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Fuzzy ideals and fuzzy bi-ideals in fuzzy semigroups
FUZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 92 (1997) 103-111
Fuzzy ideals and fuzzy bi-ideals in fuzzy semigroups K . A . Dib*, N. G a l h u m Department o[ Mathematics, Faculty of Science, Fayoum Branch. Cairo Unit~ersity. Fayoum. Et,typt
Received January 1994; revised June 1996
Abstract
The purpose of this paper is to introduce some basic concepts of fuzzy algebra such as fuzzy (left, right) ideal and fuzzy bi-ideal in fuzzy semigroup, through the new approach of fuzzy space and fuzzy group introduced by Dib (1994). Our notion of fuzzy ideal and fuzzy bi-ideal includes the (classical) concepts of fuzzy ideal and fuzzy bi-ideal of ordinary semigroup. Many counterexamples are also given. © 1997 Published by Elsevier Science B.V. Keywords: Fuzzy space; Fuzzy semigroup; Fuzzy ideal; Fuzzy bi-ideal
O. Introduction
The study of the fuzzy algebraic structures has started with the introduction of the concepts of fuzzy (subgroupoids) subgroups and fuzzy (left, fight) ideals in the pioneering paper of Rosenfeld [9]. In 1975, Negoita and Ralescu [8] considered a generalization of Rosenfeld's definition where the unit interval [0, 1] is replaced by an appropriate lattice structure. In 1979, Anthony and Sherwood [1] redefined fuzzy (subgroupoids) subgroups using the concept of triangular norm. Kuroki [4, 5] introduced and studied fuzzy (left, right) ideals and fuzzy biideals in semigroups. Several mathematicians [6-8, 10] have followed the Rosenfeld approach in investigating fuzzy algebra where a given ordinary algebraic structure on a given set X is assumed, then introducing the fuzzy algebraic structure as a fuzzy subset A of X satisfying some suitable conditions.
* Correspondingauthor.
In this paper, any fuzzy algebraic structure in the Rosenfeld approach is called a classical fuzzy structure. In 1991, new concepts of fuzzy Cartesian product, fuzzy relation, fuzzy function and a fuzzy binary operation on a set have been introduced [3]. As we know, the concept of universal set played an essential role in ordinary mathematics, e.g. any algebraic system is a universal set with one or more binary operations and the metric space is a universal set, having distance, etc. Therefore, in the absence of the concept of fuzzy universal set, the formulation of a concrete definition of any fuzzy structure is not evident. Recently, the concept of a fuzzy space is introduced in [2] as a replacement for the concept of universal set in the ordinary case. Moreover, a fuzzy subspace, fuzzy binary operation on a fuzzy space and fuzzy (subgroup) group were introduced also in [2]. In the present paper, we introduce some basic concepts of fuzzy algebra such as fuzzy (left, fight) ideals and fuzzy bi-ideals in fuzzy semigroup, through the new approach of fuzzy spaces and fuzzy groups introduced in [2]. A relation between the introduced fuzzy (left, right) ideals or fuzzy bi-ideals and the classical
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104
K.A. Dib, N. GalhumlFuzzy Sets and Systems 92 (1997) 103-111
ones is given. Moreover, many examples are given to illustrate the difference between our point of view and the classical case.
1. Definitions, notations and preliminaries Throughout the paper we adopt, the following notations, definitions and some results of [2, 3]: I: The lattice [0,1] with the usual order of real numbers. L, K, N,...: Arbitrary lattices having a least element and a greatest element. L [] K: The vector lattice L x K with the partial ordered relation defined by (i) (rt,r2)<~(shs2) iff rt ~
(ii) (st,s2) = (0, 0) whenever st = 0 or s2 = 0. J: The vector lattice I []I. When we speak of an L-fuzzy subset we mean that the associated membership function takes its values from the lattice L. The fuzzy subset A of X will be denoted by {(x, Ax): x E X}, where Ax is the membership value of the element x. Definition 1.1. A fuzzy space, denoted by XL, is the set of all ordered pairs (x,L), x E X, i.e.
XL = {(x,L): x E X}, where (x,L) is called a fuzzy element of the fuzzy space XL. Definition 1.2. A subset K of the lattice L is said to be M-sublattice of L if it satisfies the following conditions: (i) K contains at least one element more than the zero element of L, (ii) K is closed under arbitrary times of "V" (supremum) and "A" (infimum) operations of L. Definition 1.3. A fuzzy subspace U of the fuzzy space XL is a collection of ordered pairs (y, Ly), where y E U0 for a given subset U0 of X and Ly is an M-sublattice of L and denoted by f = {(y, Ly): y E U0}, where (y, Ly) is called a fuzzy element of the fuzzy subspace U.
The fuzzy subset A of X may be considered as a subset of the fuzzy subspace U = {O',Ly): y E U0} iffAyELy ifyEUo and A y = 0 if), ff U0. Moreover, any fuzzy subset A defines a fuzzy subspace (called the induced fuzzy subspace by .4)
H(A) = {(x,[O, Ax]): x E A0} Of Xl, where I = [0, 1] and A0 = {x E X: Ax 7L 0} is the support of A. Definition 1.4. A fuzzy Cartesian product of the two fuzzy spaces XL and YK is the fuzzy space
XLY(YK =---(X x Y,L~K) = {((x,y),Lr~K): (x,y) E X x Y}. For the two fuzzy subspaces U = {(x, Lx): x E U0} and V = {(y, Kv): y E II0} of the fuzzy spaces Xt. and YK, respectively, their fuzzy Cartesian product is a fuzzy subspace of XLY(YK which is denoted by
UXV = {((x,y),Lx[]Ky): (x,y) E U0 x Vo}. Definition 1.5. Let XL, YK and Z~¢ be the given nonempty fuzzy spaces. The fuzz), function F F_ from the set of all LnK-fuzzy subsets o f X x Y into the set N z of all N-fuzzy subsets of Z, characterized by the pair (F, fxy), where F : X x Y ---, Z is a function and {fxy}(x,y)EXxY is a family of the comembership functions f x y : L D K --~ N satisfying the following conditions: (i) fxy is nondecreasing on L[]K, (ii) fxy(OL, OK ) = ON and f¢y( IL, Ix) = 1N, (where 0L, 1L, OK, 1K and ON, 1N are the least and the greatest elements of L, K and N, respectively) such that the image of any L oK-fuzzy subset C o f X × Y under F_ is the N-fuzzy subset F_(C) of Z defined by
I F (C)z =
V (x,y)EF-'(:)
[ ou
fxv(C(x,y))
i f F - ' ( z ) ~ O,
"
ifF-l(z) = 0
for every z E Z. For this fuzzy function and for simplicity we shall use the notation F_ = (F, Fxy): X x Y---~Z.
K.A. Dib, N. GalhumlFuzzy Sets and Systems 92 (1997) 103-111
Remark. For simplicity and when no confusion arises we shall use 0 instead of OL and 1 instead of 1L, for any lattice L. Theorem 1.1. Every fuzzv function F_.=(F, f~y): Xx
Y---~Z with onto comembership functions fxy : L~K N, defines a function, denoted again by F_.,from the fuzz), space XL S YK tO ZN and acts as F__((x,y), L []K) = (F(x, y), fxy(L nK))
=_(xFy, N). Let U = {(x, Lx): x E U0} and V = {(y, Kr): y E V0} be fuzzy subspaces of the fuzzy spaces XL and Yx, respectively. Then the fuzzy function F = (F, fxy): X × Y--~ Z acts on the fuzzy elements of the fuzzy subspace U2 V as follows: E((x, y), Lx OKy) = (F(x, y), fxy(Lx []Ky) ) =_--(xFy, fxy(Lx c?Ky)). I f fxy( Lx r~Ky ) is an M-sublattice of N for all (x, y) E Uo × Vo, then F_ translates the fuzzy subspace UXV into a fuzzy subspace Of ZN and we have: Theorem 1.2. Let U={(x, Lx): xEUo}, V={(y, Ky): y E V0} and W= {(z,N:): z E W0} be fuzzy subspaces
of XL, YK and Zu, respectively. I f F__= (F, f~y) : X x Y --~ Z is a fuzzy function with onto comembership functions fxy : LnK ~ N, then we have the following: (1) F F_ translates UY(V to F(UXV) where F(UgV) = {(xFy, fxy(LxnK,)): (x,y) E Uo × V0} which is a fuzzy subspace Of ZN iff fxy(Lx o K v) = fu~,(LurTK~.) for all x, u E X and y, v E Y such that xFy = uFv. (2) F_. translates UY~V to the fuzzy subspace W
105
Definition 1.6. A fuzzy binary operation F = (F, f~, ) on the fuzzy space XL is a fuzzy function from X × X to X such that the comembership functions fx)," LraL --~ L satisfy: (i) f~y(r,s) = 0 iffr = 0 ors = O, (ii) the functions f~:, are onto for all (x, y ) E X × X . Definition 1.7. A fuzzy space XL with the fuzzy binary operation F_.on XL is said to be afuz-y groupoid and denoted by (XL,F_). If F is associative on XL, then (XL,F) is called a fuzzy semigroup. Moreover, if the comembership functions fxy are identical, i.e. fxy = f for all (x, y) E X x X, then (XL, F) is called a uniform fuzzy (groupoid) semigroup. Theorem 1.3. To
each fuz-r (groupoid) semigroup (XL,F) there is an associated ordinary (groupoid) semigroup (X,F) which is isomorphic to the fuzzy (groupoid) semigroup (XL,F) by the correspondence x ~ (x,L).
Definition 1.8. A fuzzy subspace U of the fuzzy groupoid (XL, F_) is called a fuz,y subgroupoid if U is closed under F__and denoted by (U, F_.). Theorem 1.4. The fuzzy subspace U = {(x, Lx): x E
Uo} of the fuzzy (9roupoid) semigroup (XL, F) is a fuzzy (subgroupoid) subsemigroup of (XL, F) iff (i) (Uo,F) is a (subgroupoM) subsemigroup of the ordinary (groupoid) semigroup (X,F), (ii) fxy(LxDLy) = LxFy for all x, y E Uo. When L=I, Theorem 1.4 can be reformulated for the induced fuzzy subspace H(A) by the fuzzy subset A of X, in the following form:
iff
Theorem 1.5. A fuzzy subspace H(A) of the fuzzy (groupoid) semioroup (Xb F__)is afuzzy (subgroupoid)
(i) F(Uo x Vo) C Wo (ii) fxy(LxnKv) = NxF),,for all (x,y) E Uo × Vo. (3) I f A, B and C are fuzzy subsets of)(, Y and Z, respectively, then F__translates H (A ) Y(H(B) to H ( C )
subsemigroup iff (i) fAg,F) is a ( subgroupoid) subsemigroup of the ordinary (groupoid) semiorou p (X,F), (ii) fxy(Ax, Ay) = A(xFy), for all x,y E Ao.
iff (i) F(Ao × Bo) C Co (ii) fxv(Ax, By)=C(xFy), for all (x,y)EAo × Bo.
For more details about fuzzy subgroupoids and fuzzy subgroups see [2].
106
K.A. Dib, N. Galhum/Fuz_-y Sets and Systems 92 (1997) 103-111
2. Fuzzy ideals
(ii) for every xEX, y E Uo we have
f~-,.(L~Lv) = Lx15. and
f.rx(L,.mL) = LyFx.
Let U = {O',Ly): y E U0} be a fuzzy subspace of the fuzzy semigroup (XL,F), where XL = {(x,L): x E X} and F__ = (F,f~y). We like to point out that the proof of the theorems are carried out for the fuzzy left ideals, and the case of fuzzy right ideals is similar.
Remark 2.1. Although in the classical fuzzy case, any fuzzy (left, right) ideal is fuzzy subsemigroup, it is not true in our case.
Definition 2.1. A fuzzy subspace U of the fuzzy semigroup (X/., F_) is said to befit=.-), left (right) ideal if for each (x,L) E XL and (y, Ly) E U, we have
Examples 2.1.
(x,L)F(y,L:) E U
((y, Ly)F(x,L) E U).
If U is both fuzzy left and right ideal, then it is called a fu='_y ideal of the fuzzy semigroup (At,_F.).
In the following we give two counterexamples:
(1) Let (X,F) be a given ordinary semigroup. Define F = (F,f~:v) he a uniform fuzzy binary operation on the fuzzy space )(1, where fvv = " . " is the ordinary product on I. Consider the following fuzzy subspace:
u = {(x,[o, 31): xEX}.
Theorem 2.1. A fuzzy subspace U of the fuzzy semigroup (Xt,F) is a ]kz~v left (right) ideal iff (i) (Uo, F) is a left (right) ideal of the ordinary semigroup (X, F), (ii)Jbr allx E X, y E Uo we have
Then we have (i) (U,..F_F)is a fuzzy ideal of the fuzzy semigroup (Xb F), since (X,F) is an ideal and for each x, y E X , we have
fxy(LaLy) = LxF,.
fv.v(laL.v) = fvv(l a [0, 3])
(fvx(LymL) = LyFx).
Proof. Let us consider the case when U is a fuzzy left ideal of the fuzzy semigroup (Xz, F). This is equivalent to:
= [O, f,:,.(1, ½)] = [o, ½] = L~F,,.
for every (x,L) E XL, (y, Ly) E U, we have
Also
(x,L)F__(y,L;,) E U.
L,,.(z,:, []1) = L..~([o, ½1al)
Therefore
( ( xF)', f~-v(L ~Ly ) ) =--(z, Lz ) ) E U which is equivalent to (a) x F y = z E U o , xEX, yEUo, (b) fxy(LC~L~,) = L: = LxFy. Condition (a) means that (U0, F ) is a left ideal of the semigroup (X,F) and (b) is (ii). [] By using Theorem 2.1, we can directly prove the following theorem: Theorem 2.2. A fuzzy subspace U of the fuzzy semigroup (XL,F) is a fuzzy ideal iff (i) (Uo,F) is an ideal of the ordinary semigroup (X, F),
= [,O f.,,,.( ~,i 1 )] = [0, ½l : L.,,F.,.. (ii) (U, F ) is not a fuzzy subsemigroup of (Xt, F), since for every (x, [0,½]), (y, [0,3]) E U, we get (x, [O, ½])F( y, [0, 31) = (xFy, [O, f,-,,(}, ½)]) = (x8', [o, ¼1) ¢ u, which means that F is not closed on U. (2) Let X = {a,b,c} and F be the binary operation defined as in Table 1. Then (X,F) is a semigroup. (i) The subset U0 = {e} is an ideal of (X,F).
K.A. Dib, N. GalhumlFuzzy Sets and Systems 92 (1997) 103-111 Table 1 F
a
b
a
a
b
c c
b
C
C
C
c
c
c
c
fxy(r,s) =
x = y = c, otherwise,
then it is easy to see that (X/, F), where E = (F, fxy), is a fuzzy semigroup. (iii) Let U = {(c, [0, ~])}. Hence (a) (U,F_) is a fuzzy left ideal of (X/,F.), for
fac(ImLc)= fac(l[][O,~]) = [o, L,,(1, =
[ 0 , 11 =
(*)
f~y(L[]Ly) = fxy(LxC3Ly).
(X,V),
r.s Max(r,s, ~)' r • s,
Proof. Firstly, suppose that U is a fuzzy left ideal of the fuzzy semigroup (XL,F) which satisfies for all x, y E U0, that
Then, by Theorem 2.1, we have: (i) (U0,F) is a left ideal of the ordinary semigroup
(ii) If for any x, y E X ,
I
107
~)1
(ii) f~v(LrqLy) = Lxry for all xE X, )'E Uo. We can see that (i) implies (Uo, F) is an ordinary subsemigroup of (X, F) and, by (i i) and (.), we have
f~y(Lx []Ly) = Lxo,
for all x, y E U0,
which is equivalent to, by Theorem 1.4, (U,__F) is a fuzzy subsemigroup of (XL, F__). Conversely, if U is a fuzzy left ideal which is a fuzzy subsemigroup of (XL, F). Then for every x, y E U0, we have
f~y(Lt~Ly) = LxFy = f~y(LxoL, ).
[]
Larc = Lc,
fbc(IC~Lc)= fbc(IC~[O,~]) =[O, fbc(1,~)]
Corollary 2.1. A fuzz}, ideal U of the fuz.Lv semi-
group (XL, F_) is a fuzzy semigroup iff the comembership functions fry satisfy one of the following conditions: for all x, y E Uo
= [o, ~] = L~c = L~,
f..(r~[]Lc) = f..(l [] [0, ~]) = [0, fcc(1, ~)1 = [0, ~1 = L,.Fc = L¢.
(b) But (U,F) is not a fuzzy subsemigroup of
fxy(L[]Ly) = f~y(Lx reLy) or fvx(Ly DL) = fyx(Ly oLx). 2. I. Fuzzy ideals induced by a fuzzy subset
0(1, F), since
(e, [o, ~ ])_e_(e, [o, -~]) = (cFc, f..([O, ~1 [] [0, ~1)) = (cFc,[O, fcc(~,-~)])
= ( c , [ O , 8 ] ) ¢ u. Theorem 2.3. A fuzzy left (right) ideal U of the
fuzzy semigroup (XL, G_F_)is a fuzzy subsemigroup iff for all x, y E Uo, we have
Let us recall that, if (XI,F) is a fuzzy semigroup and A is fuzzy subset of X. Then the fuzzy subspace induced by A is the set H ( A ) = {(x,[O, Ax]): xEao},
where A0 = {xEX: A x ¢ 0} is the support of A: For this special fuzzy subspace H(A), we may reformulate Theoremff 2.1-2.3 in the following forms: Theorem 2.1a. A fuzzy subspace H(A ) of the fuzzy
f~y(L []Ly ) = fxy(Lx []Ly) (fyx(Ly []L) = fvx(Ly mLx)).
semigroup (Xt,E) is a fuzzy left (right) ideal iff (i) (Ao,F) is a left (right) ideal of the ordinary semigroup (X, F),
K.A. Dib, N. Galhum/Fuz.2r Sets and Systems 92 (1997) 103-111
108
(ii) for allxEX,
y E
f~.r( 1, Ay) = A(xFy)
Ao, we have (fvx(A)', 1) = A(yFx)).
Theorem 2.2a. A fuzzy subspace H(A) of the fuzzy semigroup (X1,17) is a J'uz(v ideal iff (i) (Ao,F) is an ideal of the ordinary semigroup (X,F), (ii) for all xEX, y E no, we have
f~.,,(1, Ay) = A(xFy)
Table 2
and
F
a
b
a
a
b
c a
b
a
b
a
c
a
b
c
ideal of the uniform fuzzy semigroup (X/,17) in our sense as we see in the following:
fyx(Ay, 1 ) = A(yFx).
Example 2.2. Let (X/, 17), F = (F, f ) be a uniform Theorem 2.3a. Let H(A ) be a fuz-v left (right) ideal of the fuzzy semigroup (Xt, 17). Then H(A ) is a fuzzy subsemigroup of(X1,17) ifffor all x, y EAo, we have
fuzzy semigroup, where (X.F) is the semigroup as in Table 2 and f(r,s) = r As. Consider the fuzzy subset A of X defined by
A(b) = O,
A(e) = !4"
f~:,,( 1, Ay) = f~y(Ax, Ay),
A(a) = ½,
(f~.x(Ay, l) = f,.,.(Ay, Ax)).
Then we can show that (i) A is a classical fuzzy left ideal of (X,F), (ii) the support A0 = {a,c} of A is a left ideal of (X,F), (iii) although the support A0 of A is a left ideal of (X,F), we see that the induced fuzzy subspace
2.2. The relationship between fuzzy ideals and classical fuzzy ideals Let (XI, F ) be a uniform fuzzy semigroup and 17 = (F, "A"), where F is a binary operation on X and "A" is the minimum function from the vector lattice J = I o I into I.
Theorem 2.4. Ever), fuzzy subset A of X which induced a fuzzy (left, right) ideal ( H ( A ) , F ) of the uniform fuzzy semigroup (X/, 17) is a classical fuzzy (left, right) ideal of the ordinary semigroup (X,F). Proof. Suppose that A is a fuzzy subset of X such that ( H ( A ) , F ) is the induced fuzzy left ideal by A of the uniform fuzzy semigroup (Xt, 17), where 17 = (F,"A"). By Theorem 2.1a(ii), we have
A(xFy)=Ay
f o r a l l x E X , yEA0.
Since Ay = 0 for each y E X - A 0 , then
A(xFy)>_.Ay for all x, yEX, which means that A is a classical fuzzy left ideal of
(X,F).
[]
Remark 2.2. The converse of Theorem 2.4, in general, is not true, i.e. there is a classical fuzzy (left, right) ideal A o f a semigroup (X,F) such that a fuzzy subspace H(A) induced by A is not a fuzzy (left, right)
/-/(.,1) = {(a,[0, ½1), (c,[0, ¼])} is not a fuzzy left ideal of (Xt, 17) in our sense, because
A c ( l , A c ) = f.c(l, ¼) = ,
# A(aFc) = A(a) = ½. Theorem 2.5. If (Y,F) is an ordinary (left, right) ideal of the semigroup (X,F). Then for each fuzzy subset A of X with support Ao = Y, there is a fuzzy semioroup (XI, G_.)such that the induced fuzzy subspace H(A) is a fuzzy (left, right) ideal of (X/, G). Proof. Let (Y,F) be a left ideal of the semigroup (X,F) and A be a fuzzy subset of X such that A0 = Y. It is clear that (Ao,F) is a left ideal of (X,F). Now we define a fuzzy binary operation _.G = (G, #xy) o n X as follows: Take G = F and 9xy(r,s) = hxy(r A s), where
hxy(k) = { i(xF'~')k, k < Ay, Ay 1 - a(xFy) f--_--~y ] ( k - A y ) + A ( x F y ) ,
k>/Ay
K.A. Dib. N. GalhumlFuzzy Sets and Systems 92 (1997) 103-11l
for a l l x EX, v EA0. And h.,?.(k) = k i f x E X, y f{ Ao. It is clear that gxy(r,s), x, y E X are continuous functions on J and 9.~v(r,s) = 0 iff r = 0 or s = 0. Then it is easy to show that (Xt, G ) is a fuzzy semigroup. If H(A ) is the induced fuzzy subspace by A of the fuzzy semigroup (Xt,_G_G),then from the definition of g.~v, we have for all xEX, yEA•,
9.,~,,(1, Ay) = h.,;,,(Ay) = A(xFy), which implies (H(A),G__) is a fuzzy left ideal of (X~, ¢7). [] Corollary 2.2. Ever), classical fuz=y (left, right) ideal of the semigroup (X,F) induces a fuzzy (left, right) ideal relative to some.fuzzy semigroup (Xt, G_G_).
3. Fuzzy bi-ideal Definition 3.1. A fuzzy subsemigroup (U,F__) of the fuzzy semigroup (XL,F ) is called a fuzzy bi-ideal if for every (y, Ly), (z,L.)E U and (x,L)EXL we have
[(y, Ly ) F (x,L ) ] F (z, Lz ) = ( y , Ly)IV_[(x,L)F__(z,L.)]E U.
Theorem 3.1. A fuzzy subsemioroup (U,F__) of the fuzzy semigroup (XL, F) is a fuzzy bi-ideal iff (i) ( Uo,F) is a hi-ideal of the ordinary semioroup (X,F), (ii) for all y, z E U• and x EX, we have
Table 3
F a
a a
b b
b
a
b
¢e~ ((yFx)Fz, fo,F.,r.(£..~.(Ly~L )[]L: )) = (yF(xFz), .~,~xF:)(LytZf,:(L ~zL:)) =--(w,L,,,) E U ¢:~ (i) (yFx)Fz = yF(xFz) = wE Uo, y, z E Uo, xEX, (ii) f ( y V x ) z ( £ , x ( L y [] L) []L: ) = .~'L~Fz~(Ly[]f~: (L[]L:))= L.. ¢~ (i) (Uo,F) is a bi-ideal of the ordinary semigroup (X,F), (ii) LyFxF,. = f~yFx~:(~,x(Lv []L) []Lz ) = f.txF:)(Ly[]fx:(L[]L_-)). [] Remark 3.1. Although in the classical fuzzy case, every fuzzy (left, right) ideal is a fuzzy bi-ideal, in our sense there are fuzzy (left, right) ideals which are not a fuzzy bi-ideals as we see in the following:
Example 3.1. Let (X, F ) be the semigroup defined by Table 3 Then:
(I) It is easy to see that the subset U• = {a} is a left ideal and hence its a bi-ideal of (X, F). (2) Consider (XI,__F) is a fuzzy semigroup, where F_. = (F, fxy), and
LyFxFz = f ( ) . F x ) z ( f , x ( L y nL ) []L=) = f.~xFz)(Ly []f~z(L ~sL:)). Proof. Suppose that ( U , F ) is a fuzzy bi-ideal of the fuzzy semigroup (XL,F). Then for all (y,L.~.), (z, Lz ) E U and (x, L) E XL, we have
109
½rs f~y(r,s) =
1 - ½(r + s - r s ) '
x # y,
rAs,
x=y.
Then we have the following:
(i) For U = {(a, [0, t]): a E Uo} where tE(0,1], we have
[(y, L;, )F(x, L )IF(z, L: ) Ao(t,o [] La) = Aa([0, t] ~ [o, t])
= (y, Ly)F[(x,L)F(z, Lz)] E U = [O, faa(t,t)] = [0, t] = La ¢~, [(yFx ), ~.x(L y []L )]iv(z, Lz )
= (y, Ly)F[(xFz, fxz(LoL,.)] E U
and hence (U, F) is a fuzzy subsemigroup o f the fuzzy
semigroup (X, F).
K.A. Dib, N. GalhumlFuz~v Sets and Systems 92 (1997) 103-111
110 (ii) Since
f,,,(l ~La)
=
[0, .faa(l, t)]
= [0, t]
=
LaFa =
La
Then (U.__F) is not a fuzzy bi-ideal although it is a fuzzy left ideal of 0(/, F__). In the following, Theorems 3.2-3.4, we discuss the sufficient conditions for the fuzzy (left, right) ideal to be a fuzzy bi-ideals, respectively.
and
fba( l []La ) = [0, .fba(1, t)] = [0, t] = LbFa = La. Then, from (1) and Theorem 2.1, we see that ( U , F ) is a fuzzy left ideal. (iii) Now, we show that U is not a fuzzy bi-ideal, for
La = LaFaFa= f(aFa)a(faa(La ~I) []La) = faa([0, .f~a(t, 1 )] [] La)
= faa(La :~La) = [0, faa(t, t)] = [o, t] = Lo = fa(aFa)(La
nfo~(I pLy))
Theorem 3.2. Ever), fuzz), left ideal U of the fuzz), semigroup (XL,F__) is a fuzzy bi-ideal if for all y, z E Uo and x E X, we have (1) f.:(Ly ~L:) = LyF: (2) LyFxF: = f~yFx)z(f.x(Ly nL) []L.. ). Proof. Suppose that U is a fuzzy left ideal o f the fuzzy semigroup (XL, F). Then, from Theorem 2.1, we have (i) (U0,__F) is a left ideal of the ordinary semigroup
(X,F) which implies (Uo, F) is a bi-ideal o f ( X , F ) , (ii) for all x E X and z E Uo, we have
fxz(L[]L:) = LxF:. Since for a n y x E X and y,z E U0, we havexFz E Uo and yF(xFz) E Uo. Hence, by condition (1), we get
= .foo(L~ [] [0, faa( 1, t)])
£,(xFz)(Ly t3LxFz) = LyFxFz.
=f~o(LarJLa) = [O, f~o(1, t)] = L,,.
From (ii) and condition (2), we get
But
LyFxrz = f,(xFz)(Ly []f~-_(L []L=))
La = LaFara ~= Aarb)a(.fa6(La DI)DLa) = fb~([O, f~b(t, 1 )] []La)
= fao(La [] L.) = [0, faa(t, t)] =
[ ':1 0,2_2t+t
2 •
Although
fa(bF,,~(L, ~3fha(I []La )) = f,a(La [] [0, .fba( 1, t)]) = .L~(L,, n [0, t])
= [O, faa(t,t)]
= J~yFx):(fvx(Lv []L) rTL.) and by (i), we see that U is a fuzzy bi-ideal o f (XL, g ) . [] Theorem 3.3. Every fuzz), right ideal of the fuzzy semigroup (XL,F) is a fuzz)' bi-ideal if for all y, z E Uo and x E 2(, we have (i) f,:(L,, t~L-_) = LvF.., (ii) L~,rxr= = f,,~xr:)(L~, [] fx..(L taL:)). Proof. The proof is the same as for fuzzy left ideal. The combined effect of Theorems 3.2 and 3.3 is as follows:
= [0, t] = La, i.e.
La = LaFhFa = fa(hFa)(La []fba(l nLa) ) # f{aFb)a(fab(La []l)[]La).
Theorem 3.4. Every fuzzy ideal U of the fuzzy semigroup (XL,F) is a fuzzy bi-ideal if for all y,z E Uo, we have
f.z(Ly nL:) = L vF=.
[]
K.A. Dib. N. GalhumlFuzzy Sets and S.vstems 92 (1997) 103-111
Concerning the induced fuzzy subspace H ( A ) by the fuzzy subset A of X, Theorem 3.1 can be reformulated as follows:
111
Remark 3.2, The converse of the above theorem is not true, generally, as we see in the following:
Example 3.2. Let (X/, F ) be the uniform fuzzy semiTheorem 3,5. I f H ( A ) is a fuzzy subsemigroup o f the fi~zz), semigroup (XI, F) then H ( A ) is a fuzzy hi-ideal (i) (A0,F) is a bi-ideal of the ordinary semigroup (X,F), (ii) ,]'or all y,z E Ao a n d x E X, we have
A(yFxFz) = ~.,.Fxlz(f,x(Ay, 1),Az)
= f.,,~xF:~(AY,L : ( 1 , A z ) ) . In the following theorem, we give the relationship between fuzzy bi-ideals and classical fuzzy biideals:
Theorem 3.6. Let (XI, F_.) be a uniform fuzzy semigroup, where F = (F,A) and "A" the minimum /'unction from J hzto I. Then every fuzz), subset A of X which induces a fuzz)' hi-ideal ( H ( A ) , F ) of (XI, F__)is a classical fuzzy hi-ideal o f the semigroup (X,F). Proof. Suppose that A is a fuzzy subset of X such that (H(A), F_.) is a fuzzy bi-ideal o f the uniform fuzzy semigroup (XI,F_.), where F_ = ( F , " A " ) . Then, by Theorem 3.5, we have (i) (A0,F) is a bi-ideal o f the ordinary semigroup
(X,F), (ii) for all y,z E Ao and x E X, we have
(,)
A(yFxFz) = (A(y) A 1 ) A A(z) = A ( y ) AA(z).
Now, if y,z E Ao, then A ( y ) = A(z) = 0 and hence, by ( , ) , we get
A(yFxFz)>~A(y) AA(z),
for all x , y , z E X,
which means that A is a classical fuzzy bi-ideai o f
(X,F).
[]
group defined in Example 2.2, where X : {a, b, c}. If we define a fuzzy subset A o f X as A ( a ) = -J 4 ~
A ( b ) = -,~
A ( c ) = 0.
Then we can show that: (i) A is a classical fuzzy bi-ideal of (X, F ) (ii) the support A0 = {a,b} o f A is a bi-ideal o f
(X,F) (iii) although the support A0 o f A is a bi-ideal o f
(X,F) we see that the induced fuzzy subspace H ( A ) = {(a,[0, ¼]), (b,[0, ½])} by A is not a fuzzy bi-ideal in our sense, since I A(aFaFb) = A(b) = ~I ¢ A(a) A A(b) = -~.
References [1] J.M. Anthony and H. Sherwood, Fuzzy group redefined, J. Math. Anal Appl. 69 (1979) 124-130. [2] K.A. Dib, On fuzzy spaces and fuzzy group theory, 1nform. Sci. 80 (1994) 253-282. [3] K.A. Dib and N. Youssef, Fuzzy Cartesian product, fuzzy relations and fuzzy functions, Fuz:y Sets and S),stems 41 (1991) 299-315. [4] N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Paul. 28 (1980) 17-21. [5] N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuz:y Sets and Systems 5 (1981) 203-215. [6] H,V. Kumbhojkar and M.S. Bapat, Correspondence theorem for fuzzy ideals, Fuz:y Sets and Systems 41 (1991) 213-219. [7] W. Liu, Fuzzy invariant subgroups and fuzzy ideals. Fuz:y Sets and Systems 8 (1982) 133-139. [8] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to System Analysis (Wiley, New York, 1975). [9] A. Rosenfeld, Fuzzy groups, ,/. Math, Anal. AppL 35 (1971) 512-517. [10] S. Sessa, Fuzzy subgroups and fuzzy ideals, under triangular norm, Fuz:y Sets and Systems 13 (1984) 95-100. [11] L.A. Zadeh, Fuzzy sets, b![brm, and Control 8 (1965) 338-353.
sets and systems ELSEVIER
Fuzzy Sets and Systems 92 (1997) 103-111
Fuzzy ideals and fuzzy bi-ideals in fuzzy semigroups K . A . Dib*, N. G a l h u m Department o[ Mathematics, Faculty of Science, Fayoum Branch. Cairo Unit~ersity. Fayoum. Et,typt
Received January 1994; revised June 1996
Abstract
The purpose of this paper is to introduce some basic concepts of fuzzy algebra such as fuzzy (left, right) ideal and fuzzy bi-ideal in fuzzy semigroup, through the new approach of fuzzy space and fuzzy group introduced by Dib (1994). Our notion of fuzzy ideal and fuzzy bi-ideal includes the (classical) concepts of fuzzy ideal and fuzzy bi-ideal of ordinary semigroup. Many counterexamples are also given. © 1997 Published by Elsevier Science B.V. Keywords: Fuzzy space; Fuzzy semigroup; Fuzzy ideal; Fuzzy bi-ideal
O. Introduction
The study of the fuzzy algebraic structures has started with the introduction of the concepts of fuzzy (subgroupoids) subgroups and fuzzy (left, fight) ideals in the pioneering paper of Rosenfeld [9]. In 1975, Negoita and Ralescu [8] considered a generalization of Rosenfeld's definition where the unit interval [0, 1] is replaced by an appropriate lattice structure. In 1979, Anthony and Sherwood [1] redefined fuzzy (subgroupoids) subgroups using the concept of triangular norm. Kuroki [4, 5] introduced and studied fuzzy (left, right) ideals and fuzzy biideals in semigroups. Several mathematicians [6-8, 10] have followed the Rosenfeld approach in investigating fuzzy algebra where a given ordinary algebraic structure on a given set X is assumed, then introducing the fuzzy algebraic structure as a fuzzy subset A of X satisfying some suitable conditions.
* Correspondingauthor.
In this paper, any fuzzy algebraic structure in the Rosenfeld approach is called a classical fuzzy structure. In 1991, new concepts of fuzzy Cartesian product, fuzzy relation, fuzzy function and a fuzzy binary operation on a set have been introduced [3]. As we know, the concept of universal set played an essential role in ordinary mathematics, e.g. any algebraic system is a universal set with one or more binary operations and the metric space is a universal set, having distance, etc. Therefore, in the absence of the concept of fuzzy universal set, the formulation of a concrete definition of any fuzzy structure is not evident. Recently, the concept of a fuzzy space is introduced in [2] as a replacement for the concept of universal set in the ordinary case. Moreover, a fuzzy subspace, fuzzy binary operation on a fuzzy space and fuzzy (subgroup) group were introduced also in [2]. In the present paper, we introduce some basic concepts of fuzzy algebra such as fuzzy (left, fight) ideals and fuzzy bi-ideals in fuzzy semigroup, through the new approach of fuzzy spaces and fuzzy groups introduced in [2]. A relation between the introduced fuzzy (left, right) ideals or fuzzy bi-ideals and the classical
0165-0114/97/$17.00 (~) 1997 Published by ElsevierScience B.V. All rights reserved PHS0165-0114(96)00170-4
104
K.A. Dib, N. GalhumlFuzzy Sets and Systems 92 (1997) 103-111
ones is given. Moreover, many examples are given to illustrate the difference between our point of view and the classical case.
1. Definitions, notations and preliminaries Throughout the paper we adopt, the following notations, definitions and some results of [2, 3]: I: The lattice [0,1] with the usual order of real numbers. L, K, N,...: Arbitrary lattices having a least element and a greatest element. L [] K: The vector lattice L x K with the partial ordered relation defined by (i) (rt,r2)<~(shs2) iff rt ~
(ii) (st,s2) = (0, 0) whenever st = 0 or s2 = 0. J: The vector lattice I []I. When we speak of an L-fuzzy subset we mean that the associated membership function takes its values from the lattice L. The fuzzy subset A of X will be denoted by {(x, Ax): x E X}, where Ax is the membership value of the element x. Definition 1.1. A fuzzy space, denoted by XL, is the set of all ordered pairs (x,L), x E X, i.e.
XL = {(x,L): x E X}, where (x,L) is called a fuzzy element of the fuzzy space XL. Definition 1.2. A subset K of the lattice L is said to be M-sublattice of L if it satisfies the following conditions: (i) K contains at least one element more than the zero element of L, (ii) K is closed under arbitrary times of "V" (supremum) and "A" (infimum) operations of L. Definition 1.3. A fuzzy subspace U of the fuzzy space XL is a collection of ordered pairs (y, Ly), where y E U0 for a given subset U0 of X and Ly is an M-sublattice of L and denoted by f = {(y, Ly): y E U0}, where (y, Ly) is called a fuzzy element of the fuzzy subspace U.
The fuzzy subset A of X may be considered as a subset of the fuzzy subspace U = {O',Ly): y E U0} iffAyELy ifyEUo and A y = 0 if), ff U0. Moreover, any fuzzy subset A defines a fuzzy subspace (called the induced fuzzy subspace by .4)
H(A) = {(x,[O, Ax]): x E A0} Of Xl, where I = [0, 1] and A0 = {x E X: Ax 7L 0} is the support of A. Definition 1.4. A fuzzy Cartesian product of the two fuzzy spaces XL and YK is the fuzzy space
XLY(YK =---(X x Y,L~K) = {((x,y),Lr~K): (x,y) E X x Y}. For the two fuzzy subspaces U = {(x, Lx): x E U0} and V = {(y, Kv): y E II0} of the fuzzy spaces Xt. and YK, respectively, their fuzzy Cartesian product is a fuzzy subspace of XLY(YK which is denoted by
UXV = {((x,y),Lx[]Ky): (x,y) E U0 x Vo}. Definition 1.5. Let XL, YK and Z~¢ be the given nonempty fuzzy spaces. The fuzz), function F F_ from the set of all LnK-fuzzy subsets o f X x Y into the set N z of all N-fuzzy subsets of Z, characterized by the pair (F, fxy), where F : X x Y ---, Z is a function and {fxy}(x,y)EXxY is a family of the comembership functions f x y : L D K --~ N satisfying the following conditions: (i) fxy is nondecreasing on L[]K, (ii) fxy(OL, OK ) = ON and f¢y( IL, Ix) = 1N, (where 0L, 1L, OK, 1K and ON, 1N are the least and the greatest elements of L, K and N, respectively) such that the image of any L oK-fuzzy subset C o f X × Y under F_ is the N-fuzzy subset F_(C) of Z defined by
I F (C)z =
V (x,y)EF-'(:)
[ ou
fxv(C(x,y))
i f F - ' ( z ) ~ O,
"
ifF-l(z) = 0
for every z E Z. For this fuzzy function and for simplicity we shall use the notation F_ = (F, Fxy): X x Y---~Z.
K.A. Dib, N. GalhumlFuzzy Sets and Systems 92 (1997) 103-111
Remark. For simplicity and when no confusion arises we shall use 0 instead of OL and 1 instead of 1L, for any lattice L. Theorem 1.1. Every fuzzv function F_.=(F, f~y): Xx
Y---~Z with onto comembership functions fxy : L~K N, defines a function, denoted again by F_.,from the fuzz), space XL S YK tO ZN and acts as F__((x,y), L []K) = (F(x, y), fxy(L nK))
=_(xFy, N). Let U = {(x, Lx): x E U0} and V = {(y, Kr): y E V0} be fuzzy subspaces of the fuzzy spaces XL and Yx, respectively. Then the fuzzy function F = (F, fxy): X × Y--~ Z acts on the fuzzy elements of the fuzzy subspace U2 V as follows: E((x, y), Lx OKy) = (F(x, y), fxy(Lx []Ky) ) =_--(xFy, fxy(Lx c?Ky)). I f fxy( Lx r~Ky ) is an M-sublattice of N for all (x, y) E Uo × Vo, then F_ translates the fuzzy subspace UXV into a fuzzy subspace Of ZN and we have: Theorem 1.2. Let U={(x, Lx): xEUo}, V={(y, Ky): y E V0} and W= {(z,N:): z E W0} be fuzzy subspaces
of XL, YK and Zu, respectively. I f F__= (F, f~y) : X x Y --~ Z is a fuzzy function with onto comembership functions fxy : LnK ~ N, then we have the following: (1) F F_ translates UY(V to F(UXV) where F(UgV) = {(xFy, fxy(LxnK,)): (x,y) E Uo × V0} which is a fuzzy subspace Of ZN iff fxy(Lx o K v) = fu~,(LurTK~.) for all x, u E X and y, v E Y such that xFy = uFv. (2) F_. translates UY~V to the fuzzy subspace W
105
Definition 1.6. A fuzzy binary operation F = (F, f~, ) on the fuzzy space XL is a fuzzy function from X × X to X such that the comembership functions fx)," LraL --~ L satisfy: (i) f~y(r,s) = 0 iffr = 0 ors = O, (ii) the functions f~:, are onto for all (x, y ) E X × X . Definition 1.7. A fuzzy space XL with the fuzzy binary operation F_.on XL is said to be afuz-y groupoid and denoted by (XL,F_). If F is associative on XL, then (XL,F) is called a fuzzy semigroup. Moreover, if the comembership functions fxy are identical, i.e. fxy = f for all (x, y) E X x X, then (XL, F) is called a uniform fuzzy (groupoid) semigroup. Theorem 1.3. To
each fuz-r (groupoid) semigroup (XL,F) there is an associated ordinary (groupoid) semigroup (X,F) which is isomorphic to the fuzzy (groupoid) semigroup (XL,F) by the correspondence x ~ (x,L).
Definition 1.8. A fuzzy subspace U of the fuzzy groupoid (XL, F_) is called a fuz,y subgroupoid if U is closed under F__and denoted by (U, F_.). Theorem 1.4. The fuzzy subspace U = {(x, Lx): x E
Uo} of the fuzzy (9roupoid) semigroup (XL, F) is a fuzzy (subgroupoid) subsemigroup of (XL, F) iff (i) (Uo,F) is a (subgroupoM) subsemigroup of the ordinary (groupoid) semigroup (X,F), (ii) fxy(LxDLy) = LxFy for all x, y E Uo. When L=I, Theorem 1.4 can be reformulated for the induced fuzzy subspace H(A) by the fuzzy subset A of X, in the following form:
iff
Theorem 1.5. A fuzzy subspace H(A) of the fuzzy (groupoid) semioroup (Xb F__)is afuzzy (subgroupoid)
(i) F(Uo x Vo) C Wo (ii) fxy(LxnKv) = NxF),,for all (x,y) E Uo × Vo. (3) I f A, B and C are fuzzy subsets of)(, Y and Z, respectively, then F__translates H (A ) Y(H(B) to H ( C )
subsemigroup iff (i) fAg,F) is a ( subgroupoid) subsemigroup of the ordinary (groupoid) semiorou p (X,F), (ii) fxy(Ax, Ay) = A(xFy), for all x,y E Ao.
iff (i) F(Ao × Bo) C Co (ii) fxv(Ax, By)=C(xFy), for all (x,y)EAo × Bo.
For more details about fuzzy subgroupoids and fuzzy subgroups see [2].
106
K.A. Dib, N. Galhum/Fuz_-y Sets and Systems 92 (1997) 103-111
2. Fuzzy ideals
(ii) for every xEX, y E Uo we have
f~-,.(L~Lv) = Lx15. and
f.rx(L,.mL) = LyFx.
Let U = {O',Ly): y E U0} be a fuzzy subspace of the fuzzy semigroup (XL,F), where XL = {(x,L): x E X} and F__ = (F,f~y). We like to point out that the proof of the theorems are carried out for the fuzzy left ideals, and the case of fuzzy right ideals is similar.
Remark 2.1. Although in the classical fuzzy case, any fuzzy (left, right) ideal is fuzzy subsemigroup, it is not true in our case.
Definition 2.1. A fuzzy subspace U of the fuzzy semigroup (X/., F_) is said to befit=.-), left (right) ideal if for each (x,L) E XL and (y, Ly) E U, we have
Examples 2.1.
(x,L)F(y,L:) E U
((y, Ly)F(x,L) E U).
If U is both fuzzy left and right ideal, then it is called a fu='_y ideal of the fuzzy semigroup (At,_F.).
In the following we give two counterexamples:
(1) Let (X,F) be a given ordinary semigroup. Define F = (F,f~:v) he a uniform fuzzy binary operation on the fuzzy space )(1, where fvv = " . " is the ordinary product on I. Consider the following fuzzy subspace:
u = {(x,[o, 31): xEX}.
Theorem 2.1. A fuzzy subspace U of the fuzzy semigroup (Xt,F) is a ]kz~v left (right) ideal iff (i) (Uo, F) is a left (right) ideal of the ordinary semigroup (X, F), (ii)Jbr allx E X, y E Uo we have
Then we have (i) (U,..F_F)is a fuzzy ideal of the fuzzy semigroup (Xb F), since (X,F) is an ideal and for each x, y E X , we have
fxy(LaLy) = LxF,.
fv.v(laL.v) = fvv(l a [0, 3])
(fvx(LymL) = LyFx).
Proof. Let us consider the case when U is a fuzzy left ideal of the fuzzy semigroup (Xz, F). This is equivalent to:
= [O, f,:,.(1, ½)] = [o, ½] = L~F,,.
for every (x,L) E XL, (y, Ly) E U, we have
Also
(x,L)F__(y,L;,) E U.
L,,.(z,:, []1) = L..~([o, ½1al)
Therefore
( ( xF)', f~-v(L ~Ly ) ) =--(z, Lz ) ) E U which is equivalent to (a) x F y = z E U o , xEX, yEUo, (b) fxy(LC~L~,) = L: = LxFy. Condition (a) means that (U0, F ) is a left ideal of the semigroup (X,F) and (b) is (ii). [] By using Theorem 2.1, we can directly prove the following theorem: Theorem 2.2. A fuzzy subspace U of the fuzzy semigroup (XL,F) is a fuzzy ideal iff (i) (Uo,F) is an ideal of the ordinary semigroup (X, F),
= [,O f.,,,.( ~,i 1 )] = [0, ½l : L.,,F.,.. (ii) (U, F ) is not a fuzzy subsemigroup of (Xt, F), since for every (x, [0,½]), (y, [0,3]) E U, we get (x, [O, ½])F( y, [0, 31) = (xFy, [O, f,-,,(}, ½)]) = (x8', [o, ¼1) ¢ u, which means that F is not closed on U. (2) Let X = {a,b,c} and F be the binary operation defined as in Table 1. Then (X,F) is a semigroup. (i) The subset U0 = {e} is an ideal of (X,F).
K.A. Dib, N. GalhumlFuzzy Sets and Systems 92 (1997) 103-111 Table 1 F
a
b
a
a
b
c c
b
C
C
C
c
c
c
c
fxy(r,s) =
x = y = c, otherwise,
then it is easy to see that (X/, F), where E = (F, fxy), is a fuzzy semigroup. (iii) Let U = {(c, [0, ~])}. Hence (a) (U,F_) is a fuzzy left ideal of (X/,F.), for
fac(ImLc)= fac(l[][O,~]) = [o, L,,(1, =
[ 0 , 11 =
(*)
f~y(L[]Ly) = fxy(LxC3Ly).
(X,V),
r.s Max(r,s, ~)' r • s,
Proof. Firstly, suppose that U is a fuzzy left ideal of the fuzzy semigroup (XL,F) which satisfies for all x, y E U0, that
Then, by Theorem 2.1, we have: (i) (U0,F) is a left ideal of the ordinary semigroup
(ii) If for any x, y E X ,
I
107
~)1
(ii) f~v(LrqLy) = Lxry for all xE X, )'E Uo. We can see that (i) implies (Uo, F) is an ordinary subsemigroup of (X, F) and, by (i i) and (.), we have
f~y(Lx []Ly) = Lxo,
for all x, y E U0,
which is equivalent to, by Theorem 1.4, (U,__F) is a fuzzy subsemigroup of (XL, F__). Conversely, if U is a fuzzy left ideal which is a fuzzy subsemigroup of (XL, F). Then for every x, y E U0, we have
f~y(Lt~Ly) = LxFy = f~y(LxoL, ).
[]
Larc = Lc,
fbc(IC~Lc)= fbc(IC~[O,~]) =[O, fbc(1,~)]
Corollary 2.1. A fuzz}, ideal U of the fuz.Lv semi-
group (XL, F_) is a fuzzy semigroup iff the comembership functions fry satisfy one of the following conditions: for all x, y E Uo
= [o, ~] = L~c = L~,
f..(r~[]Lc) = f..(l [] [0, ~]) = [0, fcc(1, ~)1 = [0, ~1 = L,.Fc = L¢.
(b) But (U,F) is not a fuzzy subsemigroup of
fxy(L[]Ly) = f~y(Lx reLy) or fvx(Ly DL) = fyx(Ly oLx). 2. I. Fuzzy ideals induced by a fuzzy subset
0(1, F), since
(e, [o, ~ ])_e_(e, [o, -~]) = (cFc, f..([O, ~1 [] [0, ~1)) = (cFc,[O, fcc(~,-~)])
= ( c , [ O , 8 ] ) ¢ u. Theorem 2.3. A fuzzy left (right) ideal U of the
fuzzy semigroup (XL, G_F_)is a fuzzy subsemigroup iff for all x, y E Uo, we have
Let us recall that, if (XI,F) is a fuzzy semigroup and A is fuzzy subset of X. Then the fuzzy subspace induced by A is the set H ( A ) = {(x,[O, Ax]): xEao},
where A0 = {xEX: A x ¢ 0} is the support of A: For this special fuzzy subspace H(A), we may reformulate Theoremff 2.1-2.3 in the following forms: Theorem 2.1a. A fuzzy subspace H(A ) of the fuzzy
f~y(L []Ly ) = fxy(Lx []Ly) (fyx(Ly []L) = fvx(Ly mLx)).
semigroup (Xt,E) is a fuzzy left (right) ideal iff (i) (Ao,F) is a left (right) ideal of the ordinary semigroup (X, F),
K.A. Dib, N. Galhum/Fuz.2r Sets and Systems 92 (1997) 103-111
108
(ii) for allxEX,
y E
f~.r( 1, Ay) = A(xFy)
Ao, we have (fvx(A)', 1) = A(yFx)).
Theorem 2.2a. A fuzzy subspace H(A) of the fuzzy semigroup (X1,17) is a J'uz(v ideal iff (i) (Ao,F) is an ideal of the ordinary semigroup (X,F), (ii) for all xEX, y E no, we have
f~.,,(1, Ay) = A(xFy)
Table 2
and
F
a
b
a
a
b
c a
b
a
b
a
c
a
b
c
ideal of the uniform fuzzy semigroup (X/,17) in our sense as we see in the following:
fyx(Ay, 1 ) = A(yFx).
Example 2.2. Let (X/, 17), F = (F, f ) be a uniform Theorem 2.3a. Let H(A ) be a fuz-v left (right) ideal of the fuzzy semigroup (Xt, 17). Then H(A ) is a fuzzy subsemigroup of(X1,17) ifffor all x, y EAo, we have
fuzzy semigroup, where (X.F) is the semigroup as in Table 2 and f(r,s) = r As. Consider the fuzzy subset A of X defined by
A(b) = O,
A(e) = !4"
f~:,,( 1, Ay) = f~y(Ax, Ay),
A(a) = ½,
(f~.x(Ay, l) = f,.,.(Ay, Ax)).
Then we can show that (i) A is a classical fuzzy left ideal of (X,F), (ii) the support A0 = {a,c} of A is a left ideal of (X,F), (iii) although the support A0 of A is a left ideal of (X,F), we see that the induced fuzzy subspace
2.2. The relationship between fuzzy ideals and classical fuzzy ideals Let (XI, F ) be a uniform fuzzy semigroup and 17 = (F, "A"), where F is a binary operation on X and "A" is the minimum function from the vector lattice J = I o I into I.
Theorem 2.4. Ever), fuzzy subset A of X which induced a fuzzy (left, right) ideal ( H ( A ) , F ) of the uniform fuzzy semigroup (X/, 17) is a classical fuzzy (left, right) ideal of the ordinary semigroup (X,F). Proof. Suppose that A is a fuzzy subset of X such that ( H ( A ) , F ) is the induced fuzzy left ideal by A of the uniform fuzzy semigroup (Xt, 17), where 17 = (F,"A"). By Theorem 2.1a(ii), we have
A(xFy)=Ay
f o r a l l x E X , yEA0.
Since Ay = 0 for each y E X - A 0 , then
A(xFy)>_.Ay for all x, yEX, which means that A is a classical fuzzy left ideal of
(X,F).
[]
Remark 2.2. The converse of Theorem 2.4, in general, is not true, i.e. there is a classical fuzzy (left, right) ideal A o f a semigroup (X,F) such that a fuzzy subspace H(A) induced by A is not a fuzzy (left, right)
/-/(.,1) = {(a,[0, ½1), (c,[0, ¼])} is not a fuzzy left ideal of (Xt, 17) in our sense, because
A c ( l , A c ) = f.c(l, ¼) = ,
# A(aFc) = A(a) = ½. Theorem 2.5. If (Y,F) is an ordinary (left, right) ideal of the semigroup (X,F). Then for each fuzzy subset A of X with support Ao = Y, there is a fuzzy semioroup (XI, G_.)such that the induced fuzzy subspace H(A) is a fuzzy (left, right) ideal of (X/, G). Proof. Let (Y,F) be a left ideal of the semigroup (X,F) and A be a fuzzy subset of X such that A0 = Y. It is clear that (Ao,F) is a left ideal of (X,F). Now we define a fuzzy binary operation _.G = (G, #xy) o n X as follows: Take G = F and 9xy(r,s) = hxy(r A s), where
hxy(k) = { i(xF'~')k, k < Ay, Ay 1 - a(xFy) f--_--~y ] ( k - A y ) + A ( x F y ) ,
k>/Ay
K.A. Dib. N. GalhumlFuzzy Sets and Systems 92 (1997) 103-11l
for a l l x EX, v EA0. And h.,?.(k) = k i f x E X, y f{ Ao. It is clear that gxy(r,s), x, y E X are continuous functions on J and 9.~v(r,s) = 0 iff r = 0 or s = 0. Then it is easy to show that (Xt, G ) is a fuzzy semigroup. If H(A ) is the induced fuzzy subspace by A of the fuzzy semigroup (Xt,_G_G),then from the definition of g.~v, we have for all xEX, yEA•,
9.,~,,(1, Ay) = h.,;,,(Ay) = A(xFy), which implies (H(A),G__) is a fuzzy left ideal of (X~, ¢7). [] Corollary 2.2. Ever), classical fuz=y (left, right) ideal of the semigroup (X,F) induces a fuzzy (left, right) ideal relative to some.fuzzy semigroup (Xt, G_G_).
3. Fuzzy bi-ideal Definition 3.1. A fuzzy subsemigroup (U,F__) of the fuzzy semigroup (XL,F ) is called a fuzzy bi-ideal if for every (y, Ly), (z,L.)E U and (x,L)EXL we have
[(y, Ly ) F (x,L ) ] F (z, Lz ) = ( y , Ly)IV_[(x,L)F__(z,L.)]E U.
Theorem 3.1. A fuzzy subsemioroup (U,F__) of the fuzzy semigroup (XL, F) is a fuzzy bi-ideal iff (i) ( Uo,F) is a hi-ideal of the ordinary semioroup (X,F), (ii) for all y, z E U• and x EX, we have
Table 3
F a
a a
b b
b
a
b
¢e~ ((yFx)Fz, fo,F.,r.(£..~.(Ly~L )[]L: )) = (yF(xFz), .~,~xF:)(LytZf,:(L ~zL:)) =--(w,L,,,) E U ¢:~ (i) (yFx)Fz = yF(xFz) = wE Uo, y, z E Uo, xEX, (ii) f ( y V x ) z ( £ , x ( L y [] L) []L: ) = .~'L~Fz~(Ly[]f~: (L[]L:))= L.. ¢~ (i) (Uo,F) is a bi-ideal of the ordinary semigroup (X,F), (ii) LyFxF,. = f~yFx~:(~,x(Lv []L) []Lz ) = f.txF:)(Ly[]fx:(L[]L_-)). [] Remark 3.1. Although in the classical fuzzy case, every fuzzy (left, right) ideal is a fuzzy bi-ideal, in our sense there are fuzzy (left, right) ideals which are not a fuzzy bi-ideals as we see in the following:
Example 3.1. Let (X, F ) be the semigroup defined by Table 3 Then:
(I) It is easy to see that the subset U• = {a} is a left ideal and hence its a bi-ideal of (X, F). (2) Consider (XI,__F) is a fuzzy semigroup, where F_. = (F, fxy), and
LyFxFz = f ( ) . F x ) z ( f , x ( L y nL ) []L=) = f.~xFz)(Ly []f~z(L ~sL:)). Proof. Suppose that ( U , F ) is a fuzzy bi-ideal of the fuzzy semigroup (XL,F). Then for all (y,L.~.), (z, Lz ) E U and (x, L) E XL, we have
109
½rs f~y(r,s) =
1 - ½(r + s - r s ) '
x # y,
rAs,
x=y.
Then we have the following:
(i) For U = {(a, [0, t]): a E Uo} where tE(0,1], we have
[(y, L;, )F(x, L )IF(z, L: ) Ao(t,o [] La) = Aa([0, t] ~ [o, t])
= (y, Ly)F[(x,L)F(z, Lz)] E U = [O, faa(t,t)] = [0, t] = La ¢~, [(yFx ), ~.x(L y []L )]iv(z, Lz )
= (y, Ly)F[(xFz, fxz(LoL,.)] E U
and hence (U, F) is a fuzzy subsemigroup o f the fuzzy
semigroup (X, F).
K.A. Dib, N. GalhumlFuz~v Sets and Systems 92 (1997) 103-111
110 (ii) Since
f,,,(l ~La)
=
[0, .faa(l, t)]
= [0, t]
=
LaFa =
La
Then (U.__F) is not a fuzzy bi-ideal although it is a fuzzy left ideal of 0(/, F__). In the following, Theorems 3.2-3.4, we discuss the sufficient conditions for the fuzzy (left, right) ideal to be a fuzzy bi-ideals, respectively.
and
fba( l []La ) = [0, .fba(1, t)] = [0, t] = LbFa = La. Then, from (1) and Theorem 2.1, we see that ( U , F ) is a fuzzy left ideal. (iii) Now, we show that U is not a fuzzy bi-ideal, for
La = LaFaFa= f(aFa)a(faa(La ~I) []La) = faa([0, .f~a(t, 1 )] [] La)
= faa(La :~La) = [0, faa(t, t)] = [o, t] = Lo = fa(aFa)(La
nfo~(I pLy))
Theorem 3.2. Ever), fuzz), left ideal U of the fuzz), semigroup (XL,F__) is a fuzzy bi-ideal if for all y, z E Uo and x E X, we have (1) f.:(Ly ~L:) = LyF: (2) LyFxF: = f~yFx)z(f.x(Ly nL) []L.. ). Proof. Suppose that U is a fuzzy left ideal o f the fuzzy semigroup (XL, F). Then, from Theorem 2.1, we have (i) (U0,__F) is a left ideal of the ordinary semigroup
(X,F) which implies (Uo, F) is a bi-ideal o f ( X , F ) , (ii) for all x E X and z E Uo, we have
fxz(L[]L:) = LxF:. Since for a n y x E X and y,z E U0, we havexFz E Uo and yF(xFz) E Uo. Hence, by condition (1), we get
= .foo(L~ [] [0, faa( 1, t)])
£,(xFz)(Ly t3LxFz) = LyFxFz.
=f~o(LarJLa) = [O, f~o(1, t)] = L,,.
From (ii) and condition (2), we get
But
LyFxrz = f,(xFz)(Ly []f~-_(L []L=))
La = LaFara ~= Aarb)a(.fa6(La DI)DLa) = fb~([O, f~b(t, 1 )] []La)
= fao(La [] L.) = [0, faa(t, t)] =
[ ':1 0,2_2t+t
2 •
Although
fa(bF,,~(L, ~3fha(I []La )) = f,a(La [] [0, .fba( 1, t)]) = .L~(L,, n [0, t])
= [O, faa(t,t)]
= J~yFx):(fvx(Lv []L) rTL.) and by (i), we see that U is a fuzzy bi-ideal o f (XL, g ) . [] Theorem 3.3. Every fuzz), right ideal of the fuzzy semigroup (XL,F) is a fuzz)' bi-ideal if for all y, z E Uo and x E 2(, we have (i) f,:(L,, t~L-_) = LvF.., (ii) L~,rxr= = f,,~xr:)(L~, [] fx..(L taL:)). Proof. The proof is the same as for fuzzy left ideal. The combined effect of Theorems 3.2 and 3.3 is as follows:
= [0, t] = La, i.e.
La = LaFhFa = fa(hFa)(La []fba(l nLa) ) # f{aFb)a(fab(La []l)[]La).
Theorem 3.4. Every fuzzy ideal U of the fuzzy semigroup (XL,F) is a fuzzy bi-ideal if for all y,z E Uo, we have
f.z(Ly nL:) = L vF=.
[]
K.A. Dib. N. GalhumlFuzzy Sets and S.vstems 92 (1997) 103-111
Concerning the induced fuzzy subspace H ( A ) by the fuzzy subset A of X, Theorem 3.1 can be reformulated as follows:
111
Remark 3.2, The converse of the above theorem is not true, generally, as we see in the following:
Example 3.2. Let (X/, F ) be the uniform fuzzy semiTheorem 3,5. I f H ( A ) is a fuzzy subsemigroup o f the fi~zz), semigroup (XI, F) then H ( A ) is a fuzzy hi-ideal (i) (A0,F) is a bi-ideal of the ordinary semigroup (X,F), (ii) ,]'or all y,z E Ao a n d x E X, we have
A(yFxFz) = ~.,.Fxlz(f,x(Ay, 1),Az)
= f.,,~xF:~(AY,L : ( 1 , A z ) ) . In the following theorem, we give the relationship between fuzzy bi-ideals and classical fuzzy biideals:
Theorem 3.6. Let (XI, F_.) be a uniform fuzzy semigroup, where F = (F,A) and "A" the minimum /'unction from J hzto I. Then every fuzz), subset A of X which induces a fuzz)' hi-ideal ( H ( A ) , F ) of (XI, F__)is a classical fuzzy hi-ideal o f the semigroup (X,F). Proof. Suppose that A is a fuzzy subset of X such that (H(A), F_.) is a fuzzy bi-ideal o f the uniform fuzzy semigroup (XI,F_.), where F_ = ( F , " A " ) . Then, by Theorem 3.5, we have (i) (A0,F) is a bi-ideal o f the ordinary semigroup
(X,F), (ii) for all y,z E Ao and x E X, we have
(,)
A(yFxFz) = (A(y) A 1 ) A A(z) = A ( y ) AA(z).
Now, if y,z E Ao, then A ( y ) = A(z) = 0 and hence, by ( , ) , we get
A(yFxFz)>~A(y) AA(z),
for all x , y , z E X,
which means that A is a classical fuzzy bi-ideai o f
(X,F).
[]
group defined in Example 2.2, where X : {a, b, c}. If we define a fuzzy subset A o f X as A ( a ) = -J 4 ~
A ( b ) = -,~
A ( c ) = 0.
Then we can show that: (i) A is a classical fuzzy bi-ideal of (X, F ) (ii) the support A0 = {a,b} o f A is a bi-ideal o f
(X,F) (iii) although the support A0 o f A is a bi-ideal o f
(X,F) we see that the induced fuzzy subspace H ( A ) = {(a,[0, ¼]), (b,[0, ½])} by A is not a fuzzy bi-ideal in our sense, since I A(aFaFb) = A(b) = ~I ¢ A(a) A A(b) = -~.
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