Q-scale measures of fuzzy sets

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 66 (19941 59 81

Q-scale measures of fuzzy sets Kai-Yuan Cai* Department q[" Automatic Control, Bei/ing University o[ Aeronautics and Astronautics, Bei/ing 100083, People's Republic ~["China

Received January 1993; revised January 1994

Abstract In an attempt to generalize Nabmias' scale measure to apply to fuzzy sets, in this paper we introduce the notion of Q-scale measure. A Q-scale measure is slightly different from a crisp-valued possibility measure in several ways. We use a number of examples to demonstrate the validity of the notion of Q-scale measure in various circumstances and give preliminary discussions on basic properties of Q-scale measure, induced Q-scale measure, semigroup of Q-scale measures, Q-scale measure of probabilistic sets, and Q-scale measure of L-fuzzy sets. Key words: Fuzzy measure; Q-pattern space; Q-scale measure; Semigroup; Probabilistic set; L-fuzzy set

1. Introduction Studies on fuzzy measures and related fields have become a viable aspect in fuzzy set theory since Sugeno's pioneering work [25]. All the resultant related measures m a y be divided into four classes: probability-like, probability-counterpart, generalized, and others. (I) Probability-like measures: These measures attempt to give a generalization of classic probability measures in various circumstances. Let ~1 be a a-algebra of subsets of a universe f2. Sugeno defined a fuzzy measure as a m a p p i n g v : d - - * [0, 1] with the following properties: (1) v(0) = 0, v((2) = 1 (2) VA, B e ~ I , A ~ B ~ v ( A ) < ~ v(B) (3) {A.} is a m o n o t o n i c sequence of subsets in ~ ' and lim . . . . A. e d ~ v ( l i m n ~ An) = l i m . ~ , v(An). Zadeh defined fuzzy event as a fuzzy set with M-measurable membership function and probability of fuzzy event as the expectation of its membership function with respect to Lebesgue integral [28]. With an attempt to axiomate Zadeh's probability measure of fuzzy event, Klement et al., proposed the notion of fuzzy probability measure for fuzzy a-algebra [17]. Let d v be a class of fuzzy sets defined on ~2, d v is said to be a fuzzy a-algebra if (1) A c 6 d v for every c e [0,1], where/~a.(x) = c, Vxe~2 (2) A ~ d r ~ A c 6 d v ( ~ a c ( X ) "~- 1 - ll a(x), V x e f2) (3) A. e soy, n e l~ ~ ~ ) . ~ A. e soy (~ denotes the set of all positive integers,/t U.E~A°(x) supne~]AA.(X), Vx ~f2t =

* Current address: Centre for Software Reliability, City University, Northampton Square, London EC1V 0HB, UK. 0165-0114/94/$07.00 ,~ 1994 Elsevier Science B.V. All rights reserved SSDI 0165-01 1 4 ( 9 4 1 0 0 0 9 6 - P

K.-Y. Cai / Fuzzy Sets and Systems 66 (1994) 59-81

60

A fuzzy probability measure is a mapping v: d e ~ [0, 1] with the following properties: (1) v(O) = o, v ( O ) = 1 (2) VA, B G d r , v(A u B) + v(A c~ B) = v(A) + v(B) (3) A n G d F , n G ~ , A n T A e d F ~ V ( A . ) T v(A) Qiao interpreted fuzzy a-algebra ~¢r in a slightly different sense, that is, property (1) of Klement's fuzzy a-algebra is restricted as Ac e d F for every c e {0, 1} [23] Qiao then extended Sugeno's fuzzy measure v to fuzzy a-algebra d F such that (1) ~(0) = o

(2) A, B G d r , A <~ B ~ v(A) <~ v(B) (3) {An} is a monotonic sequence of fuzzy sets in d r and limn-~ An ~ d F =~ v(lim,_~ ~ A.) = limn.~ v(An) On the other hand, Dempster and Shafer define a belief function as follows [24]. Suppose f2 is a finite set, and let 9~(f2) denote the set of all subsets of f2. Suppose the function Bel:9~(f2)~ [0, 1] satisfied the following conditions: (1) Bel(O) = O. (2) Bel(f2)= 1. (3) For every positive integer n and every collection A I . . . . . An of subsets of 9~(f2), Bel(A: u ... ~ An) >~~ B e l ( A i ) - ~ Bel(Ai ~ Aj) + . . . . i

+ ( - 1)n+lBel(Ai ~ ... n An).

i
Then Bel is a belief function. From the above definition we can view a belief function as an extension of probability measure. In fact, the belief function becomes as a probability measure on the finite set if equality always holds in the condition (3). (II) Probability-counterpart measures: These measures follow operation laws that cannot be viewed as a generalization of those satisfied by classic probability measures. Instead the operation laws are just parallel with classic probability operation laws. Among probability-counter measures, Nahmias scale measure is particularly interesting [21]. Let ~R(t2) be the power set of a universe f2. Nahmias called the pair (f2, ~(t2)) pattern space and defined a scale measure as a mapping v : 9~(f2) ~ [0, 1] with the following properties: (1) v(0) = 0, v(O) = 1 (2) v(U~A~) = sup~ v(A~) for any class of subsets of f2 (finite, countable or uncountable). As noted by Nahmias himself [21], a scale measure is closely related to Choquet's capacity [10]. Let f2 be a Hausdorff space.: A (regular) capacity on f2 is a mapping v:9i(~2)--, lt~w { + ~ } such that (1) v is monotone (increasing): A c B ~ v(A) <<,v(B). (2) If A. T A is an increasing sequence of sets, then v(A.) T v(A). (3) If An J, A is a decreasing sequence of compact sets, then v(A.) ~ v(A). Another measure is called possibility measure. Zadeh was the first one to use the term 'possibility measure' [29]. However we should note that Zadeh's possibility measure is essentially identical to Nahmias' scale measure, although Zadeh defined possibility measure in a constructive manner from membership function of fuzzy set and the restriction v(I2) = 1 is not included. Klement and Weber used the same term 'possibility measure' to explain a slightly different matter [18]. Let d be a a-algebra of subsets of a universe f2. A possibility measure is a mapping v : ~¢ --* [0, M], M ~ (0, oo ]

1 See [10] for the m a t h e m a t i c a l definitions of a H a u s d o r f f space a n d a c o m p a c t set.

K.-Y. Cai / Fuzz)' Sets and Systems 66 (1994) 59 81

61

with the following properties: (1) v(O) = 0 (2) v(A w B) = max(v(A), v(B)) if A m B = 0 (3) {A,} is an increasing sequence of subsets in ~ / a n d l i m . ~ A. = A ~ d ~ v ( A . ) T v(A) This definition was then extended to apply to the collection d E of all fuzzy events on (1) v(0) = 0 (2) v(A w B) = max(v(A), v(B)) (3) {A.} is an increasing sequence of fuzzy events in d L =~ v(A.) T v(A) In this regard Klement and Weber called v crisp-valued possibility measure. (III) Generalized measures: These measures attempt to cover both probability-like measures and probability-counterpart measures. That is, both probabilityqike measures and probability-counterpart measures are just special cases of generalized measures. Generalized measures include Weber's l-decomposable measure [--27] and the measures defined in terms of pseudo-additions and multiplications by Ichihashi et al. [16]. More generally, generalized measures may be fuzzy measures if the term 'fuzzy measure' is interpreted in such a sense that the continuity property of Sugeno's fuzzy measure is dropped [1, 13]. (IV) Other measures: Besides probability-like, probability-counterpart, and generalized measures, there appear other fuzzy-related measures in literature. For example, the term 'fuzzy measure' may be defined in other different senses [11, 22]. However, in this paper we confine ourselves to Sugeno's sense of fuzzy measure. Among the four classes of measures, scale measure or possibility measure shows an attractive value in applications particularly when the size of available samples is limited. For example, in reliability area Cai showed how to use scale measure to develop a theory of fuzzy reliability (posbist reliability theory) in possibility context [4] and software reliability behavior is possibilistic in nature [5, 6]. This may be further explained by the statement 'probability stems from sample generality and possibility characterizes sample particularity' [-7]. So it is valuable to extend the notion of scale measure or possibility measure to apply to fuzzy sets (not confined to fuzzy events) in a formal framework. In resemblance with Nahmias' terminology [21], in this paper we introduce the notion of Q-scale measure. Q-scale measure is defined on Q-pattern space and can be viewed as a generalization of Nahmias' scale measure. Such a measure will play a central role in posfust reliability theory which is based on fuzzy-state assumption and possibility assumption [2]. We note that Q-scale measure is slightly different from crispvalued possibility measure, in particular, in four points. (1) Q-scale measure is defined on Q-domain. A Q-domain may cover various classes of fuzzy sets depending on choice of index set Q. However, crisp-valued possibility measure is confined to the class of fuzzy events. We note that there is a broad class of fuzzy sets with nonmeasurable membership functions. (2) Q-scale measure assign values in [-0, 1]. This makes its intuitive attraction in applications. In contrast, crisp-valued possibility measure assigns values in [0, oe ). (3) Q-scale measure a assumes a(A) = c if IrA(X) = C, Vx ~ O for every c e [0, 1], whereas crisp-valued possibility measure v only assumes v(0) = 0. (4) Q-scale measure allows Q to be finite, countable or uncountable provided that sup~Qcr(A~)= a([j~Q A~), whereas crisp-valued possibility measure only accomodates s u p . ~ v(A.) = a ( U , ~ A,). Based on the above comparison, we believe that Q-scale measure will become a flexible and attractive concept in engineering applications. Further, we note that existing discussions on crisp-valued possibility measure were mainly focused on relationship between measure and the related integral [18]. However, in this paper we disregard problems of integral and confine ourselves to other topics. Section 2 presents definitions for Q-domain and Q-scale measure and cites a number of examples to illustrate the concept validity in various circumstances. Section 3 discusses basic properties of Q-scale measure. Section 4 shows how to induce a new Q-scale measure in terms of mapping. In Section 5 we show that a class of Q-scale measures can constitute a semigroup. Properties of Q-scale measures in probabilistic sets and L-fuzzy sets are, respectively, discussed in Sections 6 and 7. Concluding remarks are contained in Section 8.

62

K.-Y. Cai / Fuzzy Sets and Systems 66 (1994) 59-81

2. Definitions and examples Let F = {x} be a universe of discourse, {A,} be a class of fuzzy sets defined on F with membership function I~a,(X) for A,. In the sequel the notation U , A , is always interpreted as a fuzzy set defined on F with membership function sup~I~A,(X ). Similarly N , A , is a fuzzy set defined on F with membership function

inf~/~a,(x). Definition 2.1. Let F be a universe of discourse, Q a nonempty index set. Let ~ be a class of fuzzy sets defined on F with the following properties:

(1) 0 ~ , r e ~ (2) VA~ ~ ~, ~ ~ Q, there holds U~QA~ ~ Then ~ is called Q-domain and the pair (F, ~) Q-pattern space. When no confusion may arise, we may prefer to simply calling ~ Q-domain and (F, ~) Q-pattern space without predetermining the index set Q. This may imply ~ is a Q-domain and (F, ~) is always a Q-pattern space for any index set Q, or ~ is a Q-domain and (F, ~) is a Q-pattern space for some choice of index set depending on specific circumstance.

Proposition 2.2. Let (F, ~) be a Q-pattern space and Qs is a subset of Q. Then (F, ~) is also a Qs-pattern space. Further, if there exists an one-to-one correspondence between Qs and the set T, then (F, ~) is also a T-pattern space. Proof. Trivial.

[]

In the following we present as examples a number of important special cases of Q-pattern space to show the extensiveness of the concept of Q-pattern space.

Example 2.3. Let ~ be a fuzzy a-algebra defined on F (in the sense implied by Qiao as indicated in Section 1). If we choose Q = N, then ~ is a Q-domain and (F, ~) is a Q-pattern space. However, in general, a Q-domain is not a fuzzy a-algebra. This is because Q may be finite or uncountable and A e ( does not certainly imply A~.

Example 2.4. Suppose (F, d , P) is a probability space, ~ = {A: A is a fuzzy event defined on F}. Let Q = ~. Then (F, ~) is a Q-pattern space. However, the positive conclusion is dependent on the choice of Q. Let us consider a counterexample. We denote [~ as the set comprising all real numbers. From real analysis theory we know that there certainly exist nonmeasurable sets in ~ [26]. Suppose K is such a nonmeasurable set and Q is such a set that there exists a one-to-one correspondence between K and Q, that is, V~ e Q, there exists a unique x~ e K. On ~ we define fuzzy set A, with membership function

I~A,(X)=

1 if x = x ~ e K , 0 if X # X , .

This implies for every ~ ~ Q,/~a,(x) is measurable or A, is a fuzzy event. However, it has been shown that sup,~Q I~A,(X) is a nonmeasurable function [26], or U,~Q A, is not a fuzzy event. So we conclude that (F, ~) is no longer a Q-pattern space with F = R. This tells us that to make (F, ~) a Q-pattern space, the choice of index set Q must be taken into account.

K.-Y. Cai / Fuzzy Sets and ~vstems 66 (1994) 59 81

63

Example 2.5. Let ~ be the universe of discourse and ~ = {A: A is a fuzzy set defined on ~ whose membership function is continuous at Xo = 0~ ~}. Suppose Q is a finite index set. Then (~,~) is a Q-pattern space. However, if Q is an infinite index set (e.g., Q = ~), then (R, ~) is no longer a Q-pattern space. In fact we can define fuzzy set A, with the following membership function

i if x~<0,

FlA,(X)=

X

if 0 < X ~ < l/n, if x > 1/n.

Obviously/~A,(x) is continuous at Xo = 0. However sup/tA,(X) = {01 if X~<0, ,~ if x > 0, is not continuous at x = 0. Example 2.6. Suppose Y is the universe of discourse and ~ = {A: A is a fuzzy set defined on F with I~a(x) 7~c, Vc ~ (0, 1)I u {0, F }. Then (F, ~) is not a Q-pattern space when IQ[ > 1 ([ Q[ signifies cardinality of Q), In fact, let A be a real subset of F and {~

if x ~ A (c ~(0,1)), if x ~ A

{0

ifx~A(c~(0,1)). if x ¢ A

l~a(X) =

/~n(x) =

Then PA ~ B(X) -- C. This make us arrive at the assertion. Example 2.7. Suppose I" is the universe of discourse and ~ = {0, F}. Then (V, ~) is a Q-pattern space for any index set Q. Example 2.8. Suppose F is the universe of discourse and ~ = ~ ( F ) (comprising all nonfuzzy subsets of F). Then (F, ~) is a Q-pattern space for any index set Q. This implies Nahmias' pattern space [21] and is a special case of Q-pattern space. Example 2.9. Suppose F is the universe of discourse and ~ = o~(F) (comprising all fuzzy sets defined on F). Then (F, ~) is a Q-pattern space for any index set Q. Example 2.10. Suppose F is the universe of discourse and ~c = {Ac: pAo(X) = C, 0 ~< C ~< 1}. Then (F, ~c) is a Q-pattern space for any index set Q. Example 2.11. Suppose F is the universe of discourse and ~ = {A: A is a normal fuzzy set } w {0 }. Then (F, ~) is a Q-pattern space for any index set Q.

Definition2.12.

Let (F,~) be a Q-pattern space. A Q-scale measure is a mapping a: ~---, [0, 1] with the following properties: (1) a ( A ) = c if I~A(X) =-- C Vc ~ [0, 1] (2) a ( U ~ o A ~ ) = s u p ~ o a ( A ~ ) V A ~ ~Q The triple (F, ~, a) is called Q-scale measure space. When no confusion can arise, a Q-scale measure is abbreviated as Q-measure and a Q-scale measure space as Q-measure space.

64

K.-Y. Cai / Fuzz), Sets and Systems 66 (1994) 59-81

Table 1. Q-scale measuresfor a gracefullydegradable system Membership function

Element of F

Q-scale measure

0

1

2

3

al

a2

as

btF Ps

0 l

~ ~

~ ~

1 0

~ ~

~ ~

l 1

#z I~c

0

0

0

0

0

0

0

1

1

1

1

1

1

1

Proposition 2.13.

Let ( F , ~ , a ) be a Q-measure space. Suppose Qs is a subset of Q, then ( F , ¢ , a ) is also a Q f p a t t e r n space. Further, (F, 3, a) is also a T-measure space when there exists a one-to-one correspondence between Qs and T.

Proof. Trivial.

[]

Example2.14. Let F be the universe of discourse, 4 = {0, F}, a(0) = 0, a(F) = 1. Then (F, ~, a) is a Q-measure space.

Example 2.15. Let F be the universe of discourse and ~ = I~(F). Further let a be Nahmias' scale measure. Obviously (F, ¢, a) is a Q-measure space. Since Zadeh's possibility measure v is identical to Nahmias' scale measure when v(F) = 1, and Q-scale measure is just a generalization of Nahmias' scale measure, then for any Q-measure space (F, ~, a) and VA e 4, it is reasonable to interpret a(A) as the possibility of A (fuzzy set or fuzzy event). Example 2.16. Let F be the universe of discourse and 4 = ~ ( F ) . Further let a(A) = supx~r/~A(X). Then it is easy to verify (F, ¢, a) is a Q-measure space. Example 2.17. Consider a gracefully degradable computing system with three computers (such a class of computing systems provides an important background for discarding the binary-state assumption in conventional system reliability theory [8]). Every computer may be in one of the two states: functioning or failed. Then the system may exhibit four nonfuzzy states: 0, 1, 2, 3, where the number denotes the number of functioning computers in the system. Let F = {0, 1, 2, 3}, 4 = {F, S, K, Z, C} be a class of fuzzy sets defined on F with membership functions as tabulated in Table 1. We also define mappings a 1, a2 and a3 in the table. Then it is easy to verify that all (F, 4, a 1), (F, 4, a2) and (F, 4, a3) are Q-measure space. This implies that more than one Q-measure can be defined on a Q-measure space. Example 2.18. Let F be the universe of discourse and 4 = {At:/~ao(x) - c, 0 < c ~< 1}. Further let a(Ac) = c. Then (F, 4, o) is a Q-measure space. Example 2.19. Let F be the universe of discourse and ~ = {A: A is normal fuzzy set} w {0}. Let {~ a(A)=

irA=0, if A ~ O .

Then (F, 4, or) is a Q-measure space.

K.-Y. Cai / Fuz,~v Sets and Systems 66 (1994) 59-81

65

3. Basic properties Proposition 3.1. Let (F,~,a) be a Q-measure space, A,B ~ 4, A <~ B. Then a(A) <~ a(B). Proof. Since A ~< B, so B = A u B. Thus a(B) = sup(a(A),a(B)). This implies a(A) ~< a(B).

[]

Proposition 3.2. Let (F,~,~r) be a Q-measure space. Then VA ~ ~, there has 0 <~ a(A) <~ 1. Proof. Since a(0) = 0, a(F) = 1, so the assertion is a direct result of Proposition 3.1.

[]

Proposition 3.3. Let (F, 4) be a Q-pattern space. Suppose {ap} is a class of Q-measures defined on (F, 4), where 6 Q~ ~ Q. Then a = supo~Q aa is also a Q-measure. Proof. VA~ ~ ~ with #ao(x) = c, c ~ [0, 1], we have a(A~) = sup aa(Ac) = sup c = c. fl~Qs

fleQs

Further, VA~ e 4, ~ e Q,

= sup sup trp(A~) #eQs aaQ

= sup sup trp(A~) = sup tr(A~). aEQ

So o is a Q-measure of (F, 4).

E3

Proposition 3.4. Let (F, 4, a) be a Q-measure space with 4 = ~ ( F). Further suppose there exists a one-to-one correspondence between F and Qs c Q. Then VA e 4, o'(A) ~< supx~r#A(X). Proof. Let Au E 4 be defined by

~A,(X)= {oA(U ) ifx=u,ifx#u. From Proposition 3.1 we have a(Au) <~ I~A(U). On the other hand A = Uu~r A, and from Proposition 2.13 we note that a is also a F-measure. So

a(A) = sup a(Au) ~< sup #A(u). ueF

[]

u~F

Proposition 3.5. Let (F, ~,tr) be a Q-measure space, A~ ~ 4, ct ~ Q. Then a ( N ~ e A~) <~ inf~Q a(A~). Proof. Obviously N~Q A~ <~ A~. So from Proposition 3.1 we have a ( n ~ Q A) <~ a(A~) for every ~ e Q. This implies a ( N ~ Q A~) ~< inf~Qtr(A~). []

K.-Y. Cai / Fuzz), Sets and Systems 66 (1994) 59-81

66

Now we discuss continuity properties of Q-measure. Given a series of fuzzy sets {A,} defined on F, let

n~oc,

n=l

B=limA,=

k~n

~) (~Ak.

n~oo

n=l

k~n

lim.++ A, exists if and only if A = B. It is well known that lim,~+ A. = U.~=I A. if A~ ~< A 2 ~< " " ~< A. ~< A,+I ~< " " , and lim,~ooA. = 0.%~A, if A1 /> A2 ~> ..- /> A,/> A.+I >/ "" • We have the following theorem.

Theorem 3.6. Let ( F , ¢ , a ) be a Q-measure space and {A,} be a series of f u z z y sets in ~ with Aa <<,Az <~ "" <~ A , <~ A,+ I <~ ".. • Further there exists a one-to-one correspondence between t~ and Qs = Q. Then a ( l i m , ~ A.) = lim . . . . a(A,). Proof. Since Aa <~ Az <~ ... <~ A,<~ A,+~ <~ ..., so a(A~)<~ a(Az)<~ .." <~ a(A,)<~ a(A.+a)<~ ... and thus l i m , ~ a ( A , ) = s u p , ~ a ( A . ) as a result of the fact that a(A,) is bounded. On the other hand, l i m , ~ A, = U,~= 1A, and cr is also a ~-measure, so a(lim,.o~ A.) = a( U.~= 1A.) = s u p . ~ cr(A,). Hence we have a(lim . . . . A.) = l i m , ~ a(A.). [] Theorem 3.6. This just implies that if the index set Q is 'large enough' (infinite), then a Q-measure is continuous from below. Combining with Proposition 3.1, we know that when ~ denotes the class of all f u z z y sets defined on F and Q is infinite, then a Q-measure defined on Q-pattern space (F, ~) is certainly a crisp-valued possibility measure. However, in general, a Q-measure is not continuous from above. Let us consider an example. Let F be the universe of discourse and ~ = { A,:/~a,(x) ~ 0 if and only if x = u. where u ~ F is predetermined} w {0, F}, {~ a(A)=

if A = O , if A #O.

Then (F, ¢,a) is a Q-measure space for any index set Q. We choose a series of fuzzy sets {A.,} with 1/n Pa. . ( x ) = ~ 0

if x = u, if x ~ u.

Obviously Atu ~> Azu >~ "" >~ A . , >1 A~,+I~. >~ "" • So lim,~o~ A, = N~=I A. = 0, or a(lim.~o~A,~) = 0. However a(A,,) = 1 and thus l i m . ~ , a(A,,) = 1. That is, ~r(lim,.~ A..) 4: lim.o~ a(A,.). This implies a is not continuous from above. From the above example we see that the property of continuity from above is not attached to a Q-measure. This makes a Q-measure essentially different from a fuzzy measure of Sugeno or Qiao's extension. We note that to make a measure to be a fuzzy measure, it should be continuous both from below and from above. So we conclude that a Q-measure is not a fuzzy measure.

K.-Y. Cai / Fuzzy Sets and Systems 66 (1994) 59 81

67

Theorem 3.7. Let ( F, ~, a) be a Q-measure space and [ A, } be a series qf.luzzy sets of ~. Further suppose there exists a one-to-one correspondence between ~ and Q.~ ~ Q. Then there hold a ( l~m A , ) <~ lim a(A,),

Proof. F r o m Propositions 2.13 and 3.5, we have

n=k

n=l

= inf s u p a ( A k ) n>~lk>~n

= lim a(A,). n ~ ~:

n=

1 k=n

= sup a n~>l

Ak k

~< sup inf a(Ak) n>~ 1 k>/n

= lim

a(Ak).

[]

4. Induced Q-scale measure Let qJ be a m a p p i n g from F onto O, that is, Vx ~ F, ~b(x) ~ O, and Vy ~ ~ there exists at least one x ~ f such that t p ( x ) = y. In this section we discuss whether a Q-measure can induce a new Q-measure through m a p p i n g 0.

Proposition 4.1. Let ( O , d ) be a Q-pattern space and 0 - l ( d ) = [A = ~b l(B): VB ~ ~1 and/IA(X ) for every x ~ F I. Then (F,~ x ( ~ ) ) is also a Q-pattern space.

= ~B(~[I(X))

Proof. (1) ¥c ~ {0, 1}, since ( I 2 , d ) is a Q-pattern space, there exists Bc ~ d with/~Bc(Y) ~- c for every y e f2. Then P~0 ,(B,)(x) = Psc(~b(x)) = c for every x ~ F. That is 0 ~ ~9- l ( d ) , F ~ ~k-t (~1). (2) N o w we should show that VA~ e 0 - l ( d ) , ~ E Q, there has ~),~e A, s ~b-1(~/). Since A , e ~ b - l ( ~ ) , there exists B ~ e a ¢ such that pA,(X)=~B~(0(X)), we note U,~eB~E~¢. So ~ - l ( ~ z o B a ) ~ l / / - l ( d ). To show ~)~¢oA, c f f l(.z/), it is sufficient to show f f - I ( U ~ o B , ) = ~),~QA,.

68

K . - Y . Cai / Fuzz), S e t s and S y s t e m s 66 (1994) 5 9 - 8 1

However /%-, (U.°~B,)(x) =/aU=.~B,(¢(x)) = sup/~n,(¢(x)) aeQ

= sup

I~a,(X)

orEQ

=/~ U,°,~A, (x). This completes the proof.

[]

Theorem 4.2. Suppose (O, sl, a) is a Q-measure space. Let a , - , ( ~ k - l ( B ) ) = a ( B ) , V B e ~ . (F, ¢ - l ( s l ) , a ~,-,) is also a Q-measure space, where a ,-~ is referred to as induced Q-scale measure.

Then

Proof. Since (F, ~ , - ~ ( d ) ) is a Q-pattern space, here it is sufficient to show ao-, is a Q-scale measure. (1) Given Ace ~-1(~¢), suppose/~a~(x) = c, c ~ [0, 1] for every x e F. Then there exists Bc e ~d such that #Ac(X) =/ts~(¢(x)) = c

for every x

Since ~ is a mapping from F onto f2, so /~Bc(Y)-=c for every y e f2. That is, a(Bc)= c. Thus a,-,(A~) = a(Bc)= c. (2) VA~ e ¢ - l ( d ) , ct e Q, there exists B~ E .~¢ such that A~ = ¢ - I(B~). From the proof of Proposition 4.1 we note that there holds

~J-I(UQBt~)=UA~=U~f-I(B~). ct

ae Q

ctciQ

So

=a

U

= sup a(B~) aeQ

= sup a~-,(A~). This implies a¢-, is a Q-scale measure.

[]

Proposition 4.3. Let ( r , ¢ ) be a Q-pattern space and if(C)= {~k(A): VA ~ ~, [dCtA)(y)= sup,tx)=r#a(x) for every y ~ [2}. Then (t2,¢(ff)) is also a Q-pattern space. Proof. (1) Vc.e {0, 1}, since ~ is a Q-domain, there exists Ace ~ with/~ac(x) = c for every x e F. Then

12(Ac)(y)

=

sup /~Ao(X)---=C for every y e f2. q~(x)=y

That is, 0 • ~(¢), 12 • ~(~).

K.-Y. Cai / Fuzz), Sets and Systems 66 (1994) 59 81

69

(2) Now we should show that Vfl, e ¢(~), ~ e Q, there has U:+Q B: + ~,(¢). This is sufficient to show that there exists A e ~ such that ~O(A)= U ~ o B , . Since B , e ~O(~), there exists A,~ ~ such that B~e qJ(A,). Let A = ~),~t2 A,. Obviously A s ~. Then Vy 6 f2, we have k/~0~a)(y) =

sup pA(X)= O~x)=y

sup /~U,oQA=(X) q~(x)=y

= sup sup/~a,(x) qJ(x)=y ~e(2

= sup

sup

pa,(x)

~(2 q~(x)=y

= sup //0(A=)(Y) ateQ

= #U=~QB=(Y)" So ~k(A)= U:+QB:. Proposition 4.4.

[]

Suppose ( F, ~, a) is a Q-measure space. Let

a~,(B) = sup a(A),

VBe¢(¢).

q,(a)=o

Then (1) If pn(y ) - c, c ~ [0, 1] for every y e f2, then a,(B) <<.c. (2) VB, e qJ(~), ~ e Q, there holds

aq,( U B,)>>.supa,(B,). \ aeQ

cteQ

Proof. (1) Suppose

I~n(y) = c, c e [0, 1], for every y e f2. Since Bce ~(~), then for any A e ~ with ~O(A) = B,

there holds kta(y) =

sup /~a(x) = c,

y e f2.

¢,(x)=y

This implies #A(X) <~C for every x ¢ F. So a(A) <~c or a,(B) = supoCA)=na(A ) ~< c. (2) Suppose B, e 0(~). V~ e Q. Let

o= =

{H:

O(H)= n~},

D*= ~ D , = { H * ' H * =

U H~ withH, eD, },

ateQ

aeQ

eEQ

First we show D* c D. In fact VH* e D * , that is, H* = U~eH~= with 0(H~,) = B~, from the proof of Proposition 4.3, we have

~9(H*) = ~O( ~

H,,) = U ~k(Ht~,)= U B~. • eQ

So D* = D .

~Q

K.-Y. Cai / Fuzzy Sets and Systems 66 (1994) 59 81

70

Then

>~ sup 0.(A) A~D*

0.( A )

sup A = U ~ Q H3," H ~ e Da

su, H,~Q [H#:O,] =

sup sup 0.(Hp~) ~ , ~ [Hp ~D,] ~ O

= sup sup

0.(Hp,)

~teQ H~ eD~

= sup % ( B , ) .

[]

5. Semigroups of Q-scale measures In P r o p o s i t i o n 3.3 we p o i n t o u t that the s u p r e m u m of a class of Q-scale measures is still a Q-scale measure. So it is valuable to further investigate the relationships a m o n g Q-scale measures. F i r s t we review a n u m b e r of definitions in s e m i g r o u p t h e o r y [-9, 15]. Let (F, ~) be a Q - p a t t e r n space. S u p p o s e a l , 0.2 are two Q-scale measures defined on (F, ~). W e say that 0.1 a n d 0"2 are equivalent, i.e., 0.1 = 0.2, if a n d only if 0.1(A) = 0 . 2 ( A ) for every A E 3. Let S be a set. S u p p o s e • is a b i n a r y o p e r a t o r defined on S. T h e n (S, * ), or simply d e n o t e d by S, is a s e m i g r o u p if a n d only if there hold (1) Va, b ~ S , a * b ~ S a n d (2) Va, b, c e S , ( a * b ) * c = a * ( b * c ) An element 69 ~ S is called zero (element) if 6} • a = a * 6} = 6} for every a e S. A n element e e S is called identity (element) if e * a = a , e = a for every a e S. It is easy to show that the zero (identity) element is unique if there exists. An element a is called i d e m p o t e n t element if a * a = a. A b a n d is a s e m i g r o u p whose elements are all i d e m p o t e n t s . A s e m i g r o u p S is c o m m u t a t i v e if Va, b 6 S, a * b = b * a. An element a of s e m i g r o u p S is said to be regular if there is x e S such that a • x • a = a, S is regular if all the elements are regular, a ' e S is said to be an inverse of a if a * a ' * a = a a n d a ' * a * a ' = a'. S is said to be inverse if Va e S, a has a unique inverse a'. W i t h the a b o v e prerequisites, it is time for us to have the following results.

Theorem 5.1. Suppose (F, ~) is a Q-pattern space. Let S = {a: a is a Q-scale measure defined on (F, ~)}. On S we define a binary operator * such that Vax, a2 ~ S, a l* a2 = s u p { a l, a2 }. Then (1) ( S , * ) is a commutative band. (2) ( S , * ) is regular. (3) ( S , * ) is inverse. (4) There exists a unique zero element in S. (5) There exists an identity element in S if and o n l y / f i n f , ~ s {0.} e S.

K . - Y . Cai /' F u z z y Sets and S y s t e m s 66 (1994) 5 9 - 8 1

Proof.

(1) From Proposition 3.3, w e

see t h a t

Vo,t, o.2 E

S, (7 1 * 0"2 ~-- s u p { o.l, 0 2 } ~ S. F u r t h e r

71

Vo.j, o.2, o- 3 E S,

(o., * o-2)* o-3 = sup{sup{o.1 ,o-2 ],cr31 = sup{o.t ,sup{o-2,o-3 }} = o-t *(o-2 * o.3). So S is a semigroup. Since o.t * o-2 = sup { o.~, o" 2 } = s u p { o" 2, o-1} = 0"2 * o-1, thus S is c o m m u t a t i v e . M o r e over, o-1 * o,1 = sup{o.l ,o-t } = o-1. This implies S is a c o m m u t a t i v e band. (2) Since Vo. ~ S, o. is an i d e m p o t e n t , we m a y choose x = o. • S such that o. * x * o. = o.. So S is regular. (3) It is evident that o. is an inverse of o. in itself. So to show S is inverse, we should show the inverse is unique. In fact, s u p p o s e x, y are two inverses of o., then tT*X*O.

= (7,

X*O.*X

:

X.

Since S is a c o m m u t a t i v e band, thus (7 =

O-*O-*X

~- O.*X,

X

~

O.*X*X

~--- O - * X

or O . = X.

Similarly o-=y SOX=y.

(4) Let o-o = s u p e r s { o-}. Then o.o • S a n d Vo- e S, Cro * o- = o- * Cro = o.o. This implies tro is the unique zero element of S. (5) S u p p o s e inf,~s{o.} • S. Then o b v i o u s l y tre = inf,~s{o-] is the identity element. O n the o t h e r hand, if there exists identity element o-e e S, then Vo- • S, o-e* o- = o- *tre = tr, or VA • ~, o-e(A) <~ O-(A). This implies o-e(A) <~ inf~s{o-(A)}. However, i n f ~ s { o - ( A ) } ~< o'e(A) as a result of o-e ~ S. So o-e = i n f ~ s { o . } . [] W i t h the a b o v e results, we can define an o r d e r relation ~< such that (S, 4 ) becomes an u p p e r semilattice. In fact, Va, b • S, let a ~< b if a n d only if a * b = b, then (1) V a • S , s i n c e a , a = a , soa~
a<~c. (4) Va • S, we have a * o.o = o-o, that is, a ~< o.o. So (S, ~< ) is a u p p e r semilattice. W e can further show that S is not a 0-simple semigroup. A s e m i g r o u p S with zero element o-o is said to be 0-simple if a n d only if S * a * S = S for every a • S, a :# o-o, or if a n d only if for every a, b • S, a, b # o.o, there existx, yeSsuchthat x*a*y=b. T h e o r e m 5.2. Suppose S is defined as that in Theorem 5.1 and Isl > 2 (ISl sienifies cardinality of S). Then S is not a O-simple semigroup. Proof. S u p p o s e S is a 0-simple semigroup, then Va, b • S (a, b # ao), there exist x, y • S such that x • a * y = b. Since S is c o m m u t a t i v e , this implies that Va, b e S (a, b :# a e ) , there exists z e S such that a * z = b. T h a t is, VA ~ , sup{a(A),z(A)l = b(A). However, we have a :# b. So w i t h o u t loss of generality we can assert a(A) > b(A) for s o m e A • ~. This m a k e s sup(a(A),z(A)) # b(A). So S is n o t 0-simple. [] N o w we discuss r e p r e s e n t a t i o n of S. Let (S, * ) a n d ( S ' , G ) be two semigroups. A m a p p i n g ~b : S --* S' is called a h o m o m o r p h i s m if $ ( a * b ) = O ( a ) G O(b) for every pair a, b • S. A h o m o m o r p h i s m becomes an

72

K.-Y. Cai / Fuzz)' Sets and Systems 66 (1994) 59-81

endomorphism when (S, * ) = (S', e ). By a representation of a semigroup S by S' we mean a homomorphism t# of S into S'. Let X = [0, 1], 5-x = { f : f i s a mapping from X to X}. On ~ x we define a binary operator o such that Vf O • Y-x, x • X, there holds ( f o g ) ( x ) = f ( g ( x ) ) . Then it is easy to verify that ( S x , o ) is a semigroup. The following theorem gives a representation of S (as defined in Theorem 5.1) by Y'x. Theorem 5.3. Let fi, • .Y-x with fo( x) = sup{re, x}, x • [0, 1]. Suppose ~ is a mappino from S to 5 x such that V0. • S, ~(0.) = f t . Then ~k is a representation of S. Proof. To show 0 is a representation of S, we should show Ve, 0. • S, there holds ~b(e• a) = ~,(e) o $(a). We note that $(e * 0.) = ~(sup(e, 0.))= f~up~,.,) and 0(e)o ~(0.) = f o f o . Now Vx • X, we have fsup(r, Ü)(X) = sup{ sup(e, 0.), x} = sup{8,0.,x}.

(L °L)(x) =L(L(x)) =fi(sup{0.,x}) = sup{e, sup{0., x} } = sup{8,0.,x}. So ¢(~ • 0.) = 4,(8)o 0(0.).

[]

Theorem 5.4. Suppose Io is a mapping from S to S such that Ve e S, Io(e) = 0. * 8. Let Sl = {1o: 0. • S}. On S~ we define a binary operator o such that Vlo, l~ • St and Va • S, (lo ° l,)(a) = lo(l,(a)). Then (1) lo is an endomorphism orS. (2) S~ is a commutative band. Proof. (1) V81,82 e S, we have /~,(81 * 8 2 ) = 0"*(81 * 8 2 ) -~-- 0 . , 0 . , 8 1

*8 2

= (0.* ~i) * (0.* ~2) = /o(el)* Io(82).

So 1, is an endomorphism of S. (2) VIo, l,,l~eSz and V a e S , we have (lo o l~)(a) = l~(l~(a)) = lo(e*a) ~- 0 . * 8 , a

= lo,~(a).

K.-Y. Cai / Fuzz)" Sets and Systems 66 (1994) 59 81

73

So I, o l, E St. Further, ((16 ° l,) o IA(a) = (l~ o lo)(e * a) = l~(a * ~ * a) = 6*a*g*a. (1~o(1. o lA)(a) = 1~((1. o l~)(a)) = 1~((1,(~* a)))

= l~(a*e*a) = ~*o'*l~*a.

So (l,~ o 1,,)o 1~ = 16 ° (lo ° IA. And it is trivial to show that 1,, o 1~ = I~ o 1o and 1~o l, = 1o. Therefore (St, ° ) is a commutative band. []

6. Q-scale measure of probabilistic sets In pattern recognition or decision-making activities, membership function of a fuzzy set is frequently not deterministic but experiences with randomness. This stimulated Hirota to use the notion of probabilistic sets [ 14]. A fuzzy set is said to be a probabilistic set if for any point of the universe of discourse, the corresponding grade of membership is a random variable defined on some probability space. So we see that a probabilistic set is essentially a special random function [12] whose values lie in the unity interval [0, 1]. Obviously the notion of Q-scale measure also applies to probabilistic sets provided that Q is properly chosen such that ~, a class of probabilistic sets, becomes a Q-domain. In the remainder of this section, let F denote the universe of discourse and ~ a class of probabilistic sets whose membership functions are random functions defined on probability space ( f 2 , d , P ) . We always assume that index set Q is properly chosen such that (F, ~) is a Q-pattern space. For example, let ~ = {A: A is a probabilistic set defined on F with respect to (f2, d , P)} and Q = t~, then membership function of 0 s~Q As is a random function defined on (f2, d , P), or 0s~t2 As ~ ~ and thus (F, ~) is a Q-pattern space. In fact in this case, Q may be uncountable provided that it is a separable metric space [12]. Then we can discuss some basic properties of Q-scale measure of probabilistic sets. Evidently they should be presented in probabilistic manner. A probabilistic set may be denoted as A whose grade of membership at x e F is pa(X, 09) depending on 09 ~ f2. For convenience, we also interchangably use A(x,09) and pA(X, 09) to denote the probabilistic set A when no confusion may arise.

Proposition 6.1. Let ( F, ~, a) be a Q-scale measure space with ~ being a class of probabilistic sets defined on F with respect to probability space (f2, z¢, P). Suppose A, B ~ ~ and P{09: #A(X, 09) <~ PD(X, O9)} = 1 for every x ~ F. Then P{09: a ( A ) <~ a(B)} = 1. Proof. Since for any 09~f2, if #A(X, 09)<~pB(X, 09 ) for every x ~ F , a ( A ) <~ a(B). So {09: a ( A ) <. a(B)} ~ {~0: t2A(X, 09) ~ liB(X, 09) for all x ~ F} x~F

from Proposition 3.1 there holds

K.-Y. Cai / Fuzzy Sets and Systems 66 (1994) 59 81

74

However, P[{~o: #A(X,O)) <. I~B(X,CO) for all x e F} ~]

= P [ x?r {~O: l~a(x, oo) <~l~B(x, cg) } c] ~< 2

P[{o,: l~a(X,~O) <<,H,(x,~o)} c]

xeF

= 2

[1 - P{~o: I~A(X,O9) <~U,(X,~O)}]

xe/= 0,

where 'c' denotes the c o m p l e m e n t of a set. Therefore P{m: a(A) <~ a(B)} = 1.

[]

Proposition 6.2. Let (F, 3, a) be a Q-scale space with ~ being a class of probabilistic sets defined on F with respect to probability space (t2, s4, P). Let

a*(A(x, og)) = sup a(A(x, og)). gel2

Then a* is a Q-scale measure defined on (F, ~). Further, if there exists a one-to-one correspondence between and Q s c Q, then a*(A(x, m)) = sup a(A(x, ~)) oeQ

= a(supA(x, Proof. Suppose A ( x , ~ o ) - c, c ~ [0, 1], for every x ~ F , ~oef2. Then a(A(x,~o)= c for every ~o ~ f2. So

a*(A(x, og)) = c. On the other hand,

= sup sup a(A,(x, co))

= sup supa(A~(x, e9)) ~eO coeQ

= sup a*(A,(x, ~o)). ~eQ

So a * is a Q-measure.

K.-Y. Cai / Fuzzy Sets and Systems66 11994)59 81

75

If there exists a one-to-one correspondence between Q and Q++ c Q, from Proposition 2.13, a is also a Q-measure. Let B~(x) = A(x,~o). Then we can view Bo,(X) as a constant random variable. In this way

++°

++° A,x

= sup

a(B,o(x))

69 E (2

= sup

a(A(x,(o))

o~EQ

This completes the proof.

[]

Let (F, ~, cr) be a Q-scale measure space with ~ = {A (x, co): A (x, co) is a probabilistic set defined on Y with respect to probability space (£2,d , P)}. Suppose there exists a one-to-one correspondence between F and Q,+c Q. Then Proposition 6.3.

P {~°: a(A(x'c°)) <~sup ltA(X'~°)} = Proof. Let

]'IB~(U~U))=fO A(X'(O)" Then

ifif U=X,u #X.

Bx(u,~) is a probabilistic set. F r o m Proposition 6.1 we arrive at P I(1): a(Bx(u,~o)) ~ l~a(x,(o)} = 1.

Immediately it is easy to verify

P {~o:~(Bx(u,~o)) <~sup#A(X,~O)} = On the other hand,

A{x,~o) = ~J Bx(u, oJ). x£F

Since there exists a one-to-one correspondence between F and Qs c Q, from Proposition 2.13, a is also a F-measure. So

a(A(x,~))= ( U Bx(u,e~)) "xcF

= sup ueF

~(B~(u,o))).

76

K.-Y. Cai / Fuzz)' Sets and Systems 66 (1994) 59-81

Thus

:

n

{~: o,.~,u.~,,.< su~,~,/,~,,}

ueF

P

[{

09: a(A(x, co)) <~ sup l~A(x, co)

--P

[( [{

}~]

m ,1,: ~(Bx(u,co))~< sup~(x,m)

ueF

xeF

co: a(Bx(u, to)) ~< sup #a(x, co)

<<. ~ P x~F

xeF

}c] }~]

=0.

Therefore P{CO: a(A(x, co))<~ sup~ua(x, co)} = 1.

[]

x¢F

Proposition 6.4. Let ( F, 4, a) be a Q-scale measure space with ~ = { A (x, co): A (x, co) is a probabilistic set defined on F with respect to probability space ([2, d , P)}. Suppose ( ~ Q A~(x, co) ~ 4. Then a

~

~nfa(A~(x, co))~ = 1.

Proof. We note

{~:ol~A~,x.o,)<.inf~,A~,x.~,,} Thus P

[{ (~oo 0 co: a

<. ~, P aEQ

A~(x, co

<. inf a(A,(x, to agQ

[{ (~0o ,) to: a

A~(x, co

,,}1 ,,}~]

~ a(A~(x, co

.

K.-Y. Cai / Fuzzy Sets and Systems 66 (1994) 59-81

77

However

and from Proposition 6.1, we have

\~eQ

/

So

Proposition 6.5.

Let (F, ~, a) be a Q-scale measure space with ~ being a class of probabilistic sets defined on F with respect to probability space ( Q , d , P ) and ~ is the power set of Q (or d is so large a a-algebra of Q to make sense all the probability operation in the following proof). Suppose {A.(x, ~)} is a series of probabilistic sets in ~ and

P{~o: A.(x, og)<<.A n + l ( X ,

fO)} =

1, V x ~ I ' ,

n = 1, 2. . . .

Further there exists a one-to-one correspondence between [~ and Qs c Q. Then

Proof. (1) Since P{~o: A,(x,o))<~ A,+l(x,o))} = 1 for every x e F with the help of Proposition 6.1, we have P{o): a(A.(x,~o) <. a(A.+l(x,~))} = 1.

On the other hand, {to: a(A1)-..< a(A2) .~< = ~

".

~<

a(A.) -..< a(A,+l)-..< .,. ~< 1}

{co: a(A,) ~< a(A.+,) <<. 1}

n>~l

P[{(o: a(At)-..< a(A2)--.< ... ~< a(A.)-..< a(A,+l)-..< ... ~< 1} c] =P[

U {°~:a(A")<''a(A"+~)<''

~< Z P[{to: a(A.) ~< a(A.+l) ~< 1} c] n~>l =0.

K.-Y. Cai / Fuzz), Sets and Systems 66 (1994) 59-81

78

Thus P{o~: a(A1) <~G(Az) <<.... <~a(A.) <~ a(A.+l) <~ ... <~ 1} -= 1.

However {o~: lim a(An)

sup

a(An)l

z3

{~,: o'(A1) ~< a(A2)

<~ "'" <~ (7(An) <~ (~(An+I) <~ •-. ~ 1 }

So P{~o: n~limo(An) = 1sup.1 a(A.)t . > ~ = (2) It is easy to verify P{~: AI(x,~o)<~ A2(x, co) ~< "'" ~< A.(x,~o)<~ An+I(X,(o) ~ ' " } = 1. So

P I~o: lira A.(x, co)= ( n~oo

U An(x,og)} = 1. n>~l

Since

we arrive at

(3) We note

te): a

lim An = lim a(A.) = sup a(An) \n~

n~

n>/ 1

{

~o: lira a(An) = nsup a(An) t. ~>l

This implies P

~o:a

E((

']

lim An = lim ~r(An \n~

)

n~oc

= supa(An n >/ 1

't~1

or

~ (~ ~ (~i~ ~n) : ~-~lim~ ' : 1suP ~ 1~ ' t :

=0,

K.-Y. Cai / Fuzzy Sets and Systems 66 (1994) 59 81

79

However

Therefore

7. Q-scale measures of L-fuzzy sets Let Y be the universe of discourse. Suppose L is some algebraic set such as semigroup, poser, lattice or Boolean a-ring. G o g u e n defined an L-fuzzy set on F is a m a p p i n g : F ~ L. Here we confine L to a complete lattice with ordering ~<. Denote 0 = inf,~L {CO}, 1 = supo,~L{~O}. An L-fuzzy set can be represented as A = {X,I~A(X), x e F } , where I~A(X)~L is the corresponding m e m b e r s h i p function. We say A = 0 (null set) if I~A(X)=--0 and A = F (universal set) if t~A(X)----1. A = A1 u A 2 implies/IA(x) = sup{/~A,(x), #a~(X)} for every x c Y. In the following we generalize the notion of Q-scale measure to apply to L-fuzzy sets.

Definition 7.1. Let F be the universe of discourse and ~ be a class of L-fuzzy sets defined on Y. Suppose Q is an index set and there hold (1) 0 ~ , F ~ (2) V A ~ , ~ Q , then ( J ~ o A ~ . Then ~ is referred to as a Q - d o m a i n of L-fuzzy sets, or simply, a QL-domain. Accordingly, (Y, ~)is referred to as a Q-pattern space of L-fuzzy sets, or simply, a QL-pattern space.

Definition 7.2. Let (F, ~) be a QL-pattern space. A Q-scale measure of L-fuzzy sets, or simply, a QL-scale measure is a m a p p i n g a : ~ ~ L with the following properties: (1) a(A) = a, if/~A(X) -= a for every x (2) VA~ ~ ~, ~ ~ Q, there holds a ( U ~ o A ) = sup~Qa(A~). Then (F, ~, a) is referred to as a Q-scale measure space of L-fuzzy sets, or simply, a QL-scale measure space. For QL-Scale measure we have the following basic properties.

Proposition 7.3. Let (Y, ~, a) be a QL-scale measure space. (1) l f a, B ~ , A <<.B, then a(A)<-., a(B). (2) VA ~ ~, there holds 0 <~ a(A) <<. 1. (3) Suppose { a a ~ is a class Of QL-measures defined on ( F, ~ ), then a* = supa~Q, c Q a a is also a Qt-measure defined on ( F, ~ ). (4) Suppose ~ contains all the L-fuzzy sets defined on F and there exists a one-to-one correspondence between F and Q~ c Q, then VA E ~, a(A) <~ supx~r/~a(x). (5) V A~ e ~, ~ • Q, there holds a( ( ] ~ o A~) <<.inf~Qa(A~).

80

K.- Y. Cai / Fuzzy Sets and Systems 66 (1994) 59-81

(6) Suppose there exists a one-to-one correspondence between N and Q~ c Q and {An } is a class of series oj L-fuzzy sets in ~ with A1 <~ A2 <~ "" <-%An <%An+ l <~ "" • Then a ( l i m n ~ An) = lim,_~ a(An). Proof. Similar to those presented in Section 3.

[]

8. Concluding remarks Nahmias' scale measure is an attractive notion both in theory and in application [3, 19, 20-l. So it is natural to generalize this notion to apply to fuzzy sets. Consequently in this paper we introduce the notion of Q-scale measure. A Q-scale measure is slightly different from a crisp-valued possibility measure in several ways. We define a Q-scale measure in terms of Q-domain and Q-pattern space. This provides a formal framework for theoretical investigations. In application, Q-scale measure may serve as a basis for a theory of posfust reliability [2]. In this article, we have used a number of examples to demonstrate the validity of the notions of Q-pattern space and Q-scale measure in various circumstances. We also give preliminary discussions on basic properties of Q-scale measure, induced Q-scale measure, semigroup of Q-scale measures, Q-scale measure of probabilistic sets, and Q-scale measure of L-fuzzy sets. However, we should note that every topic mentioned above must be further investigated and integral with respect to Q-scale measure remains to be defined and explored.

References [1] R.C. Bassanezi, On functions representable by fuzzy measures, d. Math. Anal. Appl. 133 (1988) 44-56. 1-2] K.Y. Cai, Fuzzy reliability theories, Fuzzy Sets and Systems 40 (1991) 510-511. 1,3] K.Y. Cai, Parameter estimations of normal fuzzy variables, Fuzzy Sets and Systems 55 (1993) 179 185. [4] K.Y. Cai, C.Y. Wen and M.L Zhang, Fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context, Fuzzy Sets and Systems 42 (1991) 145-172. I-5] K.Y. Cai, C.Y. Wen and M.L. Zhang, A critical review on software reliability modeling, Reliability Engr 9. System Safety 32 (1991) 357-371. 1,6] K.Y. Cai, C.Y. Wen and M.L. Zhang, A novel approach to software reliability modeling, Microelectronics Reliability 33(15) (1993) 2265 2267. 1,7] K.Y. Cai, C.Y. Wen and M.L. Zhang, Fuzzy states as a basis for a theory of fuzzy reliability, Microelectronics Reliability 33(15) (1993) 2253 2263. [8] K.Y. Cai, C.Y. Wen and M.L. Zhang, Fuzzy reliability modeling of gracefully degradable computing systems, Reliability Engr9. System Safety 33 (1991) 141-157. [9] A.H. Clifford and G.B. Preston, The Algebraic Theory ofSemiyroups, Vol. 1 (American Mathematical Society, Providence, RI, 1961). [I0] G. Choquet, Lectures on Analysis, Vol. 1 (Benjamin, New York, 1969). [11] M.de Glas, Fuzzy a-fields and fuzzy measures, J. Math. Anal. Appl. 124 (1987) 281-289. 1,12] I.I. Gihman and A.V. Skorokod, The Theory of Stochastic Processes I (Springer Berlin, 1974). [13] M. Grabisch, T. Murofushi and M. Sugeno, Fuzzy measure of fuzzy events defined by fuzzy integrals, Fuzzy Sets and Systems 50 (1992) 293 313. [14] K. Hirota, Concepts of probabilistic sets, Fuzzy Sets and Systems 5 (1981) 31-46. [15] J.M. Howie, An Introduction to Semigroup Theory (Academic Press, New York, 1976). 1,16] H. Ichihashi, H. Tanaka and K. Asai, Fuzzy integrals based on pseudo-addition and multiplications, J. Math. Anal. Appl. 130 (1988) 354-364. [17] E.P. Klement, W. Schwyhla and R. Lowen, Fuzzy probability measures, Fuzzy Sets and Systems 5 (1981) 21-30. [18] E.P. Klement and S. Weber, Generalized measures, Fuzzy Sets and Systems 40 (1991) 375-394. [19] H. Kwakernaak, Fuzzy random variables 1, Inform. Sci. 15 (1978) 1-29. [20] H. Kwakernaak, Fuzzy random variables I1, Inform. Sci. 17 (1979) 253-278. 1,21] S. Nahmias, Fuzzy variables, Fuzzy Sets and Systems (1978) 97-110.

K.-Y. Cai / Fuzzy Sets and Systems 66 (1994) 59-81 [22] [23] [24] [25] [26] [27] [28] [29]

M.L. Puri, Convergence theorem for fuzzy martingales, J. Math. Anal. Appl. 160 (1991) 107-122. Z. Qiao, On fuzzy measure and fuzzy integral on fuzzy set, Fuzzy Sets and Systems 37 (1990) 77-92. G. Sharer, A Mathematical Theory of Evidence (Princeton University Press, Princeton, N J, 1976). M. Sugeno, Theory of fuzzy integral and its applications, Ph.D. Thesis, Tokyo Institute of Technology (1974). L. Wang, Counterexamples in Real Analysis (Advanced Education Press, 1988) in Chinese. S. Weber, ±-decomposable measures and integrals for archimedean t-conorms ±, J. Math. Anal. Appl. 101 tl984) l l4 138. L.A. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl. 23 i1968) 421 427. L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3 28.

8