Fuzzy algebra in triangular norm system

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems93 (1998) 331 335

Short Communication

Fuzzy algebra in triangular norm system Song Xiaoqiu, Pan Zhi* Department of Mathematics and Mechanics, China Universityof Mining & Technology,Jiangsu, Xuzhou, 221008. People's Republic of China

ReceivedJanuary 1993; revisedJune 1996

Abstract

Triangular norm is a powerful tool in the theory research and application development of fuzzy sets. In this paper, using the triangular norm, we introduce some concepts such as fuzzy algebra, fuzzy a algebra, fuzzy monotone class, fuzzy class and fuzzy )~class, and discuss the relations among them, obtaining a series of conclusions, main ones of which are fuzzy monotone class theorems (cf. Theorems 1-3 ). © 1998 Elsevier Science B.V. Keywords: Fuzzy algebra; Fuzzy cr algebra; Fuzzy monotone class; Fuzzy ~ class; Fuzzy 2 class; Triangular norm system

1. Preliminaries

The triangular norm, T-norm and S-norm, originated from the studies of probabilistic metric spaces in which triangular inequalities were extended using the theory of T-norm and S-norm. Later, Hohle [5], Alsina et al. [1] introduced the T-norm and the S-norm into fuzzy set theory and suggested that the T-norm and the S-norm be used for the intersection and union of fuzzy sets. Since then, many other researchers have presented various types of T-norms and S-norms for particular purposes [3, 6]. In practice, Zadeh's conventional T-norm and S-norm,/~ and V, have been used in almost every design for fuzzy logic controllers and

* Corresponding author.

even in the modelling of other decision-making processes. However, some theoretical and experimental studies seem to indicate that other types of T-norms and S-norms may work better in some situations, especially in the context of decisionmaking processes. In this paper, using the triangular norms, we introduce some concepts such as fuzzy algebra, fuzzy a algebra, fuzzy monotone class, fuzzy ~ class and fuzzy 2 class in norm system (I, S, T, C), and discuss the relations among them, obtaining the following results: (1) In the norm system (I, S, T, C), i f d is a fuzzy algebra, then d is a fuzzy a algebra if and only if d is a fuzzy monotone class; (2) If cp c Y ( X ) is a fuzzy algebra, then m(cp) = a(cp), here m(~o) and a(~o) represent the fuzzy monotone class and fuzzy a algebra generated by ~p, respectively; (3) If d is a fuzzy ~ class, then ~(d) = ~(d).

0165-0114/98/$19.00 © 1998 Elsevier ScienceB.V. All rights reserved PII S0165-01 14(96j00195-9

Song Xiaoqiu, Pan Zhi / Fuzzy Sets and Systems 93 (1998) 331-335

332

D e f i n i t i o n 1. Let I = [0, 1]. A T - n o r m is a function

T : I x I ~ I satisfying for every x, y and z in I (I) T(x, 1) = x; (II) T(x,y)= T(y,x); (III) T(xi,yl) <~ T(x2,Y2), ifx~ ~< x2, yi ~< Y2; (IV) T(x, T(y,z)) -- T(T(x,y),z); (V) lim.~ ~ T(x,, y,) = T(lim.~ o0x,, lim.~ ~ y,), if x , , y , e I , n = l , 2 , . . . , and lim,.ooX, and lim._, ~ y, exist. An S-norm is a function S : I x I ~ I satisfying (II)-(V) and (I)' S(x, O) = x; F o r a e I , let a ¢ = ~ 1 - a ~ I , we say that a c is a complement o f a e I. It is easy to see that (a¢) ~ = a, Va ~ I, (I, S, T, C) is called a triangular n o r m system. Given T - n o r m T(x, y), let

S(x,y)&l -- T(1 - - x , 1 -y)&T¢(x~,y¢),

(1)

obviously, S(x, y) is an S-norm. We say that S and T are dual. 2. Let X be universe, VA, B e ~ ( X ) ( ~ ( X ) represents all fuzzy sets of X ) define Definition

S(A,B)~-S(A(x),B(x)),

Vx ~ X ,

(2)

T(A,B)& r(A(x),S(x)),

Vx ~ X

(3)

S &S

i=t

i=1

i=i

Ai,A3 ;

i=

A i & S \ i = l Ai,an ;

Ai&T

T Ai, A3 ,

i=1

"F A i Z ~ T \ i = l Ai,An

i=1

Generally, for {A .}.= ~ 1 c ~ ( X ) , we define S A.-~lim n=l

n ~

S Ai,

"F A i g l i m

i=1

i=1

n ~

~" Ai

(4)

i=1

if the limits of right-hand sides of (4) exist. Definition 3. { A , }.~= 1 c ~ ( X ) is monotonicallyincreasing (denoted by A, T) if Ai c A 2 ~ A 3 ~ ...; {A,}.=I~ c ~ ( X ) is monotonically decreasing (denoted by A, l ) if A1 = A2 ~ A3 ~ ." {A,} is called a m o n o t o n i c sequence if it is either monotonically increasing or monotonically decreasing. In the triangular n o r m system (I,S,T,C), if {A,},= i , ° ° {Bn}n~=l ~ ~ ( X ) a n d A , T,B,$,wedefine lim n~m

A.& S A,, n=l

lim n~m

B. ~= T B,.

(5)

n=l

and say that S(A, B) and T(A,B) are the S-union and T-intersection.

Let E be an ordinary subset of X and ZE be a characteristic function of E, we define e'ZE as follows:

L e m m a 1. In the norm system (I,S, T,C), the Sunion and T-intersection defined by (2) and (3) satisfy

{~, if x e E , (~'Z~)(x)~=~ A ZE(x)= O, if xCE,

the following properties: (I) S(A,B) = S(B,A), T(A,B) = T(B,A); (II) S(S(A, B), C) = S(A, S(B, C)), T(T(A, B), C) = T(A, T(B, C)); (III) T(A, el)) = ~, T(A,X) = A, S(A, el)) = A, S ( A , X ) = X;

(IV) [S(A, B)] ~= T(A ¢,Be), [T(A, B)] ¢ = S(A ~,B~). Proof. By the definitions of S-norm and T-norm, S-union and T-intersection, and (1), we get the lemma at once. [] For Ai e ~ ( X ) , i = 1, 2,..., n, we define recursively the finite S-union and T-intersection as follows: 2

2

S Ai~S(A1,A2);

T Ai~--T(A1, Az),

i=1

i=1

here ~ e [0, 1]. D e f i n i t i o n 4. Suppose ~¢¢ c ~-(X), if d

satisfies (I) ~- ;~E e d for all ~ ~ [0, 1] and E c X; (II) A c e d f o r a l l A ~ d ; (III) S(A, B) ~ d for all A, B ~ d , then we say d is a fuzzy algebra in the (I, S, T, C). oo t c d , If d satisfies (I), (II) and (III)' for {A ,},= n=l AnE d ,

then d is called a fuzzy a algebra. Using " 0 " to represent the ordinary intersection symbol, we have the following lemma. L e m m a 2. I f {do}O~k is a family offuzzy a algebrae

then Na~k do is also a fuzzy a algebra.

Song Xiaoqiu, Pan Zhi / Fuzzy Sets and Systems 93 (1998) 331-335

Proof. Because for all fl c k, N~ is a fuzzy a algebra, we have: (I) For all ~ e [0, 1], fl c k, since ~'Zr e ~'p, therefore ~. ZE ~ ( ~ k ~'~ (II) Suppose A e 0 ~ k d e , then A e d~ for all fl ~ k, hence A ~ e ~/~, therefore A ~ ~ ~ k ~ " (III) Suppose A , e ( ~ k ~ , n = 1,2 . . . . . then A,~'¢ for all f l ~ k and n = 1,2,3,..., hence, S~,~-~A , e ~ , thus S,~=~A , ~ O ~ k ~ . F r o m (I)-(III), we know that ( ~ k d~ is a fuzzy algebra. [] For d ~ ~ ( X ) , from Lemma 2, the intersection of all fuzzy a algebra containing d is a fuzzy a algebra, and is the least fuzzy a algebra containing d , represented by ~r(~'). We call a ( d ) a fuzzy a algebra generated by fuzzy set family d . D e f i n i t i o n 5. Suppose ~ c J ( X ) , if for any fuzzy monotonic sequence {A. },% 1 ~ ~, lim,~ o~A, e g, then we call ~ a fuzzy monotone class. It is evident that the following lemma holds.

Lemma 3. I f {d~t~}p~k is a family of fuzzy monotone classes, then O,~k ~ is also a fuzzy monotone class. From Lemma 3, we known that for every ~¢ ~ W(X), the intersection of all fuzzy monotone classes containing d is a monotone class, and is the least fuzzy monotone class containing ~¢, represented by re(d). We call m ( d ) a fuzzy monotone class generated by fuzzy set family ~¢. D e f i n i t i o n 6. Suppose d

~ i f ( X ) , if for A, B e ~ , T(A,B) e d , and for ~ e [ 0 , 1 ] and E c X , ~'ZE ~ o~, then we call d a fuzzy n class.

Lemma 5. / f {A,}p~k is a family of fuzzy 2 class, then (~p~k ~eJ~is also a fuzzy 2 class. Lemma 4 can be proved by Definitions 4, 6 and 7, and Lemma 5 can be proved similar to Lemma 2, From Lemma 5, we known that for every ~ ' c W(X), the intersection of all fuzzy 2 classes containing d is a fuzzy 2 class, and is the least fuzzy 2 class containing d , represented by 2 ( d ) . We call 2(A) is a fuzzy 2 class generated by fuzzy set family d .

2. M a i n t h e o r e m s

In this section, we will discuss the relations among fuzzy tr algebra, fuzzy monotone class, fuzzy class and fuzzy 2 class. First we prove the following theorem. 1. Let (I,S, T,C) be a triangular norm system, S and T be dual norm. Then (I) If ~¢ is a fuzzy a algebra, then ~¢ is also a fuzzy monotone class; (II) If a fuzzy algebra ~¢ is a fuzzy monotone class, then ~¢ is also a fuzzy a algebra.

Theorem

Proof. (I) Let {A,}.~=I and {B,}2_ 1 be arbitrary monotonic sequence in d such that A, T and B, +. Since d is a fuzzy a algebra, hence S,% 1 A, e ~4. But l i m . ~ A . = S ,=1 ~ A., we obtain that lim, ~ ~ A. ~ ~ . Notice that B c ~ A, n = 1, 2 ..... and B,~ T, from the above process, we have l i m , ~ B~ = S ~.=IB. d . Using the definition of fuzzy o- algebra and Lemma 1, we know that limB,=

D e f i n i t i o n 7. Suppose d

~ ~ ( X ) , if d satisfies

(I) A~ d f o r a l l A c d ; (II) S(A, B) ~ d for all A, B e d and T(A, B) -- 0; (III) if {A . }, = 1 ~ d and A. T, then lim,~ o0A, e d , then we say d is a fuzzy 2 class in the (I, S, T, C). Lemma 4. ~ ' ~ ~ ( X ) is a fuzzy a algebra if and only if d is both fuzzy 2 class and fuzzy 7z class.

333

n~oc

T B,=

n= l

()c S B~

n=l

ed.

Therefore, ~¢ is a fuzzy monotone class. (II) To prove d is a fuzzy a algebra, we only need to prove that d satisfies countable additivity since ~¢ is a fuzzy algebra. For every {A , },~c= 1 c ~ ' and fixed m ~> 1, we have S"~k=lAke~. Let Bm -- S m k= 1 Ak, then B,, c ~4, m = 1, 2 . . . . and Bm TNoticing that s¢ is a fuzzy monotone class, we

Song Xiaoqiu, Pan Zhi / Fuzzy Sets and Systems 93 (1998) 331-335

334

which shows that lim.~oo B, ~ D(A), namely, D(A) is a fuzzy monotone class. For A, B e ¢p, since ~ois a fuzzy algebra, S(A, B) ~o ~ m(q~), which tells us that B ~ D(A) and hence

get A k = lim ( ~ k=l

m ~

k=l

A k ) = lim Bme ~¢, m ~

which shows that d is a fuzzy a algebra.

[]

Theorem 1 shows that if d is a fuzzy algebra, then d is a fuzzy a algebra if and only if d is a fuzzy monotone class. We will further discuss another problem: for every q~ c ~ ( X ) , what is the relation between m(~0) and a(q))? The following theorem answers the question. Theorem 2. I f q~ ~ o~(X) is a fuzzy algebra, then m(q~) = a(q~), here m(q)) and a(q~) represent the fuzzy

monotone class and fuzzy a algebra generated by q), respectively. Proof. First, we will prove m(~0) ~ a(~o). Since a(~p) is a fuzzy a algebra, Theorem 1 tell us a(~p) is a fuzzy monotone class. In addition, since m(q~) is the least fuzzy monotone class containing q~ and ~o ~ a(q~), therefore, m(q~) ~ a(~p). Second, we will prove m(~0)~ a(q)). If we can prove m(q)) is a fuzzy a algebra, then m(q)) = a(~0) because a(cp) is the least fuzzy a algebra containing q~. From Theorem 1, we only need to prove m(~o) is a fuzzy algebra. Consider the following steps: (I) For all ~ ~ [0, 1], since q~ is a fuzzy algebra, ~'Z~ ~ q~; And since ~o ~ m(~0), we have ~'Z~ m(cp). (II) Given A e re(go), let

D(A) = {B: B ~ m(~p),S(A,B) ~ m(~p)},

(6)

we want to prove m(q~) = D(A). By definition of D(A), we know D(A) ~ m(q~). ~ x ~ D(A) is a fuzzy Conversely, suppose {B ,},= monotonic sequence, then lim,~oo B, ~ re(d) and S(A, B,) e m(~o), n = 1, 2..... Notice that S(A, B,) is a fuzzy monotonic sequence and S(A,B,) is continuous to B,, we know that

n--*~

99 ~ D(A). From the above process, we obtain m(~o) = D(A), which demonstrates that m(cp) is closed under Sunion operation. (III) Let D={A:A~m(qg)

and

A ¢~m(q~)}.

(7)

By (7), we know ~p c D c m(q0). Conversely, let {A,},~ 1 c D such that A, T and l i m , ~ o o A . = S , % 1 A , = A . Since m(q~) is a fuzzy monotone class and D ~ m(~o), thus A ~ m(~o). And ~ An~ = A ¢, ~ ¢1o~ c D since A~+ and hmn~ " ( A n)n=l m(~o), we have A ¢ 6 m(~o), hence A 6 D. Similarly, if {B,},%1 ~ D is a fuzzy monotone sequence such that B, $ and lim,~o B, = B, then B e D. Therefore, D is a fuzzy monotone class, thus m(~o) ~ D since re(q0) is the least monotone class containing ~o. From the above process, we get D = m(~o) which shows that m(q~) is closed under complementary operation "c". The above three steps show that m(q~) is a fuzzy algebra. [] Theorem 3. If d

is a fuzzy

~ class, then

Proof. Since a ( d ) is a fuzzy a class (from Theorem 1), Lemma 4 tells us £ ( d ) is a fuzzy £ class. In addition, since 2 ( d ) is the least fuzzy A class containing d and d c a ( d ) , 2 ( d ) c a ( d ) . Conversely, we need only to prove 2 ( d ) is a fuzzy ~ class, from which we can get 2 ( d ) is a fuzzy a algebra and 2 ( d ) ~ a ( d ) by Lemmas 4 and 5. In order to prove 2(~¢) is a fuzzy 7z class, let 91 = {A: A ~ 2 ( d ) and for every B e d ,

T(A, B) ~ 2 ( d ) } then ~x ~ d . For A ~ 91, B ~ d , since T(A¢,B) = [S(A, Be)] °, A ~ ~ 91, and from which we conclude that ~1 is a fuzzy 2 class, so ~1 ~ 2(~¢), this is to

Song Xiaoqiu. Pan Zhi ,/Fuzzy Sets and Systems 93 (1998) 331-335

say that for every B e d ,

;+~').

A•2(d),

T(A,B)•

Again, let ~2 = {A: A • 2 ( d ) and for every B • 2 ( d ) ,

T(A, B) • 2 ( d ) } , then ~2 ~ ,52~. From the process similar to the above, we can get Be is a fuzzy 2 class, therefore ~ 2 = "~(~:~), which shows that 2 ( d ) is closed under T-intersection, so 2 ( d ) is a fuzzy rc class. [] Theorems 2 and 3 can be called fuzzy monotone class theorems, which generalizes the classical situations in measure theory to fuzzy set theory.

335

References [-1] C. Alsina et al., On some logical connectives for fuzzy set theory, J. Math. Anal. Appl. 93 (1983) 15-26. [2] A. Dvurecenskij, The Radon Nikodym theorem for fuzzy probability spaces, Fuzzy Sets and Systems 45 (1992) 62-78. I-3] M.M. Gupta and J. Qi, Theory of T-norms and fuzzy inference methods, Fuzzy Sets and Systems 40 (1991) 431-450. 1-4] P.R. Halmos, Measure Theory (van Nostrand Company, New York, 1967). I-5] U. Hohle, Probabilistic uniformization of fuzzy topologies, Fuzzy Sets and Systems 1 (1978) 311 332. [6] Yu yandong, Triangular norms and TNF-sigma algebras, Fuzzy Sets and Systems 16 (1985) 251 264. [7] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338 353.