fuzzy sets and systems volume 48 issue 1 1992 [doi 10.1016%2f0165-0114%2892%2990258-6] r.g. mclean;...

8/13/2019 Fuzzy Sets and Systems Volume 48 Issue 1 1992 [Doi 10.1016%2F0165-0114%2892%2990258-6] R.G. McLean; H

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Fuzzy Se t s and Sys tem s 48 (1992) 137-14 0 137N o r t h - H o l l a n d

uzzy ideals in semigroupsR G M c L e a nDepartment of Mathematics, City University, NorthamptonSquare , London, EC I V O H B, U KH K u m m e rDepartmen t of M athematics and Statistics, Queen sUniversity, Kingston, Ontario, Canada KTL 3N6Rece ived Ju ly 1990Re v i s e d J a n u a r y 1 9 91

Abstract: Th e s e t o f a l l f u z z y i d e a l s o f a s e m i g r o u p i s s h o wnt o b e a c o n v e x s e t h a v i n g t h e ( c r i s p ) id e a l s a s e x t r e m e p o i n t s .S o m e c o n n e c t i o n s b e t we e n f u z z y i d e a l s a n d Gr e e n ' s r e l a t i o n sa re d i scussed .Keywords: F u z z y s e t s ; s e m i g r o u p s ; c o n v e x s e t s ; Gr e e n ' sr e l a t ions .

1 IntroductionF u z z y i d ea l s in s e m i g r o u p s w e r e i n t r o d u c e d i n

[ 7] a nd d i sc us se d f u r the r i n [ 3 , 4 , 5 ] . I n t h i s no t ew e s h a l l d e s c r i b e t h e c o n v e x s t r u c t u r e o f t h e s e to f a ll fu z z y id e a l s o f a s e m i g r o u p a n d i n v e s t i g a t es o m e c o n n e c t i o n s b e t w e e n f u z z y id e a ls a n dG r e e n s r e la t i o n s .

W e b e g i n b y r e c a ll i n g s o m e d e f i n it i o n s f r o m[7 ] a n d [8 ]. F o r t h e b a s i c s o f s e m i g r o u p t h e o r y ,a n d d e f in i t io n s o f t h e c o n c e p t s u s e d i n o u re x a m p l e s , t h e r e a d e r i s r e f e r r e d t o [ 2 ] o r [ 6 ] .

A f u z z y s u b s e t o f a s e t S i s d e f i n e d t o b e am a p 6 f r om S in to t he i n t e r va l [ 0 , 1 ] o f . I f S isa s e m i g r o u p t h e n a f u z z y s u b s e t 6 o f S is ca l l e d :a fu zz y r ight idea l of S i f 6 ( x y ) > 6 ( x ) f o r a l lx , y ~ S ;

a fu zz y l e ft idea l of S i f 6 ( x y ) >i 6 ( y ) f o r a l lx , y ~ S ;a f u z z y i de a l ( o r f u z z y t w o - s i d e d i d e a l ) i f i t i sb o t h a f u z z y l e f t i d e a l a n d a f u z z y r i g h t i d e a l .E a c h s u b s e t I o f S m a y b e r e g a r d e d a s a f u z z y

s u b s e t b y i d e n t i f y i n g it w i t h i t s c h a r a c t e r i s t i cf u n c t i o n X I. I f I i s a n y n o n e m p t y s u b s e t o f St h e n I i s a r i g h t ( l e f t, t w o - s i d e d ) i d e a l i f a n d o n l yi f X t i s a f uz z y r i gh t ( l e f t , tw o- s id e d) i de a l ( se e[7]).

Example 1 . A m a p f r o m a s e m i l a t t i c e L i n t o[ 0, 1 ] i s a fuz z y i de a l o f L i f a n d on ly i f i t i s o r d e rr e ve r s ing .Example 2 . L e t S b e a n i n v e r s e s e m i g r o u p .T h e n b y d e f i n i t i o n [ 2 , p. 1 2 9 ] f o r e a c h a ~ St h e r e i s a u n i q u e a * ~ S w i t haa*a = a a n d a*aa* = a*.L e t E b e t h e s e t o f id e m p o t e n t s o f S , o r d e r e d b yt h e r e l a t i o na < - b :> a b = a ,a n d l e t 6 b e a m a p f r o m S i n t o [ 0 , 1 ]. T h e n 6 i s af u z z y r i g h t i d e a l i f a n d o n l y i f t h e r e s t r i c t i o n o f 6t o E i s o r d e r r e v e r s i n g a n d 6(a ) = 6 (aa*) f o r a l la ~ S .

T o p r o v e t h i s , f i r s t s u p p o s e t h a t 6 i s a f u z z yr igh t i de a l , t he n , f o r a l l a ~ S ,6(aa*) >i 6(a) = 6(a a*a ) >i 6(aa* ) ,a n d f o r a n y p , q ~ E ,p < ~q ~ p = q p ~ 6 ( p ) = 6 ( q p ) > 1 6 ( q ) .C o n v e r s e l y s u p p o s e t h a t 6 is o r d e r r e v e r s in g a n dt h a t 6 ( a ) = 6(aa*) f o r a ll a 6 S . T h e n f o r a n yx , y 6 S w e h a v e xyy*x* i 6 x x * ) = 6 x ) .

Example 3 . L e t S b e a s e m i l a t t ic e o f g r o u p s{Gr [ r ~ L } [ 2 p . 8 9 ] a n d f o r e a c h r i n t h es e m i l a t t i c e L l e t l r b e t h e i d e n t i t y o f t h e g r o u pG ~ . T h e n b y d e f i n i t i o n S i s t h e d i s j o i n t u n i o n o ft h e g r o u p s G , (r e L ) a n d f o r e a c h r , s i n t h es e m i l a t t i c e L w e h a v e GrG ~ ~_ Gr~. I f 6 i s a m a pf r o m S i n t o [ 0 , 1 ] t h e n t h e s e a r e e q u i v a l e n t :

( 1 ) 6 i s a f uz z y r igh t i de a l ;( 2 ) 6 i s a f uz z y l e f t ide a l ;( 3 ) 6 is c o n s t a n t o n e a c h G ~ a n d r ~ < s

6 ( l r ) / > 6 ( l s ) f o r a l l r , s c L .W e s h a l l p r o v e t h e e q u i v a l e n c e o f ( 1 ) a n d ( 3 ) ;

t h e r e m a i n i n g e q u i v a l e n c e f o l l o w s f ro m a s i m i la ra r g u m e n t . F i r s t s u p p o s e t h a t 6 i s a fu z z y ri g h tide a l . I f x ~ G~ a n d x - I i s t he i nve r se o f x i n Gr

0 1 6 5 - 0 1 1 4 / 9 2 / 0 5 . 0 0 (~ ) 1 9 9 2 - - E l s e v i e r S c i e n c e P u b l i s h e r s B . V . A l l r i g h t s re s e r v e d


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138t h e n6 ( 1 , ) ~< 6 ( l r X ) = 6(X ) 6 ( l r ) = 6 ( X ) .H e n c e 6 i s a f u z z y r ig h t i d e a l .

2 T h e c o m p a c t c o n v e x s e t o f f u z z y i d ea l sW e s h a l l n o w s h o w t h a t a n a r b i t r a r y f u z z y

i d e a l c a n b e a p p r o x i m a t e d b y a s u i t a b l e l i n e a rc o m b i n a t i o n o f c h a r a c t e r i s t ic f u n c t i o n s o f i d e a ls .T o m a k e t h i s s t a t e m e n t p r e c i s e w e s h a l l e n d o wt h e s e t ~ o f a ll fu z z y id e a l s o f a s e m i g r o u p Sw i t h t w o t o p o l o g i e s . T h e t opo logy o f po in twi seconvergence o n ~ is t h e s u b s p a c e t o p o l o g y o ni n d u c e d b y t h e p r o d u c t t o p o l o g y o n [ 0, 1 ] s. T h enorm topo logy o n ~ : is t h e s u b s p a c e t o p o l o g yi n d u c e d o n ~ b y th e B a n a c h s p a c e g ~ ( S ) o f a llb o u n d e d r e a l v a l u e d f u n c t i o n s o n S , e q u i p p e dw i th t h e su p r e m u m n o r m d e f in e d b y11611 = s u p { 1 6 ( x ) l : x e s } v 6 ~ t = ( s ) .

P r o p o s i t i o n 1 . Le t ~ be the se t o f a l l f u zz y r igh tideals o f a sem igroup S . Then ~ i s a com pactconv ex se t in t he topo logy o f po in twi seconvergence and the ex t reme po in t s o f ~ are thecharacteris t ic funct ions o f the r ight ideals o f Stoge ther wi th the zero func t ion . I f 6 i s a f u zz yright ideal o f S then for every e > 0 there i s ac o n v e x c o m b i n a t io n y o f e x tr e m e p o i n t s o fwith 0


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R.G. McLea n, H. Kumm er / Fu zzy ideals in semigroups 1393 . F u z z y i d e a l s a n d G r e e n s r e l a ti o n s

W e b e g i n t h i s s e c t i o n b y i n t r o d u c i n g s o m en o t a t i o n a n d r e v i e w i n g t h e d e f i n i t io n s o f G r e e n sr e l a t i o n s ( o r e q u i v a l e n c e s ) . F u r t h e r d e t a i l s m a yb e f o u n d i n [ 2 , 6 ] .L e t S b e a s e m i g r o u p . T h e n f o r e v e r y a S w ed e f i n ea S l = {a} tO { a x I x S } ;S l a = {a) tO {x a I x S ) ;S~aS ~ = S~a t3 aS ~ O {xa y Ix , y S} .Gre e n ' s re la t i ons a r e t h e e q u i v a l e n c e r e l a t i o n sG , 5 ~, J , ~ ( a n d ~ d e f i n e d f o r a l l a , b S b ya ~ b ~ a S l = b S l,a 5Eb 1 0 ( b ) - e for a l l e > 0 soO(a)> O(b) . []

T h e f o l l o w i n g c o r o l l a r i e s a r e r o u t i n e c o n s e -q u e n c e s o f t h e l a st p r o p o s i t i o n .

T h e f o l l o w i n g r e s u l t i s e a s i l y p r o v e d . C o r o l l a r y 2 . I f a a n d b a r e e le m e n t s o f a


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140 R.G. McLean , H. Kum mer / Fuzzy ideals in semigroupss e m i g r o u p S th e n t h e f o l l o w i n g a r e eq u i v a l e n t :

(1) a ~ b ;(2) Z A b ) = 1 = z B a ) w h e r e A = a S 1 a n d B =

b S l ;(3) x t a ) = z t b ) f o r a l l p r i n c i p a l r i g h t id e a l s 1o f s ;4 ) xt a ) = t b) fo r al l r ights ideals I of S;

5 ) 6 a ) = 6 b ) f o r a ll f u z z y r ig h t i d ea ls 6o r S .W e s h a l l s a y t h a t a s e t o o f f u n c t i o n s o n a

s e m i g r o u p S i s s e p a r a t i n g i f , f o r a l l d i s t i nc t x a ndy in S , t h e r e i s a n f e ~ w i th f x ) : / : f y ) .Corollary 3 . L e t S b e a s e m i g r o u p . T h e n t h e s ea r e e q u i v a l e n t :

(1) t h e r el a ti o n < ~ ~ ) i s a n o r d e r o n S ;(2) e a c h Z - c l a s s i s a s in g l e t o n ;(3) t h e s e t o f a l l c h a r a c t e r i s t i c f u n c t i o n s o f

p r i n c i p a l r i g h t i d e a l s o f S i s s e p a r a t i n g ;(4) t h e s e t o f a l l c h a r a c t e r i s ti c f u n c t i o n s o f r i g h t

i d e a l s o f S i s s e p a r a t i n g ;(5) t h e s e t o f a l l f u z z y r i g h t id e a ls o f S i s

s e p a r a t i n g .

T h e r e a r e c o r r e s p o n d i n g p r o p o s i t i o n s a n dc o r o l l a r ie s c o n n e c t i n g t h e r e l a t i o n ~ 7 w i t h f u z z yl e f t i d e a l s a n d c o n n e c t i n g t h e r e l a t i o n J w i t hf u z z y i d e a l s . W e s h a l l s u m m a r i s e a f e w o f t h e s er e s u l t s i n t h e n e x t p r o p o s i t i o n .

Example 5 . T h e f u z z y r i g h t i d e a l s o f a l e f tc a n c e ll a ti v e m o n o i d a r e s e p a r a t i n g i f a n d o n l y i f1 i s t h e o n l y e l e m e n t h a v i n g a r i g h t i n v e r s e .( T h is f o ll o w s f r o m C o r o l l a r y 3 a n d E x a m p l e 4 ) .Example 6 . L e t S b e a c a n c e l l a t i v e s e m i g r o u pw i t h o u t i d e n t i t y . T h e n i t i s k n o w n t h a t t h e~ - c l a s s e s o f S a r e s i n g l e t o n s a n d t h a t t h e r e i s ap a r t i c u l a r c a n c e l l a t i v e s e m i g r o u p S w i t h o u ti d e n t i ty f o r w h i c h t h e r e is o n l y o n e J - c l a s s ( s ee[2 , E x e r c i s e 1 o n p . 5 3 ] ) . I t f o l l o w s f r o mC o r o l l a r y 3 t h a t t h e f u z z y r ig h t i d e a l s o f S a r es e p a r a t i n g , w h i l e P r o p o s i t i o n 4 , p a r t ( 1 ) , i m p l i e st h a t t h e f u z z y i d ea l s o f S n e e d n o t b e s e p a r a t in g .I f S i s a f r e e s e m i g r o u p t h e n i t i s e a s i l y v e r if i e dt h a t t h e J - c l a s s e s o f S a re s i n g l e t o n s a n d i tf o l l o w s t h a t , i n th i s p a r t i c u l a r c a s e , t h e f u z z yi d e a l s o f S a r e s e p a r a t i n g .

cknowledgementsT h i s w o r k w a s s t a r t e d w h i l e t h e f i r s t a u t h o r

w a s v is i ti n g Q u e e n ' s U n i v e r s i t y w i t h t h e s u p p o r to f a g ra n t f r o m t h e L e v e r h u l m e T r u s t . T h es e c o n d a u t h o r h a s b e e n s u p p o r t e d b y t h eN a t u r a l S c i e n c e s a n d E n g i n e e r i n g R e s e a r c hC o u n c i l o f C a n a d a .

ReferencesProposition 4 . L e t a a n d b b e e l e m e n t s o f as e m i g r o u p S . T h e n

(1) a J b i f a n d o n ly i f 6 a ) = 6 b ) f o r a llf u z z y i de a ls 6 o f S ;

(2) a ~ b i f a n d o n l y i f th e r e i s a n x E S w i t hC p a ) = c p x ) f o r a l l f u z z y r i g h t i d e a l s dp o f S a n d~ p x) = l p b ) f o r a l l f u z z y l e f t i d e a l s ~p o f S ;(3) a Y b i f a n d o n l y i f 6 a ) = 6 b ) w h e n e v e r6 i s a f u z z y l e ft id e a l o r a f u z z y r i g h t i d e a l o f S .P r o o f . T o p r o v e ( 2 ) u s e th e f a c t t h a t ~ = ~ o( s e e [ 2, p . 3 9 ]) . E v e r y t h i n g e l s e is a c o n s e q u e n c eo f t h e r e m a r k s a b o v e . [ ]

[1] K.H. Hofm ann and P .S. Mostert, Elements of CompactSemigroups (Charles E. Me rrill, Co lumb us, OH , 1966).[2] J.M. Howie, An Introduction to Sem igrou p Theory(Academic Press, Lond on, 1976).[3] N. Ku rok i, Fu zzy bi-ideals in semigroups, Comment .M ath. U niv. St . Pa uli . 28 (1979) 17-21.[4] N. Ku roki, On fuzz y ideals and fuz zy bi-ideals insemigroups, Fuzzy Sets and Systems 5 (1981) 203-215.[5] N. Ku roki, Fuzzy semiprim e ideals in sem igroups, FuzzySets and Systems 8 (1982) 71-79.[6] M. Petrich, Introduct ion to Semigroups (Charles E.Merrill, Colum bus, OH , 1973).[7] A. Rosenfeid, Fuzzy groups, J . M ath. Anal . A ppl . 35(1971) 512-517.[8] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965)338-353.

fuzzy sets and systems volume 48 issue 1 1992 [doi 10.1016%2f0165-0114%2892%2990258-6] r.g. mclean;...

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