all x, y • X , [Q-'T(Qcpg)l(x, y) = sup [T(Q-'(x, z), (Qqggl(z, y)] zeX
<~sup [T(Q(z, x), cp(Q(z, x), R(x, y)))] ~
(b) For all x, y • X, [Qq)(Q-1TS)](x, y ) = min{~nf [q0(Q(x, z), (Q-ITS)(z, y))], ( A x C)(x, Y)I = min/tz~xinf[qv(Q(x, z), sup,~x[T(Q(t, z), S(t, Y))])l' ( A x C)(x, y)} =min/~z,xinf[q0(Q(x, z), max{T(Q(x, z), S(x, y)), sup,,xT(Q(t, z), S(t, Y))/)]' ( A x C)(x, y)} ~>min{~nfx [qg(Q(x, z), T(Q(x, z), S(x, y)))], (Z x C)(x, Y)I /> min{S(x, y), (A x C)(x, y)} = S(x, y).
[]
Lemma 5.2. Let Q • F(A x B), R • F(B x C) and S • F(A x C). Then (a) Q = (QcpR)cpR -1, (b) S c (ScpR-')qgR.
332
M.J. Ferndndezet al. / Equationsof fuzzy relationsdefinedon fuzzy subsets
Proof.
(a) For all x, y ~ X,
[(QqvR)cpR-l](x, y)= min{inf [cp((QcpR)(x, z), R-l(z, y))], (Ax B)(x, y)}
= min{~nf [q~(min{inf [q0(Q(x, t), R(t, z)), (Zx C)(x, z)}, R(y, z))], (Zx B)(x, y)} i> min(!nf [q0(inf [qg(Q(x, t), R(t, z))], R(y, z))], (Zx B)(x, y)} ~>min{!nf [qg(q0(Q(x, y), R(y, z)), R(y, z))], (Zx B)(x, y)} ~>min{~nf Q(x, y), (Z x B)(x, y)} = min{Q(x, y), (Z x B)(x, y)} = Q(x, y). (b) For all x, y ~ X, [(ScpR-l)epR](x, y)= min{~nf [qg((Sq~R-l(x, z), R(z, y))], (Ax C)(x, y)}
= min{~nf [q~(min{inf [q0(S(x, t), R(z, t))], ( A x B ) ( x , z)}, R(z, y))], ( Z x C ) ( x , y)} i> min{inf [qg(infL \t~x [qg(S(x, t), R(z, t))], R(z, Y)) t' (Zx C)(x, y)} ~>min{inf [qg(q0(S(x, y), R(z, y)), R(z, y))], (Z x C)(x, y)} /> min{S(x, y), (A x C)(x, y)} = S(x, y).
[]
With these lemmas we prove the following theorems that give us the smallest solution of the problem (dl) and the greatest solution of the problem (d2). Theorem 5.3. Let Q ~ F(A x B), S E F(A x C) and ~ = {R • F(B x C) I Qq~R = S}. If ~ 4=O, then the fuzzy relation Q-1TS e F(B x C) is the minimum of ~. Proof. Let R e ~. From (a) of Lemma 5.1, R ~ Q-~TS. Then S = QcpR ~ Qqg(Q-~TS) ~ S (from (b) of Lemma 5.1). []
Analogously for the problem (d2). Theorem 5.4. Let R ~ F(B × C), S • F(A x C) and Q = {Q ~ F(A x B) I QqoR = S}. If Q 4=O, then the fuzzy relation ScpR-1 ~ F(A × B) defined by
(ScpR-')(x, y)= min{~nf [q~(S(x, z), R-l(z, y))], (A x B)(x, y)} for allx, y ~ X is the maximum of Q. Proof. Let Q e Q. From (a) of Lemma 5.2, Q c ScpR -~. Then S = Qq~R ~ (ScpR-l)cpR ~ S (from (b) of Lemma 5.2). []
Now we are interested in adjoint fuzzy relation equations for which inf-q0 composition is replaced by sup-fl composition already introduced, when the fuzzy relations are defined on fuzzy subsets, as in Section 4 (Definition 4.1): QflR = S,
M.J. Ferndndez et al. / Equations of fuzzy relations defined on fuzzy subsets
333
i.e,~
S(x, y)= min/sup [13(Q(x, z), R(z, y))], (A x C)(x, y)~ for all x, y tzeX
• X,
J
where Q • F(A x B), R • F(B x C) and S • F(A x C). Once again, we formulate the two problems: (d'l) Determine R if Q and S are given. (d'2) Determine Q if R and S are given. The following lemma is the basis for solving the problem (d'l). 5.5. Let Q • F(A x B) and S • F(A x C). Then S ~ Q[I(Q-IT'S).
Lemma Proof.
For all x,
y • X,
[Q~(Q-~T'S)](x, y ) = man{sup [/3(Q(x, z), (Q-1T'S)(z, Y))], (A x C)(x, y)}
= min/suptz~x[/3(Q(x, z), minlinf[T'(Q(t,t,~x
z), S(t, y))], ( B x C ) ( z , y)})], ( A x C ) ( x , y)}
~
[]
2. Let R • F(B x C). The following counterexample shows that R c~ Q-1T'(Q[3R). Let
Remark
A = (0.7, 0.3), AxB=
With
B = (0.5, 0.9),
C = (0.2, 0.4),
041
3 0.3 '
0.2 0.4 '
AxC=
04]
0.2 0.3 "
o6] R=[Ol 04]
Q=
0.4 0.3 0 . 1 '
0.2 0.3
we have
and Q-'*(Q~,R)=[~I2
2131]* [00.2 00.3]=[00122 00133].
In a special case, when the fuzzy subsets A, B and C verify that sup B(z) ~ min(A(x), C(y))
for all x, y • X
zEX
then from Proposition 4.4 it follows (QjgR)(x, y) = sup [~(Q(x, z), R(z, y))]. z~x Lemma
5.6. Let Q • F(A x B) and R • F(B x C). If (5.1) holds, then R c Q-IT'(Q~R).
(5.1)
334
M.J. Ferndndezet al. / Equations of fuzzy relations defined on fuzzy subsets
Proof. For all x, y • X, [Q-1T'(QflR)](x, y ) = min{~nf [T'(Q-I(x, z), (QflR)(z, y))], (B x C)(x, y)}
=min/tz,xinf L[T'(Q(z', x), sup,~x[fl(Q(z, t), R(t, y))])], ( B x C)(x, y)} ~>min{}nf [T'(Q(z, x), fl(Q(z, x), R(x, y)))], ( B x C)(x, y)} ~>min{}nf R(x, y), (B x C)(x, y)} = min{R(x, y), ( B x C)(x, y)} = R(x, y).
[]
Theorem 5.7. Let Q • F(A x B), S • F(A x C) such that (5.1) holds and ~ = {R • F(B x C) I QflR = S}. I f R q~O, then the fuzzy relation Q-1T'S • F(B x C) defined by (Q-'T'S)(x, y) = mintinft~,x[T'(Q-'(x, z), S(z, y))], (B x C)(x, y)} for all x, y • X is the maximum of ~.
Proof. Let R • E. From the previous lemma R 71Q-1T'S. Then S = QflR c Qfl(Q-~T'S) c S (from Lemma 5.5). [] The problem (d'2) is solved here in a very restrictive context because the following inclusions between relations for Q • F(A × B), R • F(B x C) and S • F(A x C), are in general false: (a) Q = (aflR)flR -1. (b) S = (SflR-1)flR. Counterexamples. Let
A = (0.7, 0.3),
B = (0.5, 0.9),
C = (0.2, 0.4),
o4] AxB= With
0.3 0.3 '
0.2 0.4 '
[Ol 02] R__[Ol o4] Q=
0.3 0.1 '
0.2 0.4
we have 0 0.4] QfllR= 0.2 0.3 and [0 (Qfl'R)fllR-'
and with
.__
=
o4 0.4]'
0.2
s__[o 1 03]
we have S f l l R - ' = [ 0"40.3 0.30"4]
0.2 0.1
o4] AxC=
0.2 0.3 "
335
M.J. Ferndndezet al. / Equations of fuzzy relations defined on fuzzy subsets
and 04
0
In the next proposition we give a sufficient condition in order to guarantee that (a) and (b) hold. Proposition 5.8. (a) If A, B and C • F(X) verify
sup B(z) <<-min(A(x), C(y))
for all x, y • X
(5.2)
for all x, y • X
(5.3)
zEX
then Q = (QflR)flR -1 (b) If A, B and C • F(X) verify
sup C(z) <<-min(A(x), B(y)) zEX
then S = (SflR-1)flR.
Proof. (a) For all x, y • X, [(QflR)flR-1](x, y)=min{sup[fl((QflR)(x, z), R-'(z, y))], (A × B)(x, y)}
= min{SUxP[fl(SUxP [fl(Q(x, t), R(t,z))], R ( y , z ) ) ] , ( Z x B ) ( x , y ) } ~
[(SflR-')flR](x, y ) = min{szup [fl((SflR-1)(x, z), R(z, y))], (m x C)(x, y)}
= min/su p [fl(sup [fl(S(x, t), R(z, t))], R(z, y ) ) ] , (Ax C)(x, y)} tz~X
L
\t~X
<~min/SzU p _ [fl(fl(S(x, y), R(z, y)), R(z, y))], (Z x C)(x, y)} <~min{S(x, y), (A x C)(x, y)} = S(x, y).
[]
Remark 3. Note that (5.2) and (5.3) remain true if and only if B = C is a fuzzy subset with a constant membership function, and B c A. Theorem 5.9. Let R • F(B x C), S • F(A x C) such that A, B and C verify (5.2) and (5.3). Let
0 = {Q • F(A × B) IQflR = S}. If Q 4=~J, then the fuzzy relation StiR -1 • F(A x B) defined by (SflR-l)(x, y) = sup [fl(S(x, z), R(y, z))] for all x, y e X zEX
is the minimum of Q.
Proof. Let Q e Q. By (a) of Proposition 5.8, Q ~ StiR -1. Then S = QflR c (SflR-~)flR ~ S (by (b) of Proposition 5.8). [] Conclusions The solution of fuzzy relational equations when the fuzzy relations are defined on fuzzy sets has been investigated.
336
M.J. Ferndndez et al. / Equations of fuzzy relations defined on fuzzy subsets
We have given the procedures for solving different types of fuzzy relation equations. We hope that their diversity allows us to find a suitable model for real-world problems. These results may be applicable to fuzzy inference under compositional rules of inference; in this sense, if two propositions P and Q, whose predicates are defined on universes of discourse A and B, fuzzy sets, then an implication P---~ Q is defined, in general, as a fuzzy relation on A x B [3, 12].
References [1] M.K. Chakraborty and M. Das, Studies in fuzzy relations over fuzzy subsets, Fuzzy Sets and Systems 9 (1983) 79-89. [2] M. Delgado, J.L. Verdegay and M.A. Vila, El problema del ~irbol minimal para grafos difusos, Trabajos de LO. 2 (1) (1987) 3-20. [3] A. Di Nola, S. Sessa, W. Pedrycz and E. Sanchez, Fuzzy Relation Equations and their Applications to Knowledge Engineering (Kluwer Academic Publishers, Dordrecht-Boston, 1989). [4] J. Drewniak, Note on fuzzy relation equations, Busefal 12 (1982) 50-51. [5] D. Dubois and H. Prade, Additions of interactive fuzzy numbers, 1EEE. Trans. Automat. Control 26 (1981) 926-936. [6] S. Gottwaid, Fuzzy set theory with T-norms and ¢p-operators, in: A. Di Nola and A. Ventre, Eds., Topics in the Mathematics of Fuzzy Systems (Verlag TUV Rheinland, K61n, 1986) 143-196. [7] K. Izumi, H. Tanaka and K. Asai, Adjointness of fuzzy systems, Fuzzy Sets and Systems 20 (1986) 211-231. [8] C. Kimberling, On a class of associative functions, Publ. Math. Debrecen 20 (1973) 21-39. [9] E.P. Klement, Construction of fuzzy o-algebras using triangular norms, J. Math. Anal. Appl. 85 (1982) 543-565. [10] K. Menger, Statistical metric spaces, Proc. Nat. Acad. Sci. USA 28 (1942) 535-537. [11] M. Miyakoshi and M. Shimbo, Composite fuzzy relations with T-norms, Trans. IECE Japan D 67 (4) (1984) 391-398. [12] M. Miyakoshi and M. Shimbo, Solutions of composite fuzzy relational equations with triangular norms, Fuzzy Sets and Systems 16 (1985) 53-63. [13] W. Pedrycz, Fuzzy relational equations with triangular norms and their resolutions, Busefal 11 (1982) 24-32. [14] W. Pedrycz, Some aspects of fuzzy decision-making, Kybernetes 11 (1982) 297-301. [15] W. Pedrycz, Fuzzy Control and Fuzzy Systems (Wiley/Research Studies Press, London-New York, 1989). [16] E. Sanchez, Resolution of composite fuzzy relation equations, Inform. and Control 30 (1976) 38-48. [17] E. Sanchez, Compositions of fuzzy relations, in: M.M. Gupta, R.K. Ragade and R.R. Yager, Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, Amsterdam, 1979) 421-433. [18] B. Schweizer and A. Sklar, Probabilistic Metric Spaces (North-Holland, Amsterdam, 1983). [19] E. Trillas, Conjuntos Borrosos (Editorial Vicens-Vives, Barcelona, 1980). [20] L.A. Zadeh, Similarity relations and fuzzy orderings, Inform. Sci. 3 (1971) 177-200. [21] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Parts I, II, III, Inform. Sci. 8 (1975) 199-249; 301-357; 9 (1975) 43-80.
319
Equations of fuzzy relations defined on fuzzy subsets M.J. Fern~indez, F. Su~irez and P. Gil Department of Mathematics, University of Oviedo, Spain Received August 1991 Revised November 1991
Abstract: In this paper some results of the theory of fuzzy relation equations are generalized, when we consider fuzzy relations defined on fuzzy sets instead of crisp sets. If the extension of classical definitions for composition of relations [12, 13] is not straighforward, we provide suitables counterexamples illustrating the necessity of change in definitions. We find the greatest (or the least) solutions of fuzzy relation equations with different types of compositions, like the sup-T, inf-T', inf-q0 or sup-fl compositions.
Keywords: Fuzzy relations; fuzzy relation equations; types of equations.
The theory of fuzzy relation equations was introduced by Sanchez [16], obtaining the author the greatest solution of the equation Q o R = S (for the sup-min composition), knowing Q and S or knowing R and S, where the 'greatest' is viewed like the maximum of the set of solutions of the equation (in the sense of set-theoretic inclusion). Since Sanchez, many papers dealing with such equations have appeared (see Chapter 15 of [3] for an extensive bibliography until 1989). But, in all of them, the fuzzy relations are defined on crisps sets. This paper deals with the study of fuzzy relational equations, when the fuzzy relations are defined on fuzzy sets instead of classical sets. The necessity of defining fuzzy relations over fuzzy subsets was already observed by various authors, see [2, 19, 20]. Obviously, since the crisp sets are a particular case of fuzzy sets, certain results of this paper are generalizations of some results in works previously refered. This paper is organized as follows: Basic definitions and notations are given in Section 1, introducing in this section the properties that we need in the next sections. In Section 2 we introduce the definitions of s u p - T and inf-T' compositions of two fuzzy relations defined on fuzzy sets as a generalization of the definitions given when the fuzzy relations are defined on crisp sets [13]. In Section 3 the fuzzy equations of the form Q T R = S, where Q T R is the s u p - T composition defined in the section before are proposed and solved. In an analogous way, in Section 4 we propose and solve under restrictive conditions the fuzzy equations of the form Q T ' R = S, with the inf-T' composition defined in Section 2. Finally, Section 5 studies the adjoint fuzzy relation equations.
1. Introduction
We will use the standard notation of fuzzy sets, employing capital letters to denote fuzzy sets and relations, and denoting the family of fuzzy sets defined in a set X (crisp or fuzzy) by F ( X ) . Correspondence to: Prof. Dr. P. Gil, Departamento de Matematicas, Facultad de Ciencias, 2a. Planta, Calvo Sotelo, s/n, 33007 Oviedo, Spain. 0165-0114/92/$05.00 (~) 1992--Elsevier Science Publishers B.V. All rights reserved
M.J. Ferndndez et al. / Equations of fuzzy relations defined on fuzzy subsets
320
Let I = [0, 1]. A triangular norm (t-norm) [10, 18] is a real-valued function of two variables such that T(x, 1)=x, T(x, y ) = T ( y , x ) , T[T(x, y ) , z ] = T[x, T(y,z)] and T(x, y)<~
T:I×I~I
T(x, z) if y <~z; for all x, y, z • l. A triangular conorm (t-conorm) is a real-valued function of two variables T ' : I x I--->I 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) i f y ~ < z ; for all x, y, z • L Given any t-norm T, the corresponding t-conorm T', dual of T, is defined by setting
T'(x, y) = 1 - T(1 - x, 1 - y)
for all x, y • 1.
Some examples of t-norms and related dual t-conorms are reported below: Tl(x, y) = min(x, y),
T'l(X, y) = max(x, y),
T2(x, y) =x . y,
T~(x, y) =x + y - x . y,
T3(x, y) = max(0, x + y - 1),
T~(x, y) = min(1, x + y).
It is immediate that, for any t-norm and t-conorm, the following inequalities hold for all x, y • I:
T(x, y) <~Tl(x, y), T~(x, y) <~T'(x, y). We now recall the definition of the pseudocomplement of a t-norm T (resp. t-conorm T'):
Definition 1.1 [14]. An operator tp : I × I---~ I (resp. /3) is associated with a t-norm T (resp. t-conorm T') if (a) (b) (c)
for all a, b, c • I: ~(a, max{b, c}) = max{qg(a, b), qg(a, c)} (resp. /3(a, min{b, c}) = min{/3(a, b),/3(a, c)}), T(a, qg(a, b)) <~b (resp. T'(a, /3(a, b)) >! b), cp(a, T(a, b)) >! b (resp. fl(a, T'(a, b)) <~b).
If T = T1 and T' = T'I these operators were introduced by Sanchez [16]. Gottwald has proved in [6] that: " A n operator q~ (resp. fl) is associated with a t-norm T (resp. t-conorm T') iff T (resp. T') is lower semicontinuous (resp. upper semicontinuous)". Then from now on, we consider lower semicontinuous triangular norms and upper semicontinuous triangular conorms. All t-norms (resp. t-conorms) mentioned in the above examples are lower (resp. upper) semicontinuous. The corresponding operators q0 and/3 are listed below:
{1,
a<-b, a >b,
tpl(a , b) = b,
{0, /31(a, b) = b,
~2(a, b)=
/32(a, b)= ,
cp3(a,b)={lb, -
a>b, a+ l,
a<-b, a>b,
/33(a,b)=
a >~b, a
b-a 1 a {~'_
a,
,
a
a>~b' a
The operator tp verifies a lot of properties that we need in this work. We recall some of them in:
Proposition 1.2 [3, 4, 11, 12]. Let T be a lower semicontinuous t-norm and cp the operator associated with T. Then: (a) q0(a, bl) ~< qg(a, bE) if b~ ~ b2, (b) cp(al, b) >~cp(a2, b) if al <~a2, (c) tp(qg(a, b), b)/> a, (d) ~(a, b) = sup{c • I I T(a, c) <~b}.
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In the same way, we have:
Proposition 1.3. Let T' the dual t-conorm of T and cp (resp. /3) be the operator associated with T (resp. T'). Then /3(a,b)=l-q~(1-a,l-b)
foralla, b • l .
Proof. We have /3(a, b) = inf{c • I I T'(a, c) >/b} = inf{c • I ] 1 - T(1 - a, 1 - c) ~> b}
=inf{ccIlT(1-a,
1-c)~< l-b}
=inf{1-c
•I I T(1-a,
c)~< l - b }
= 1 - sup{c • ! [ T ( 1 - a , c)~< 1 - b} = 1 - q0(1- a, 1 - b).
[]
The next is an immediate consequence of the two previous propositions.
Proposition 1.4. (a) /3(a, b 0 <~/3(a, b2) if bl <<-b2, (b) /3(al, b)~>/3(az, b) ifal<~a2, (c) /3(/3(a, b), b) ~< a. Let now Q • F ( X x Y), R • F ( Y × Z), T be any t-norm and T' be any t-conorm. Definition 1.5 [13]. The s u p - T (resp. i n f - T ' ) composition of Q and R, denoted by QTR (resp. QT'R) is defined as a fuzzy relation on X x Z whose membership function is given by:
(QTR)(x, z) = sup [T(Q(x, y), R(y, z))]
resp.
yEY
(QT'R)(x, z) = inf [T'(Q(x, y), n(y, z))]. yeY
for all x • X, z • Z .
Definition 1.6 [12]. Let q0 (resp. /3) be the operator associated with a t-norm T (resp. t-conorm T'). The inf-q9 composition (resp. sup-/3 composition) of Q and R, denoted by QcpR (resp. Q/3R) is defined as a fuzzy relation on X × Z whose membership function is given by: (QcpR)(x, z) = inf [qg(Q(x, y), R(y, z))] y~Y
resp.
(Q/3R)(x, z) = sup [/3(Q(x, y), R(y, z))] y~Y
for all x • X, z • Z. Let X be the initial set and A, B • F(X). We will use the next definition of Cartesian product proposed by Zadeh [20]: A × B is the fuzzy subset of X × X defined by
(A × B)(x, y) = min(A(x), B(y))
for all x, y • X
and hence the following: Definition 1.7 [1, 20]. Let R • F(A x B), i.e. R(x, y) <~min(A(x), B(y)) for all x, y • X. Then R is a fuzzy relation from A to B.
2. (sup-T) fuzzy relation equations The max-min (sup-min) composition of two fuzzy relations is closely linked with the extension principle [21], as it was stated by Sanchez [17] and Pedrycz [15].
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We propose a generalization of this composition, using t-norms, as in Dubois and Prade [5] to define the fuzzy operations between fuzzy numbers. Due to the associative property of the t-conorms, in [8] it is defined:
T'{xl, x2 . . . . , x,} = T'{T'{xl, x2 . . . . , x~-l}, Xn}
for n >i 2.
As well, by the monotonity property of t-conorms and by its definition on the closed interval [0, 1], we can define on a countable set, see [9]:
T'
{Xi}
=
i~N
lim T'{xl,
X 2 ....
, Xn}.
n--~
Let A, B and C be fuzzy subsets of X (finite or countable crisp sets), we can define the composition of fuzzy relations by means of: Definition 2.1. Let Q • F(A × B) and R • F(B × C). The T ' - T composition of Q and R, in symbols Qr,rR, is the fuzzy subset given by the membership function
(Qr,rR)(x, y) = T' [T(Q(x, z), R(z, y))]
for all x, y • X
z~X
where T and T' are any t-norm and t-conorm respectively, not necessarily duals. When T' -- max and T -- rain, we have the classical definition of max-min composition between two fuzzy relations. The next theorem proves that we cannot be very ambitious in the definition of the composition of fuzzy relations defined on fuzzy sets, provided that we wish that the result continues to be a fuzzy relation too. Theorem 2.2. For any t-norm T, Qr,rR will be a fuzzy relation on fuzzy subsets, for any Q and R fuzzy
relations on fuzzy subsets if and only if T'(x, y) = max(x, y)
for all x, y • [0, 1].
Proof. Let Q e F(A × B) and R e F(B x C). We have
T(Q(x, z), R(z, y)) <- min(Q(x, z), R(z, y)) ~
for all x, y, z • X
whatever X be. Consequently,
(Qr,rR)(x, y) = max [T(Q(x, z), R(z, y))] ~
for all x, y • X
zeX
and the sufficiency condition is proved; let us regard its necessity. By hypothesis,
T'z~xtT(Q(x, z), g(z, Y))I <~min(A(x), C(y))
for all x, y • X
(2.1)
for each t-norm T and for any fuzzy relations Q • F(A × B) and R • F(B x C), where A, B and C are fuzzy subsets of a referential X. Let us suppose that there exists a t-conorm T', different from the maximum, such that (2.1) is verified. Then there exists at least two elements a, b • [0, 1] such that T'(a, b) > max(a, b) and let a > b. Then
T'(a, a) >I T'(a, b) > a.
(2.2)
Let X = {x, y} and A, B and C be three fuzzy subsets of X such that
a(x)=a,
B(x)= B ( y ) - - 1 ,
C ( y ) = l.
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We define Q ~ F(A x B) and R ~ F(B x C) such that
Q(x, x) = min(A(x), B(x)) = a,
Q(x, y) = min(A(x), B(y)) = a,
R(x, y) = min(B(x), C(y)) = 1,
n(y, y) = min(B(y), C(y)) = 1.
Then, from (2.1), we have
T' [T(Q(x, z), R(z, y)) = T'[T(Q(x, x), R(x, y)), T(Q(x, y), R(y, y))]
z~S
= T'[T(a, 1), T(a,
1)1 =
T'(a, a) <~min(A(x), C(y)) = a,
which is in opposition to (2.2). The theorem is proved. Therefore, we will use the t-conorm T' = max to define the composition between two fuzzy relations that, furthermore, can be extended to any set X, not necessarily countable or finite. In this way, we give the following definition, that will be used for infinite X as well as in the finite case.
Definition 2.3. Let Q ~ F(A x B) and R ~ F(B x C). The s u p - T composition of Q and R is the fuzzy relation QTR ~ F(A × C) defined by
(QTR)(x, y) = sup [T(Q(x, z), R(z, y))] z~X
for all x, y • X
This definition coincides, when A, B and C are crisp sets, with that proposed by various authors, like [13]. When the fuzzy relations are defined on crisp sets, the s u p - T composition is associated with the i n f - T ' composition (in symbols QT'R, where T' is the dual t-conorm of T) according to the relationship
(QT'R)C= (QCTRC). In fact, this was one of the reasons for attempting the extension of the i n f - T ' definition to the case of fuzzy relations defined on fuzzy subsets. Unfortunately, the extension cannot be made so straightforward, because, in general, the i n f - T ' composition of two fuzzy relations defined on fuzzy sets is not a fuzzy relation on fuzzy subsets, as we can see in the next counterexample (where the asterisk is the min-max composition).
Counterexample. Let X = {x, y} and let A, B and C be defined by A(x) = 0.3,
A ( y ) = 0.5,
B(x)=0.1,
B(y)=0.7,
C(x)=0,
C(y) =0.1.
Then
AxB=
BxC=
1 0.5 '
Assuming that Q = [~.1
0.1 0.5]'
R=
O.
=AxC.
01]
0.1 '
we have
0.
'
and we see that Q • F(A x B), R • F(B x C), whereas Q * R ~ F(A x C).
324
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Then, we need to modify the definition of inf-T' composition in a manner such that the composition continues to be a fuzzy relation on fuzzy subsets and coincides with the classical definition when the subsets are crisp. Thus we propose: Definition 2.4. Let A, B and C • F(X). The inf-T' composition of Q • F(A x B) and R • F(B × C) is
the fuzzy relation, QT'R • F(A x C), given by the membership function
(QT'R)(x, y ) = min{~nfx [T'(Q(x, z), R(z, y))], (A × C)(x, y)}. The following proposition gives the main properties of the inf-T' composition. Proposition 2.5. (a) QT'(R f3 S) = (QT'R) fq (aT'S), O • F(A x B) R, S • F(B × C).
(b) If Q c R then QT'S c RT'S, Q, R • F(A × B), S • F(B x C).
(C)
R-1T'Q -1, Q • F(A × B), R • F(B x C). (d) Q , ( R * S ) = ( Q , R ) * S , Q , R , S • F ( A x A ) . (aT'R)
-1 =
Proof. (a) For all x, y • X, (
[QT'(R f3 S)](x, y ) = min[~nf[T'(Q(x,z), min{R(z, y), S(z, y)})], ( A x C)(x, Y)I = min{~nf [min{T'(Q(x, z), R(z, y)), T'(Q(x, z), S(z, y))}], ( Z x C)(x, y)} =
min/min/inf {T'(Q(x, z), R(z, y))}, inf {T'(Q(x, z), S(z, y))}], (A x C)(x, Y)I I
LzeS
)
z~X
= min/minlinf_ L... (T'(Q(x, z), R(z, y))}, (a× C)(x, y)], min/inf~.~..(r'(a(x, z), S(z, y))), (a× C)(x, y)]} = min[(QT'R)(x, y), (QT'S)(x, y)] = [(QT'R) A (QT'S)](x, y). (b) For all x, y • X,
(QT'S)(x, y)= min{infx [T'(Q(x, z), S(z, y))], ( Z x C)(x, y)} ~ min{~nf [T'(R(x, z), S(z, y))], (A x C)(x, y)} =(RT'S)(x, y). (c) For all x, y • X,
(QT'R)-I(x, y)= (QT'R)(y, x)= min{~nf [T'(Q(y, z), R(z, x))], (a × C)(y, x)} = min{~nfx [T'(Q-X(z, y), R-l(x, z))], ( C x Z)(x, y)} = (R-1T'Q-1)(x, y). (d) For all x, y • X, [Q * (R * S)](x, y) = inf [max(a(x, z), (R *S)(z, y))] zEX
= inf
lmax(a(x, z),
z~X L
\
inf teX
[max(R(z, t), S(t, Y))l)/ /-I
= inf inf max[Q(x, z), max(R(z, t), S(t, y))] z ~X t ~ S
= inf inf max[max(O(x, z), R(z, t)), S(t, y)] tEg z~S
= i n f / m a x ( i n f [max(Q(x, z), R(z, t))], S(t, t~X L
\z~iX
y))/ /J
=inf[max((Q*R)(x,t),S(t,y))]=[(Q*R)*S](x,y). t~X
[]
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3. Solving equations on fuzzy relations with the sup-T composition Let X be a set; A, B and C be three fuzzy subsets of X, and let T be a lower semicontinuous t-norm. Consider the fuzzy relation equations
QTR = S where Q • F(A x B), R • F(B x C) and S • F(A x C) and point out the following typical problems: (pl) Determine R if Q and S are given. (p2) Determine Q if R and S are given. Let cp be the operator associated with the t-norm T. To broaden these questions we extend the definition of inf-q~ composition [12], to fuzzy relations defined on fuzzy subsets, in the following way:
Definition 3.1. Let Q • F(A x B) and R • F(B × C). Define the fuzzy relation Qq~R • F(A x C) by (Qq~R)(x, y) = minlinf/z~X[qv(Q(x, z), R(z, y))], (Z x C)(x, y)}
for all x, y • X.
Note that we have modified the definition of [12] in order to get QcpR becoming a fuzzy relation on the fuzzy set A x C. The previous definition verifies the following properties:
Proposition 3.2. (a) Let Q • F(A x B) and R, S • F(A x C) be such that R c S. Then QqgR c QcpS. (b) Let Q, R • F(A x B) such that Q c R and S • F(B x C). Then QcpS D RqgS. The proof is a straightforward verification. In the next theorem we characterize the solutions of the problem (pl). It is based on:
Lemma 3.3. Let Q • F(A × B), R • F(B x C) and S • F(A × C). Then: (a) R c Q-'cp(QTR). (b) S = QT(Q-'cpS). Proof. (a) For all x, y • X,
IQ ¢¢Q .>ICx
z> CQ .>
=min
y,}
min z~xinf[q~(Q(z, x), T(Q(z, x), R(x, y)))], (B x C)(x, y)) min{R(x, y), (B x C)(x, y)} = R(x, y). (b) For all x, y • X,
[QT( Q-1cpS)I(x, y) = sup [T(Q(x, z ), ( Q-' cpS)(z, Y))I zEX
~ sup [T(Q(x, z), q~(Q(x, z), S(x, y)))l <~S(x, y). zEX
[]
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Equations of fuzzy relations defined on fuzzy subsets
Theorem 3.4. Let ~ = {R • F(B × C) QTR = S}. If ~ --/=~then the fuzzy relation Q-lqgS • F(B × C) is the maximum of ~. Proof. Let R • ~. By Lemma 3.3(a), R ~ Q-topS. Then S = QTR = QT(Q-IqgS) ~ S, from (b) of the lemma. [] When in (pl) we have A = B = C and Q = S is immediate to prove that the fuzzy relation 0, x4=y, R(x,y)=
supQ(t,x), x = y ,
is a minimal solution of the equation QTIR = Q (Tl=min). Moreover R • F ( A ×A), because R(x, x) = supt~xQ(t, x) <~A(x). Thus • :/:0 and we have: Corollary 3.5. Let A • F(X) and Q • F(A × A). Then the fuzzy relation Q-lCpl Q • F(A × A) defined
by (Q-~cp~Q)(x, y) = min~ inf [q91(Q-~(x, z), O(z, y))], (Z x A)(x, y)~ for all x, y • X )
tzeX
is the greatest solution of QT1R = Q.
In an analogous way, by means of the following lemma we will prove results that allow us to obtain the greatest solution of the problem (p2). Lemma 3.6. Let Q • F(A x B), R • F(B x C) and S ~ F(A x C). Then (a) Q c (Rcp(QTR)-')-L (b) S ~ (RcpS-')-ITR. Proof. (a) For all x, y • X,
(RcR(QTR)-')-I(x, y) = (Rcp(QTR)-I)(y, x)
= min(~nfx [(p(R(y, z), (QTR)-I(z,x))], (BX Z)(y,x)} =min/tz~xinf[(p(R(y, z), supz~x{T(Q(x, t), R(t, z)})], ( a x B ) ( x , y)} = min{}nf [qo(R(y, z), max{ T(Q(x, y), R(y, z)), sup T(Q(x, t), R(t, z))})], ( Z x B)(x, y)} ~>min{}nf [(i0(R(y, z), T(Q(x, y), R(y, z)))], (Ax B)(x, y)[ I> min{O(x, y), (A x B)(x, y)} = Q(x, y). (b) For all x, y • X, ((Rq~S-1)-ITR)(x, y) = sup [T((RqgS-1)-'(x, z), R(z, y))] z~g
=sup [T(min[inf [qg(R(z, t), S(x, t))], ( B x A)(z, x)}, R(z, y))] z~S
L
\
~.t~S
<~sup[T(inf[qg(R(z,t),S(x,t))],R(z, y))] z~X
L
\t~X
~
[]
z~S
Theorem 3.7. Let Q= {Q • F ( A x B) I QTR =S }. If Q4~O, then the fuzzy relation (RcpS-1)-I • F(A x B) is the maximum of Q.
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327
Proof. Let Q ~ ~. By L e m m a 3.6(a), Q c (Rq~S-L) -l. Then S = QTR c (RqgS-~)-ITR ~ S (from Lemma 3.6(b)). []
4. On inf-T' composition of fuzzy relations on fuzzy subsets Let X be a set; A, B and C be three fuzzy subsets of X and T' be an upper semicontinuous triangular conorm. Parallel to the results established in Section 3 for s u p - T composition, we will study in this section the solutions of equations:
QT'R = S, that is
S(x, y) = min/inf [T'(Q(x, z), R(z, y))], (Z x C)(x, Y)I (t~g
for all x, y ~ X,
J
where Q e F(A x B), R ~ F ( B x C) and S ~ F(A x C), and we will point out again the following two problems: ( p ' l ) Determine R if Q and S are given. (p'2) Determine Q if R and S are given. Let fl be the operator associated with the conorm T'. To broaden these questions we extend the sup-fl definition to fuzzy relations defined on fuzzy subsets. If we define the sup-fl composition of fuzzy relations on fuzzy subsets like when the fuzzy relations are defined on crisp sets, that is
( QflR )(x, y) = sup [fl(Q(x, z ), R(z, y))], z~X
then the result can be a fuzzy relation that is not defined on the fuzzy subsets, as we can see in: Example. Let X = {x, y} and A = (0.7, 0.3),
AxB=
0.3
B = (0.5, 0.9),
o7]
0.3 '
BxC=
C = (0.7, 0.4), 0.7
o4] 0.4 '
[o7 o4] AxC=
0.3
0.3 "
If we consider T' = max, fi = fl~ and
o.3
o.4
LU.I
then
We see that QfllR ~ F(A × C). Therefore we need to change the definition and we give the following
Definition 4.1. Let Q ~ F(A x B) and R ~ F(B x C). We define the fuzzy relation QflR e F(A × C) by means of
(Qflg)(x, y) = min/su p [fl(Q(x, z), g(z, y))], (A × C)(x, Y)I t z~X
The sup-fl composition have the following properties:
J
for all x, y e X.
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M.J. Ferndndezet al. / Equations of fuzzy relationsdefined on fuzzy subsets
Proposition 4.2. (a) Let Q • F(A x B), R, S • F(B x C) with R c S. Then Q~R c QflS.
(b) Let Q, R ~ F(A x B) with Q ~ R and S • F(B x C). Then QflS ~ RflS. The proof is straightforward. To obtain the smaller solution (p'l) we need analogous results to those of Subsection 3.3. Here we have: Lemma 4.3. Let Q • F(A x B) and R • F(B x C). Then R ~ Q-Ifl(QT'R). Proof. For all x, y • X,
[Q-Ifl(QT'R)](x, y ) : min{szup [fl(Q-l(x, z), (QT'R)(z, y))], (B X C)(x, y)}
= min{SzUp[fl(Q(z, x), min{inf [T'(Q(z, t), R(t, y))], ( Z x C)(z, y)})], (Bx C)(x, y)} ~
~
[]
Remark 1. Let S • F(A x C). The following counterexample shows that S c~ QT'(Q-lflS). Let A = (0.7, 0.9), B = (0.8, 0.3), C = (0.4, 0.6), [0.7 0.3] B × C = [ 0"4 0.6] AxC=[0.4 0.6] A x B = 0.8 0.3 ' 0.3 0 . 3 ' 0.4 0.6" If we take
[06
Q = 0.7 0.2 ' then Q-lfllS=
0.4 0.5 '
[00] 0.3 0.3
and QT'I(Q-1fl1S)=
[06 0 l ,r0 0] 0.7 0.2J 1[0.3 0.3 =.0.3
0.3 "
To solve this trouble, we will restrict ourselves to the case in which supA(z)<-min(B(x), C(y)) for all x, y •X. z~X
Proposition 4.4. Let Q • F(A x B) and S • F(A x C). If (4.1) holds, then
(Q-lflS)(x, y) = sup [fl(Q(z, x), S(z, y))] ~ (B x C)(x, y) for all x, y • X. zEX
Proof. For all x, y • X,
sup [fl(Q(z, x), S(z, y))] ~
z~S
~
= sup A(z) <~min(B(x), C(y)). zEX
z~X
[]
(4.1)
M.J. Fern~indez et al. / Equations of fuzzy relations defined on fuzzy subsets
4.5. Let Q ~ F(A x B) and S ~ F(A × C), such that (4.1) holds. Then S c QT'(Q-IflS).
Lelnma Proof.
329
For all x,
y ~ X,
[QT'(Q-t~S)](x, y)= min{infx [T(Q(x, z), (Q-I~S)(z, y))], (Ax C)(x, y)} =min / inf [T'(Q(x, z), sup [fl(Q(t, z), S(t, y))])], ( Z x C ) ( x , y)} kzeX L
t~X
~>min{~nf [T'(Q(x, z), fl(Q(x, z), S(x, y)))], ( Z x C)(x, y)} ~>min{inf S(x, y), (m× C)(x, y)} = min{S(x, y), (Z x C)(x, y)} = S(x, y).
[]
Now we can give: 4.6. Let Q s F ( A x B), S eF(A × C) such that (4.1) holds, and let ~ = {R e F(B × C) QT'R = S}. Then, if ~ 4:0, the fully relation Q-lfls e F(B × C) is the minimum of ~, where
Theorem
(Q-'flS)(x, y) = sup [fl(Q-'(x, z), S(z, y))] for all x, y ~ X. zeX
follows immediately from Lemma 4.3 that R ~ Q-~flS. Then S = QT'R ~ QT'(Q-IflS) ~ S (from Lemma 4.5), that is QT'(Q-~flS) = S. [] Proof. Let R e E. It
Now we solve the problem (p'2) in an analogous way. 4.7. Let Q c F ( A x B ) , Nevertheless S dF (RflS-1)-IT'R.
Lemma
Proof.
R6F(BxC)
and S e F ( A x C ) .
Then Q~[Rfl(QT'R)-I] -~
For all x, y • X,
[R#(QT'R)-']-'(x, y) = [R[3(QT'R)-q(y, x)
min/sup [/~(R(y, z), (QT'R)-~(z, x))], (B x A)(y, x)/ Lz~X
J
=min{sup[fl(R(y, <~min{:up[fl(R(y, z),
t~xinf [T'(Q(x,
t), R(t, z))])], ( Z × B)(x, y)}
=min{sup[fi(R(y, z), man{ T'(Q(x, y), R(y, z)), inf T'(Q(x, t), R(t, z))})], (A× B)(x, y)} t~:y
f min]sup [fl(R(y, z), T'(Q(x, y), R(y, z)))], (A x B)(x, Y) I ~
i07 0.8 0.3 o3] 0.2 0.2
AxB= With R=
'
BxC=
' S=
{o3 0.2 0.4
[04 06] 0.3 0.3 '
AxC=
i04 06] 0.4 0.6 "
330
M.J. Ferndndezet al. / Equationsof fuzzy relationsdefined on fuzzy subsets
we have
[050.3 0.3°4] and
0.31T,[0.1
0.3,
0.3
0.3 0.3
o.3]
[]
Let A, B and C be three fuzzy subsets of X such that sup C(z) <- min(B(x), A(y))
for all x, y • X.
(4.2)
zEX
Proposition 4.8. Let R • F(B × C) and S • F(A × C). If (4.2) holds, then
(RflS-1)(x, y) = sup [fl(R(x, z), S(y, z))] ~<(B × A)(x, y)
for all x, y • X.
zEX
Proof. For all x, y • X,
sup [fl(R(x, z), S(y, z))] ~
zeX
[]
~
zeX
Lemma 4.9. Let R • F(B × C) and S • F(A × C), such that (4.2) holds. Then S c (RflS-1)-IT'R. Proof. For all x, y • X,
[(RflS-1)-IT'R](x, y ) = min[inf [T'((RflS-1)(z, x), R(z, y))], ( A x C)(x, y)}
= min{~nf [T'(sup [fl(R(z, t), S(x, t))], R(z, y))], (A× C)(x, y)} ~>mintinf [T'(fl(R(z, y), S(x, y)), R(z, y))], (A x C)(x, y)} kzeX
t> min / inf S(x, y), (A x C)(x, y)} = min{S(x, y), (Z x C)(x, y)} = S(x, y).
[]
tzeX
The next theorem characterizes the minimum solution of the problem (p'2). Theorem 4.10. Let R • F(B × C) and S • F(A × C) such that (4.2) holds. Let Q = {Q • F(A × B) I Q T ' R = S}. Then, if Q ¢ fl, the fuzzy relation ( R f l S - 1 ) -1 • F(A × B) defined by
(RflS-1)(x, y) = sup [fl(R(x, z), S(y, z))] for all x, y • X zEX
is the minimum of Q. Proof. Let Q • Q. From Lemma 4.7, Q ~ (RflS-1) -t. Then S = Q T ' R D (RflS-1)-'T'R ~ S (from Lemma 4.9). [] 5. Adjoint fuzzy relation equations
Let q0 be the operator associated with a t-norm T which is lower semicontinuous. The adjoint fuzzy relation equations were studied by Miyakoshi and Shimbo [12] and Izami, Tanaka and Asai [7], when
M.J. Fermindezet al. / Equationsof fuzzy relationsdefinedon fuzzy subsets
331
the fuzzy relations are defined on crisp subsets, dealing with fuzzy relation equations of the form: QqoR = S, i.e. S(x, y) = inf [q0(Q(x, z), R(z, y))]
for all x, y • X.
z~X
Here we are interested in this type of fuzzy relation equations for which the fuzzy relations are defined on fuzzy subsets. In this way our equations are of the form Oq~R = S where Q • F(A x B), R • F(B x C) and S • F(A x C), and S(x, y) = min/inf [¢p(Q(x, z), R(z, y))], (A x C)(x, Y)I )
tz~X
for all x, y • X.
And we formulate, as usual, the two problems: (dl) Determine R if Q and S are given. (d2) Determine Q if R and S are given. The questions (dl) and (d2) are solved with the help of the following lemmas: Lemma 5.1. Let Q • F(A x B), R • F(B x C) and S • F(A × C). Then (a) R = Q-1T(QcpR). (b) S ~ Qqg(Q-'TS). Proof. (a) For all x, y • X , [Q-'T(Qcpg)l(x, y) = sup [T(Q-'(x, z), (Qqggl(z, y)] zeX
<~sup [T(Q(z, x), cp(Q(z, x), R(x, y)))] ~
(b) For all x, y • X, [Qq)(Q-1TS)](x, y ) = min{~nf [q0(Q(x, z), (Q-ITS)(z, y))], ( A x C)(x, Y)I = min/tz~xinf[qv(Q(x, z), sup,~x[T(Q(t, z), S(t, Y))])l' ( A x C)(x, y)} =min/~z,xinf[q0(Q(x, z), max{T(Q(x, z), S(x, y)), sup,,xT(Q(t, z), S(t, Y))/)]' ( A x C)(x, y)} ~>min{~nfx [qg(Q(x, z), T(Q(x, z), S(x, y)))], (Z x C)(x, Y)I /> min{S(x, y), (A x C)(x, y)} = S(x, y).
[]
Lemma 5.2. Let Q • F(A x B), R • F(B x C) and S • F(A x C). Then (a) Q = (QcpR)cpR -1, (b) S c (ScpR-')qgR.
332
M.J. Ferndndezet al. / Equationsof fuzzy relationsdefinedon fuzzy subsets
Proof.
(a) For all x, y ~ X,
[(QqvR)cpR-l](x, y)= min{inf [cp((QcpR)(x, z), R-l(z, y))], (Ax B)(x, y)}
= min{~nf [q~(min{inf [q0(Q(x, t), R(t, z)), (Zx C)(x, z)}, R(y, z))], (Zx B)(x, y)} i> min(!nf [q0(inf [qg(Q(x, t), R(t, z))], R(y, z))], (Zx B)(x, y)} ~>min{!nf [qg(q0(Q(x, y), R(y, z)), R(y, z))], (Zx B)(x, y)} ~>min{~nf Q(x, y), (Z x B)(x, y)} = min{Q(x, y), (Z x B)(x, y)} = Q(x, y). (b) For all x, y ~ X, [(ScpR-l)epR](x, y)= min{~nf [qg((Sq~R-l(x, z), R(z, y))], (Ax C)(x, y)}
= min{~nf [q~(min{inf [q0(S(x, t), R(z, t))], ( A x B ) ( x , z)}, R(z, y))], ( Z x C ) ( x , y)} i> min{inf [qg(infL \t~x [qg(S(x, t), R(z, t))], R(z, Y)) t' (Zx C)(x, y)} ~>min{inf [qg(q0(S(x, y), R(z, y)), R(z, y))], (Z x C)(x, y)} /> min{S(x, y), (A x C)(x, y)} = S(x, y).
[]
With these lemmas we prove the following theorems that give us the smallest solution of the problem (dl) and the greatest solution of the problem (d2). Theorem 5.3. Let Q ~ F(A x B), S E F(A x C) and ~ = {R • F(B x C) I Qq~R = S}. If ~ 4=O, then the fuzzy relation Q-1TS e F(B x C) is the minimum of ~. Proof. Let R e ~. From (a) of Lemma 5.1, R ~ Q-~TS. Then S = QcpR ~ Qqg(Q-~TS) ~ S (from (b) of Lemma 5.1). []
Analogously for the problem (d2). Theorem 5.4. Let R ~ F(B × C), S • F(A x C) and Q = {Q ~ F(A x B) I QqoR = S}. If Q 4=O, then the fuzzy relation ScpR-1 ~ F(A × B) defined by
(ScpR-')(x, y)= min{~nf [q~(S(x, z), R-l(z, y))], (A x B)(x, y)} for allx, y ~ X is the maximum of Q. Proof. Let Q e Q. From (a) of Lemma 5.2, Q c ScpR -~. Then S = Qq~R ~ (ScpR-l)cpR ~ S (from (b) of Lemma 5.2). []
Now we are interested in adjoint fuzzy relation equations for which inf-q0 composition is replaced by sup-fl composition already introduced, when the fuzzy relations are defined on fuzzy subsets, as in Section 4 (Definition 4.1): QflR = S,
M.J. Ferndndez et al. / Equations of fuzzy relations defined on fuzzy subsets
333
i.e,~
S(x, y)= min/sup [13(Q(x, z), R(z, y))], (A x C)(x, y)~ for all x, y tzeX
• X,
J
where Q • F(A x B), R • F(B x C) and S • F(A x C). Once again, we formulate the two problems: (d'l) Determine R if Q and S are given. (d'2) Determine Q if R and S are given. The following lemma is the basis for solving the problem (d'l). 5.5. Let Q • F(A x B) and S • F(A x C). Then S ~ Q[I(Q-IT'S).
Lemma Proof.
For all x,
y • X,
[Q~(Q-~T'S)](x, y ) = man{sup [/3(Q(x, z), (Q-1T'S)(z, Y))], (A x C)(x, y)}
= min/suptz~x[/3(Q(x, z), minlinf[T'(Q(t,t,~x
z), S(t, y))], ( B x C ) ( z , y)})], ( A x C ) ( x , y)}
~
[]
2. Let R • F(B x C). The following counterexample shows that R c~ Q-1T'(Q[3R). Let
Remark
A = (0.7, 0.3), AxB=
With
B = (0.5, 0.9),
C = (0.2, 0.4),
041
3 0.3 '
0.2 0.4 '
AxC=
04]
0.2 0.3 "
o6] R=[Ol 04]
Q=
0.4 0.3 0 . 1 '
0.2 0.3
we have
and Q-'*(Q~,R)=[~I2
2131]* [00.2 00.3]=[00122 00133].
In a special case, when the fuzzy subsets A, B and C verify that sup B(z) ~ min(A(x), C(y))
for all x, y • X
zEX
then from Proposition 4.4 it follows (QjgR)(x, y) = sup [~(Q(x, z), R(z, y))]. z~x Lemma
5.6. Let Q • F(A x B) and R • F(B x C). If (5.1) holds, then R c Q-IT'(Q~R).
(5.1)
334
M.J. Ferndndezet al. / Equations of fuzzy relations defined on fuzzy subsets
Proof. For all x, y • X, [Q-1T'(QflR)](x, y ) = min{~nf [T'(Q-I(x, z), (QflR)(z, y))], (B x C)(x, y)}
=min/tz,xinf L[T'(Q(z', x), sup,~x[fl(Q(z, t), R(t, y))])], ( B x C)(x, y)} ~>min{}nf [T'(Q(z, x), fl(Q(z, x), R(x, y)))], ( B x C)(x, y)} ~>min{}nf R(x, y), (B x C)(x, y)} = min{R(x, y), ( B x C)(x, y)} = R(x, y).
[]
Theorem 5.7. Let Q • F(A x B), S • F(A x C) such that (5.1) holds and ~ = {R • F(B x C) I QflR = S}. I f R q~O, then the fuzzy relation Q-1T'S • F(B x C) defined by (Q-'T'S)(x, y) = mintinft~,x[T'(Q-'(x, z), S(z, y))], (B x C)(x, y)} for all x, y • X is the maximum of ~.
Proof. Let R • E. From the previous lemma R 71Q-1T'S. Then S = QflR c Qfl(Q-~T'S) c S (from Lemma 5.5). [] The problem (d'2) is solved here in a very restrictive context because the following inclusions between relations for Q • F(A × B), R • F(B x C) and S • F(A x C), are in general false: (a) Q = (aflR)flR -1. (b) S = (SflR-1)flR. Counterexamples. Let
A = (0.7, 0.3),
B = (0.5, 0.9),
C = (0.2, 0.4),
o4] AxB= With
0.3 0.3 '
0.2 0.4 '
[Ol 02] R__[Ol o4] Q=
0.3 0.1 '
0.2 0.4
we have 0 0.4] QfllR= 0.2 0.3 and [0 (Qfl'R)fllR-'
and with
.__
=
o4 0.4]'
0.2
s__[o 1 03]
we have S f l l R - ' = [ 0"40.3 0.30"4]
0.2 0.1
o4] AxC=
0.2 0.3 "
335
M.J. Ferndndezet al. / Equations of fuzzy relations defined on fuzzy subsets
and 04
0
In the next proposition we give a sufficient condition in order to guarantee that (a) and (b) hold. Proposition 5.8. (a) If A, B and C • F(X) verify
sup B(z) <<-min(A(x), C(y))
for all x, y • X
(5.2)
for all x, y • X
(5.3)
zEX
then Q = (QflR)flR -1 (b) If A, B and C • F(X) verify
sup C(z) <<-min(A(x), B(y)) zEX
then S = (SflR-1)flR.
Proof. (a) For all x, y • X, [(QflR)flR-1](x, y)=min{sup[fl((QflR)(x, z), R-'(z, y))], (A × B)(x, y)}
= min{SUxP[fl(SUxP [fl(Q(x, t), R(t,z))], R ( y , z ) ) ] , ( Z x B ) ( x , y ) } ~
[(SflR-')flR](x, y ) = min{szup [fl((SflR-1)(x, z), R(z, y))], (m x C)(x, y)}
= min/su p [fl(sup [fl(S(x, t), R(z, t))], R(z, y ) ) ] , (Ax C)(x, y)} tz~X
L
\t~X
<~min/SzU p _ [fl(fl(S(x, y), R(z, y)), R(z, y))], (Z x C)(x, y)} <~min{S(x, y), (A x C)(x, y)} = S(x, y).
[]
Remark 3. Note that (5.2) and (5.3) remain true if and only if B = C is a fuzzy subset with a constant membership function, and B c A. Theorem 5.9. Let R • F(B x C), S • F(A x C) such that A, B and C verify (5.2) and (5.3). Let
0 = {Q • F(A × B) IQflR = S}. If Q 4=~J, then the fuzzy relation StiR -1 • F(A x B) defined by (SflR-l)(x, y) = sup [fl(S(x, z), R(y, z))] for all x, y e X zEX
is the minimum of Q.
Proof. Let Q e Q. By (a) of Proposition 5.8, Q ~ StiR -1. Then S = QflR c (SflR-~)flR ~ S (by (b) of Proposition 5.8). [] Conclusions The solution of fuzzy relational equations when the fuzzy relations are defined on fuzzy sets has been investigated.
336
M.J. Ferndndez et al. / Equations of fuzzy relations defined on fuzzy subsets
We have given the procedures for solving different types of fuzzy relation equations. We hope that their diversity allows us to find a suitable model for real-world problems. These results may be applicable to fuzzy inference under compositional rules of inference; in this sense, if two propositions P and Q, whose predicates are defined on universes of discourse A and B, fuzzy sets, then an implication P---~ Q is defined, in general, as a fuzzy relation on A x B [3, 12].
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