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Fuzzy Rees congruences on semigroups
FUIZY
sets and systems Fuzzy Sets and Systems 102 (1999) 353-359
ELSEVIER
Fuzzy Rees congruences on semigroups Xiang-Yun Xie Department of Mathematics and Physics, Wuyi University, Jiangmen, Guangdong, 529020, People's Republic of China
Received May 1996; received in revised form April 1997
Abstract Fuzzy congruences on semigroups and groups have been introduced and studied by Samhan (1993) and Samhan and Ahsanullah (1994), among others. Here, we introduce fuzzy Rees congruences on semigroups and fuzzy Rees congruences semigroups. Using these ideas, we establish a relation between fuzzy ideals of a semigroup S and fuzzy ideals of a quotient semigroup of S. As an important result, we prove that the lattice of all fuzzy congruences on a fuzzy congruences semigroup is a distributive lattice. Moreover, we obtain that a homomorphic image of a fuzzy Rees congruences semigroup is a fuzzy Rees congruences semigroup, as well. © 1999 Elsevier Science B.V. All rights reserved. A M S classification: 20MlO Keywords. Semigroup; Fuzzy ideal; Fuzzy Rees congruence; Fuzzy Rees congruences semigroup
I. Introduction Crisp congruence relations and ideals on semigroups play an important role in studying algebraic structures of semigroups [2, 4]. Fuzzy ideals of a semigroup S were first defined by Kuroki [5]. Fuzzy relations on a set have also been studied since the concept of fuzzy relations was introduced by Zadeh [14]. Fuzzy equivalent relations and fuzzy ordering have been studied in the literature of fuzzy mathematics [9, 15]. Recently, fuzzy congruence relations on semigroups and groups appeared in [7, 13]. Samhan a m o n g others proved that normal fuzzy subgroups and fuzzy congruence relations determined each other in groups. Moreover, he proved that the lattice of all fuzzy congruences on a group G (resp. ring R) is isomorphic to the lattice of fuzzy
normal subgroups of G (resp. fuzzy ideals of R) [6, 8, 13]. For a semigroup S, Samhan obtained that the lattice of fuzzy congruences on a semigroup S is a complete lattice. In particular, if S is a group, then the lattice of all fuzzy congruences on S is a modular lattice [12]. For some special algebraic structures of semigroups, such as inverse semigroups and T*-pure semigroup, A1-Thukair characterized fuzzy congruences on an inverse semigroup by the concept of fuzzy congruence pair. This result is analogous to the characterization of crisp congruences on inverse semigroups [10]. Kuroki characterized fuzzy group congruences and fuzzy semilattice congruences on T*-pure semigroups [7]. The purpose of this paper is to introduce fuzzy Rees congruences on a semigroup S, and prove that if p is a fuzzy ideal of a semigroup S, then there
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354
exists an inclusion-preserving bijection from the set of ideals of S containing the set supp p onto the set of ideals of quotient semigroup S/Or. As an application of this result, we obtain the Rees homomorphism theorem [4, Proposition 3.7]. In Section 4, we define FRC-semigroups and prove that if S is an FRC-semigroup, then S has a zero element and the lattice FI(S) of all fuzzy ideals of S is isomorphic to the lattice FC(S) of all congruences on S. As a corollary of this result, we have the lattice FC(S) of an FRC-semigroup is a distributive lattice. Furthermore, we show that if S is an FRC-semigroup, then S is fuzzy congruence free if and only if S is fuzzy 0-simple. If S is an FRC-semigroup, then we also have that any homomorphic image of S is an FRCsemigroup.
Their composition is denoted by 0 o 4) and defined as
Oo 4)(x, y) = V (O(x, z) A 4)(z, y)) (Vx, y e X) [9]. z~X
Definition 2.1 (Murali [9]). A fuzzy relation 0 on X is called a fuzzy equivalent relation on X if (i) O(x, x) = 1 (V x ~ X) (reflexive); (ii) O(x, y) = O(y, x) (Vx, y ~ X) (symmetric); (iii) 0 o 0 ~< 0 (transitive).
Definition 2.2 (Samhan [12]). A fuzzy equivalent relation 0 on a semigroup S is called fuzzy compatible if (Vx, y, z, t • S)
O(x, y) A O(z, t) <~O(xz, yt).
0 is called fuzzy left (right) compatible if
2. Preliminaries
(Vx, y, z e S )
In this paper, S will mean a semigroup. If A is a nonempty subset of S, A is called an ideal of S if AS, SA ~_ A [4]. We denote by I the unit interval [0, 1]. For x,y ~ I, x V y = max{x, y} and x A y = min{x,y}. All fuzzy sets of a set X are maps p : X ~ I. A fuzzy set # of a semigroup S is called fuzzy ideal if (Vx, y e S)
~t(xy) i- p(x) V p(y) [5].
Clearly, the set of all fuzzy ideals of S, denoted by FI(S), is a distributive lattice with respect to the meet "A" and union " V " defined as follows:
(Vx e S)
(~ v
la2)(x) = ~ ( x ) v/~2(x),
(~1 A ~/2)(X) = ~/I(X) A ~/2(x). For the lattice FI(S), it has the greatest element p : p(x) = 1 (Vx e S), denoted by is, and has the least element p: p(x) = 0 (Vx ~ S) denoted by 0 if S has no zero element, otherwise, the least element is p: /~(x) = 0 (V(0 # ) x ~ S) denoted by 0s. If S has a zero element 0, then it is not difficult to show that #(0)/>/4x) (Vx e S). In this paper, we define p(0) = 1. All fuzzy relations on a set X are maps: X × X I. If 0 and 4) are two fuzzy relations on a set X, then 0 ~< 4) means that O(x, y) <~ 4)(x, y) (V x, y ~ X).
O(x, y) <~O(zx, zy), (0(x, y) <~O(xz, yz)). Definition 2.3 (Samhan [12]). A fuzzy equivalent relation 0 on a semigroup S is called fuzzy congruence if it is fuzzy compatible.
Proposition 2.4 (Samhan [12]). Let 0 be a fuzzy equivalent relation on a semigroup S. Then 0 is called fuzzy congruence if and only if O is fuzzy left and right compatible. We denote the set of all fuzzy congruences on a semigroup S by FC(S). Then Samhan [5] proved that FC(S) is a complete lattice with respect to " ' " and " + " defined as follows:
(VO, 4) ~ FC(S)) 0+4)=(0v4))
o, 0 . 4 ) = 0 A 4 ) .
Furthermore,
0 + 4) = (0o 4))°0. Let elements Vs and As be fuzzy congruences on S defined by Vs(x,y)= 1 for all x, y e S , and A s ( x , y ) = O if x ¢ y and A s ( x , y ) = l , if x = y . Then it is clear that Vs and As are the greatest
£ - Y . Xie / Fuzzy Sets and Systems 102 (1999) 353 359
element and least element of the lattice (FC(S), +, .), respectively.
(2) If x # y, then
O. o O,(x, y)
Definition 2.5. A semigroup S is called fuzzy congruences free, if S has no fuzzy congruences other than Vs and As.
=
Definition 2.6. A semigroup S is called fuzzy 0simple, if S 2 ~ {0} and 0s and ls are the only fuzzy ideals.
= O.(x, y) v
V
(O.(x,z) AO.(y,z))
zeS-{x,y}
v (O.(x, x) A O.(x, y)) V (0,,(x, y) A O.(y, y)) V
(Va, b e S )
(/~(x) A/~(z) A ~(z) A #(y))
z ~ S - Ix, y}
V
<-GO,(x, y) V z~S
Let 0 be a fuzzy congruence on S, we denoted by S/O the set {aO:a ~ S} of all fuzzy congruence classes of S. We define a binary operation "*" on S/O as follows:
355
(#(x)
A it(y))
',x,y l
= G(x, y) V O.(x, y) = O.(x, y). By (1) and (2) 0, is transitive. Furthermore, 0, is left compatible. In fact, for x, y, c e S, if cx = cx, then
O.(cx, cy) = 1 > G(x, y).
aO*bO=(ab)O.
Then S/O is a semigroup [1, 7]. For other definition and terminology not given in this paper, we employ those in [2, 4, 11, 12].
If cx ./= cy, then x -~ y. Thus,
O.(cx, cy) = ~(cx) A #(cy) > #(x)
A ~(y) =
O.(x, y).
In the same way, we can also prove that O. is right compatible. []
3. Fuzzy Rees congruences Let/~ be a fuzzy ideal of a semigroup S. Then, we can assign a fuzzy relation 0, on S defined as follows: (Vx, y e S )
O.(x,y)=tt(x)Att(y),
ifx=~y,
O.(x,x)= 1, i f x = y . Clearly, 0, is well-defined. Moreover, we have
Proposition 3.1. Let p be a fuzzy ideal of a semigroup S. Then O, is a fuzzy congruence on S. Proof. By definition of 0,, it is obvious that 0, is reflexive and symmetric. Since for all x, y e S,
0.o O.(x, y) = V (O.(x, z) A O.(z, y)). zES
We consider the cases: (1) If x = y, then
O, o O,,(x, x) = V (0,(x, z) >>,O,(x, x)) =- 1. z~S
Thus, 0, o O,(x, x) = O,(x, x).
In this paper, we shall call a fuzzy congruence 0, determined by a fuzzy ideal /~( ~ 0) of S a fuzzy Rees congruence. Let p be a fuzzy ideal of a semigroup S, and let
supp# = {x e S:/t(x) = 1}. Then it is not difficult to verify that supp ~ is an ideal of S. Similar to the case of crisp Rees congruences on S, we have Theorem 3.2. Let i~ be a fuzzy ideal of a semigroup S. Let sd be the set of ideals of S containing supp I~ and let ~ be the set of ideals of the quotient semigroup (S/O,, *). Then the mapping
f:J~-"-~JO u ( J e ~ ) is an inclusion-preserving bijection from ~ onto ~. Proof. Let J be an ideal of S. Then it is clear that JO. is an ideal of the semigroup (S/O..*). Let J 1 , J 2 6 .N' and J1 4= J2. Then there exists a ~ S such that a e J1 \J2 or a ~ J 2 \ J 1 . Ifa ~ J1 \ J 2 , then we have JlO. =/=J20.. In fact, if JlO. = J20,, then
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X.-Y. Xie / Fuzzy Sets and Systems 102 (1999) 353-359
there exists b ~ J2 such that aO. = bO.. Since a ¢ J 1 , then we have a # b. Thus, (aOu)(b) = O.(a, b) = #(a) A #(b) = (bO.)(b) = 1.
Proof. Suppose that #1,#2 i~ FI(S) and #1 # #2, then there exists x ~ S such that # 1 ( x ) ~ #2(X). Obviously, x # 0. Thus, O.I(X , 0) = #1(X) A #1(0) = #l(X),
It implies that # ( a ) = # ( b ) = 1. Then a E s u p p # J2. Impossible. In the same way, if a ~ J 2 \ J 1 , then we also have JlO. # J20.. This proves that the mapping f is injective. Let KO. be an ideal of S/O. and
O,2(x, 0) = r e ( x ) A #2(0) = #2(x). It follows that 0., # 0.2. Therefore, the mapping g is injective. It is easily seen that g is orderpreserving. []
K1 = {x ff S: xO, E KO.}.
Then, for all x ~ K1,
4. Fuzzy Rees congruence semigroups
(Sx)O. = SOu o xO. ~_ SOu o KO. ~_ KO u.
Thus, Sx ~ K1. Similarly we have x S ~ K I . It follows that K1 is an ideal of S. On the other hand, s u p p # ~_ K I . Indeed: For any a ~ s u p p # and let x ~ g x , we have (A) If a = ax, obviously, a e K l . (B) If a ~ ax, for all z 6 S, we consider the cases (1) If z # a, ax, then (aO.)(z) = O.(a, z) = #(a) A #(z)
(since #(a) = 1)
= #(ax) A #(z)
= ((ax)O,)(z). (2) If z = a, then
Definition 4.1. A semigroup S is called fuzzy Rees congruence semigroup (abbr. FRC-semigroup) if every fuzzy congruence on S is a fuzzy Rees congruence.
Proposition 4.2. Let S be a FRC-semigroup. Then (1) S has a zero element 0; (2) I f 0 is a f u z z y congruence on S, then O, = O, where #(x) = O(x, O) for all x ~ S. Proof. To prove (1), observe that As is a fuzzy congruence on S. Since S is a FRC-semigroup, there exists a fuzzy ideal # ( # 0 ) of S such that As = 0,. Since p # 0, we have # ( x ) # 0 for some x e S. Then for any y e S such that y # x, we have As(y, x) = O,(x, y) = #(x) A #(y) = O.
(aO.)(a) = 1 = #(a) A #(z) = ((ax)O.)(a).
(3) If z = ax, by (2), we have (aOu)(ax)= ((ax) O.)(ax). It follows that (aO,)= (ax)O, ~_ KO,. By definition of K1, then a e K1. Obviously, KIOu = KOu. It proves that f is onto. The inclusion-preserving o f f is only a routine verification. []
Proposition 3.3. Let S be a semigroup with O. Then
Thus, #(y) = 0. Since # is a fuzzy ideal of S, for all z E S, we have #(zx) >1 p(x), #(xz) ~ #(x), this implies that z x = x z = x. This proves that x is a zero element of S. To prove (2), let 0 be a fuzzy congruence on S. Then, by hypothesis, there exists a fuzzy ideal # ( # 0 ) such that 0 = 0,. Let (Vx ~ S),
v(x) = O(x,O)
(by (1)).
Then for all y e S,
the mapping
v(yx) = O(yx, O) >t O(x, O) = v(x), g : # w-~O.
(# ~ VI(S))
v(yx) = O(yx, 0)/> 0(y, 0) = v(y), is an order-preserving FC(S).
injective from FI(S) into v(O)
= o(o, o ) =
1.
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X.-Y. Xie / Fuzzy Sets and Systems 102 (1999) 353-359
Thus, v is a fuzzy ideal of S. Moreover, we have v = #. In fact, since for any y e S such that y ~ 0,
vCy) = 0(y, 0) = 0,(y, 0) = #(y) Then, we have # = v.
A
< (o//, o o,,2)~(x, y) = (o//, v o//)(x, y)
#(0) = #Cy).
[]
(2) and
Theorem 4.3. Let S be a FRC-semigroup. Let FI(S)
be the lattice of all fuzzy ideals of S and FC(S) the lattice of all fuzzy congruences on S. Then the mapping g:#~-+O.
#2(X) A #2(Y) = O//2(x, Y) ~ CO#, o 0u2)(x' y)
#1(x) A #2(Y) ~< #,(xy) A #2(xy) A #1(X) A #2(Y),
(3) #2(x) A ill(Y) ~< #l(xy) A #2(xy) A #2(x) A #I(Y). (4)
(peEl(S)) We consider (3) in the cases. (1) If xy = x, then
is an isomorphism from FI(S) onto FC(S). Proof. By Propositions 3.3 and 4.2, we have the mapping g is an order-preserving bijective. T o prove that g is lattice-order preserving, we only show that
#1(X) A #2(Y) ~< #1(x) A #2(x) A #2(Y)
(VPl, #2 ~: FI(S))
0.~ A 0.2 = 0//, A//~,
(2) If xy = y, then
0//, V 0//2 = 0., v//~.
#1CX) A #2(y) < #1(Y) A #2(Y) A #1(x)
~< #2(x) A #2@) <~ (0., V O,2)(x, y)
(by (2)).
Since for all x, y e S such that x 4: y, we have
4 #l(x) A #l(y)
(0//, A O//)(x, y) = O.,(x, y) A O//2(x, y)
~< (0//, V 0,2)(x, y) (by C1)).
=
#1(X)
A
#1(Y)
A
#2(X)
A
#2(Y)
= (#1 A m)Cx) A (#1 A m)(Y)
(3) If xy # x, y, then
#1(x) A #2(Y) <~ O//~(x, xy) A O//2(xy, y)
< C0.,o %)(x, y)
= 0//, A /12(X' Y)
and obviously, 0,, A O//~(x, x) = 0//, A ,2(x, x) = 1. This implies that 0//, A 0//2 = 0//, A//~. Since Pl, #2 ~< #1 V #2, by T h e o r e m 4.3, then we have 0//1 ~ 0/A1 V ~/2and 0//2 ~< 0//, v//2" Thus, 0//, V 0//2 <~ 0//, v u~. On the other hand, for all x, y e S such that x va y,
~< ( % ° 0.) ~(x, y) = (0., V O//2)(x,y)
In the same way, we consider (4) in the cases (4) If xy = x, then
#~(x) A #~Cy) < #l(x) A #,Cy) <~ (0//, V O//2)(x, y)
0,, v ,2( x, Y) ----(#1 V #2)(x) A (ill V #2)(Y)
= (#,(x) V #2(x)) A (#1(Y) V #2(Y)) =
(#l(x)
A
V (#ICy)
#I(Y)) A
V
(#1(x)
A
#2(X)) V (#2(x)
#2(Y)) A
#2(Y)).
(by (2)).
(by (1)).
(5) If xy = y, then
#2(X)
A
#I(Y) < #2(X)
A
#2(Y)
<~ CO~~,V O.2)(x, y)
(by (2)).
Furthermore,
(6) If xy # x, y, then
#l(x)
#2(x) A #I(Y) <~ O//,(y, xy) A O.2(xy, x)
A
#iCY) = 0//,(x, y) ~< (0/2, o 0//2)(x. y)
< (%0 %)~(x, y) = (0//, v %)(x, y), (1)
< (0//, v 0//)(y, x) ~< (0//, o % ) ~ ( y , x) = (0//, v O//)tx, y).
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X.-Y. Xie / Fuzzy Sets and Systems 102 (1999) 353-359
In view of all situations, we have
let 0 be a fuzzy congruences on T. Define
Oul v u~(x, Y) <~(0ul V Ou2)(x, y),
O(x, y) = O(f(x),f(y))
thus, 0u, v ~2 ~ (0., V 0~2). Consequently, we prove that
Then we have q~ is a fuzzy congruence on S. In fact, the reflexivity and symmetry of~b are obvious, since
0., v .2 = (0., V 0.2).
~o q~(x, y) = V (q~(x, z) A ¢(z, y))
[]
(Vx, y ~ S).
zeS
Since FI(S) is a distributive lattice, by T h e o r e m 4.3, we have Corollary 4.4. Let S be an FRC-semigrou p. Then
the lattice FC(S) is distributive. Theorem 4.5. Let S be a FRC-semigroup and S 2 ~ {0). Then S is fuzzy congruences free, if and
= V (o(f(x),f(z)) A O(f(z),f(y))) z~iS
V (o(f(x), z) A O(z,f(y))) g~S
= (0o O)(f(x),f(y)) O(f(x), (f(y))) = 49(x, y).
only if S is fuzzy O-simple.
Then it implies that ~b is transitive. On the other hand,
Proof. (=~) Suppose that S is fuzzy congruences
(Vx, y, z, t e S)
free and # ( ¢ 0 ) is a fuzzy ideal of S. Then 0, is a fuzzy congruence on S. By Definition 2.5, then either 0u = x7s or 0, = As. If 0u = Vs, then for all x e S (x ¢ 0),
4)(xz, yt) = O(f (xz),f (yt)) = 0 (f(x)f(z),f(y)f(t))
0.(0, x)= Vs(0, x)= 1 =u(0) A ~(x)=~(x).
>~O(f(x),f(y)) A O(f(z),f(t))
It implies that # = ls. If 0. = As, then, by Proposition 4.2(1), for all x e S (x ¢ 0),
= 4,(x, y) A 4)(y, t).
0.(0, x ) = ~ ( x ) = A s ( O , x)=0. It implies that 0, = 0s. ( ~ ) Suppose that S is fuzzy 0-simple and let 0 be a fuzzy congruence on S. By Definition 2.6, there exists a fuzzy ideal # ( ¢ 0 ) of S such that 0 = 0,. Since S is fuzzy 0-simple, then either p = 0s or kt = ls. If # = ls, then for all x, y e S (x ~ y),
Therefore, q~ is compatible. Since q~ is a fuzzy congruence on S, by hypothesis, then there exist a fuzzy ideal p ( ¢ 0 ) of S such that ~b = ~b,. Let f(p):f(/a)(x) =
V
/t(z),
Vx e S.
zEf-l(X)
Then it is clear that f(~t) is a fuzzy ideal of T. We now claim that 0 = 0z¢u). Since for all x, y e T, if x = y, then O(x,y)=Of(,)(x,y)= 1; if x # y, then there exist a,b E S such that x = f ( a ) , y = f ( b ) . Moreover,
O(x, y) = O.(x, y) = u(x) A u(y) = 1, it implies that 0 = Vs. I f p = 0s, by a routine verification, we have 0 u = As. [] Proposition 4.6. The homomorphic image of a FRC-
O(x, y) = O(f(a),f(b))
(obviously, a # b)
= 49(a, b) = ¢ , ( a , b)
= ,(a) A #(b)
semigroup is a FRC-semigroup, Proof. Let S be a FRC-semigroup and let f : S w-, T
be a h o m o m o r p h i s m of S onto a semigroup T and
=f(/~)(x) Af(/~)(y) = Of(u)(x, y).
X-Y. Xie / Fuzzy Sets and Systems 102 (1999) 353-359
Thus, we have 0 ~< OA.~. On the other hand,
Os(u)(x, y) = f(p)(x) A f(l~)(y) =
V
zef
A
l(x)
V
:
V
w~f-I(y)
(/~(z)
z E f t ( x ) . w e f l(y)
A/~(w))
(obviously, z # w)
V
=
zef-t(x),wef
w)
a(y)
V
=
4)(z,w)
z~ f-l(x), we f - l ( y )
=
~/ zef
O(f(z),f(w))
l ( x ) , w e f l(y)
<~O(x, y). Thus, OI~.) ~< O. Consequently, Oy~.) = O.
[]
References [1] F. A1-Thukair, Fuzzy congruence pairs of inverse semigroups, Fuzzy Sets and Systems 56 (1993) 117 122.
359
[2] G. Birkhoff, Lattice Theory, Vol. 25, Amer. Math. Soc. Coll. Publ., 1976. [3] J. Garcia, The congruence extension property for algebraic semigroups, Semigroup Forum 43 (1991) 1-18. [4] J.M. Howie, An Introduction to Semigroup Theory, Academic Press, New York, 1976. [5] N. Kuroki, Fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems 5 (1981) 203-215. [6] N. Kuroki, Fuzzy congruences and fuzzy normal subgroups, Inform. Sci. 60 (1992) 247-259. [7] N. Kuroki, Fuzzy congruences on T*-pure semigroups, Inform. Sci. 84 (1993) 1239-1246. [8] B.B. Mrkamba, V. Murali, Normality and congruence in fuzzy subgroups, Inform. Sci. 59 (1992) 121-129. [9] V. Murali, Fuzzy equivalence relations, Fuzzy Sets and Systems 30 (1989) 155-163. [10] M. Petrich, Inverse Semigroups, Wiley, New York, 1984. [11] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512 517. [12] M. Samhan, Fuzzy congruences on semigroups, Inform. Sci. 74 (1993) 165 175. [13] M. Samhan, T.M.G. Ahsanullah, Fuzzy congruences on groups and rings, Internat. J. Math. Math. Sci. 17 (1994) 469-474. [14] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338 353. [15] L.A. Zadeh, Similar relations and fuzzy ordering, Inform. Sci. 3 (1971) 117-200.
sets and systems Fuzzy Sets and Systems 102 (1999) 353-359
ELSEVIER
Fuzzy Rees congruences on semigroups Xiang-Yun Xie Department of Mathematics and Physics, Wuyi University, Jiangmen, Guangdong, 529020, People's Republic of China
Received May 1996; received in revised form April 1997
Abstract Fuzzy congruences on semigroups and groups have been introduced and studied by Samhan (1993) and Samhan and Ahsanullah (1994), among others. Here, we introduce fuzzy Rees congruences on semigroups and fuzzy Rees congruences semigroups. Using these ideas, we establish a relation between fuzzy ideals of a semigroup S and fuzzy ideals of a quotient semigroup of S. As an important result, we prove that the lattice of all fuzzy congruences on a fuzzy congruences semigroup is a distributive lattice. Moreover, we obtain that a homomorphic image of a fuzzy Rees congruences semigroup is a fuzzy Rees congruences semigroup, as well. © 1999 Elsevier Science B.V. All rights reserved. A M S classification: 20MlO Keywords. Semigroup; Fuzzy ideal; Fuzzy Rees congruence; Fuzzy Rees congruences semigroup
I. Introduction Crisp congruence relations and ideals on semigroups play an important role in studying algebraic structures of semigroups [2, 4]. Fuzzy ideals of a semigroup S were first defined by Kuroki [5]. Fuzzy relations on a set have also been studied since the concept of fuzzy relations was introduced by Zadeh [14]. Fuzzy equivalent relations and fuzzy ordering have been studied in the literature of fuzzy mathematics [9, 15]. Recently, fuzzy congruence relations on semigroups and groups appeared in [7, 13]. Samhan a m o n g others proved that normal fuzzy subgroups and fuzzy congruence relations determined each other in groups. Moreover, he proved that the lattice of all fuzzy congruences on a group G (resp. ring R) is isomorphic to the lattice of fuzzy
normal subgroups of G (resp. fuzzy ideals of R) [6, 8, 13]. For a semigroup S, Samhan obtained that the lattice of fuzzy congruences on a semigroup S is a complete lattice. In particular, if S is a group, then the lattice of all fuzzy congruences on S is a modular lattice [12]. For some special algebraic structures of semigroups, such as inverse semigroups and T*-pure semigroup, A1-Thukair characterized fuzzy congruences on an inverse semigroup by the concept of fuzzy congruence pair. This result is analogous to the characterization of crisp congruences on inverse semigroups [10]. Kuroki characterized fuzzy group congruences and fuzzy semilattice congruences on T*-pure semigroups [7]. The purpose of this paper is to introduce fuzzy Rees congruences on a semigroup S, and prove that if p is a fuzzy ideal of a semigroup S, then there
0165-0114/99/$ see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S0 1 65-0 1 1 4(97)00 1 56-5
X.-Y. Xie / Fuzzy Sets and Systems 102 (1999) 353-359
354
exists an inclusion-preserving bijection from the set of ideals of S containing the set supp p onto the set of ideals of quotient semigroup S/Or. As an application of this result, we obtain the Rees homomorphism theorem [4, Proposition 3.7]. In Section 4, we define FRC-semigroups and prove that if S is an FRC-semigroup, then S has a zero element and the lattice FI(S) of all fuzzy ideals of S is isomorphic to the lattice FC(S) of all congruences on S. As a corollary of this result, we have the lattice FC(S) of an FRC-semigroup is a distributive lattice. Furthermore, we show that if S is an FRC-semigroup, then S is fuzzy congruence free if and only if S is fuzzy 0-simple. If S is an FRC-semigroup, then we also have that any homomorphic image of S is an FRCsemigroup.
Their composition is denoted by 0 o 4) and defined as
Oo 4)(x, y) = V (O(x, z) A 4)(z, y)) (Vx, y e X) [9]. z~X
Definition 2.1 (Murali [9]). A fuzzy relation 0 on X is called a fuzzy equivalent relation on X if (i) O(x, x) = 1 (V x ~ X) (reflexive); (ii) O(x, y) = O(y, x) (Vx, y ~ X) (symmetric); (iii) 0 o 0 ~< 0 (transitive).
Definition 2.2 (Samhan [12]). A fuzzy equivalent relation 0 on a semigroup S is called fuzzy compatible if (Vx, y, z, t • S)
O(x, y) A O(z, t) <~O(xz, yt).
0 is called fuzzy left (right) compatible if
2. Preliminaries
(Vx, y, z e S )
In this paper, S will mean a semigroup. If A is a nonempty subset of S, A is called an ideal of S if AS, SA ~_ A [4]. We denote by I the unit interval [0, 1]. For x,y ~ I, x V y = max{x, y} and x A y = min{x,y}. All fuzzy sets of a set X are maps p : X ~ I. A fuzzy set # of a semigroup S is called fuzzy ideal if (Vx, y e S)
~t(xy) i- p(x) V p(y) [5].
Clearly, the set of all fuzzy ideals of S, denoted by FI(S), is a distributive lattice with respect to the meet "A" and union " V " defined as follows:
(Vx e S)
(~ v
la2)(x) = ~ ( x ) v/~2(x),
(~1 A ~/2)(X) = ~/I(X) A ~/2(x). For the lattice FI(S), it has the greatest element p : p(x) = 1 (Vx e S), denoted by is, and has the least element p: p(x) = 0 (Vx ~ S) denoted by 0 if S has no zero element, otherwise, the least element is p: /~(x) = 0 (V(0 # ) x ~ S) denoted by 0s. If S has a zero element 0, then it is not difficult to show that #(0)/>/4x) (Vx e S). In this paper, we define p(0) = 1. All fuzzy relations on a set X are maps: X × X I. If 0 and 4) are two fuzzy relations on a set X, then 0 ~< 4) means that O(x, y) <~ 4)(x, y) (V x, y ~ X).
O(x, y) <~O(zx, zy), (0(x, y) <~O(xz, yz)). Definition 2.3 (Samhan [12]). A fuzzy equivalent relation 0 on a semigroup S is called fuzzy congruence if it is fuzzy compatible.
Proposition 2.4 (Samhan [12]). Let 0 be a fuzzy equivalent relation on a semigroup S. Then 0 is called fuzzy congruence if and only if O is fuzzy left and right compatible. We denote the set of all fuzzy congruences on a semigroup S by FC(S). Then Samhan [5] proved that FC(S) is a complete lattice with respect to " ' " and " + " defined as follows:
(VO, 4) ~ FC(S)) 0+4)=(0v4))
o, 0 . 4 ) = 0 A 4 ) .
Furthermore,
0 + 4) = (0o 4))°0. Let elements Vs and As be fuzzy congruences on S defined by Vs(x,y)= 1 for all x, y e S , and A s ( x , y ) = O if x ¢ y and A s ( x , y ) = l , if x = y . Then it is clear that Vs and As are the greatest
£ - Y . Xie / Fuzzy Sets and Systems 102 (1999) 353 359
element and least element of the lattice (FC(S), +, .), respectively.
(2) If x # y, then
O. o O,(x, y)
Definition 2.5. A semigroup S is called fuzzy congruences free, if S has no fuzzy congruences other than Vs and As.
=
Definition 2.6. A semigroup S is called fuzzy 0simple, if S 2 ~ {0} and 0s and ls are the only fuzzy ideals.
= O.(x, y) v
V
(O.(x,z) AO.(y,z))
zeS-{x,y}
v (O.(x, x) A O.(x, y)) V (0,,(x, y) A O.(y, y)) V
(Va, b e S )
(/~(x) A/~(z) A ~(z) A #(y))
z ~ S - Ix, y}
V
<-GO,(x, y) V z~S
Let 0 be a fuzzy congruence on S, we denoted by S/O the set {aO:a ~ S} of all fuzzy congruence classes of S. We define a binary operation "*" on S/O as follows:
355
(#(x)
A it(y))
',x,y l
= G(x, y) V O.(x, y) = O.(x, y). By (1) and (2) 0, is transitive. Furthermore, 0, is left compatible. In fact, for x, y, c e S, if cx = cx, then
O.(cx, cy) = 1 > G(x, y).
aO*bO=(ab)O.
Then S/O is a semigroup [1, 7]. For other definition and terminology not given in this paper, we employ those in [2, 4, 11, 12].
If cx ./= cy, then x -~ y. Thus,
O.(cx, cy) = ~(cx) A #(cy) > #(x)
A ~(y) =
O.(x, y).
In the same way, we can also prove that O. is right compatible. []
3. Fuzzy Rees congruences Let/~ be a fuzzy ideal of a semigroup S. Then, we can assign a fuzzy relation 0, on S defined as follows: (Vx, y e S )
O.(x,y)=tt(x)Att(y),
ifx=~y,
O.(x,x)= 1, i f x = y . Clearly, 0, is well-defined. Moreover, we have
Proposition 3.1. Let p be a fuzzy ideal of a semigroup S. Then O, is a fuzzy congruence on S. Proof. By definition of 0,, it is obvious that 0, is reflexive and symmetric. Since for all x, y e S,
0.o O.(x, y) = V (O.(x, z) A O.(z, y)). zES
We consider the cases: (1) If x = y, then
O, o O,,(x, x) = V (0,(x, z) >>,O,(x, x)) =- 1. z~S
Thus, 0, o O,(x, x) = O,(x, x).
In this paper, we shall call a fuzzy congruence 0, determined by a fuzzy ideal /~( ~ 0) of S a fuzzy Rees congruence. Let p be a fuzzy ideal of a semigroup S, and let
supp# = {x e S:/t(x) = 1}. Then it is not difficult to verify that supp ~ is an ideal of S. Similar to the case of crisp Rees congruences on S, we have Theorem 3.2. Let i~ be a fuzzy ideal of a semigroup S. Let sd be the set of ideals of S containing supp I~ and let ~ be the set of ideals of the quotient semigroup (S/O,, *). Then the mapping
f:J~-"-~JO u ( J e ~ ) is an inclusion-preserving bijection from ~ onto ~. Proof. Let J be an ideal of S. Then it is clear that JO. is an ideal of the semigroup (S/O..*). Let J 1 , J 2 6 .N' and J1 4= J2. Then there exists a ~ S such that a e J1 \J2 or a ~ J 2 \ J 1 . Ifa ~ J1 \ J 2 , then we have JlO. =/=J20.. In fact, if JlO. = J20,, then
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X.-Y. Xie / Fuzzy Sets and Systems 102 (1999) 353-359
there exists b ~ J2 such that aO. = bO.. Since a ¢ J 1 , then we have a # b. Thus, (aOu)(b) = O.(a, b) = #(a) A #(b) = (bO.)(b) = 1.
Proof. Suppose that #1,#2 i~ FI(S) and #1 # #2, then there exists x ~ S such that # 1 ( x ) ~ #2(X). Obviously, x # 0. Thus, O.I(X , 0) = #1(X) A #1(0) = #l(X),
It implies that # ( a ) = # ( b ) = 1. Then a E s u p p # J2. Impossible. In the same way, if a ~ J 2 \ J 1 , then we also have JlO. # J20.. This proves that the mapping f is injective. Let KO. be an ideal of S/O. and
O,2(x, 0) = r e ( x ) A #2(0) = #2(x). It follows that 0., # 0.2. Therefore, the mapping g is injective. It is easily seen that g is orderpreserving. []
K1 = {x ff S: xO, E KO.}.
Then, for all x ~ K1,
4. Fuzzy Rees congruence semigroups
(Sx)O. = SOu o xO. ~_ SOu o KO. ~_ KO u.
Thus, Sx ~ K1. Similarly we have x S ~ K I . It follows that K1 is an ideal of S. On the other hand, s u p p # ~_ K I . Indeed: For any a ~ s u p p # and let x ~ g x , we have (A) If a = ax, obviously, a e K l . (B) If a ~ ax, for all z 6 S, we consider the cases (1) If z # a, ax, then (aO.)(z) = O.(a, z) = #(a) A #(z)
(since #(a) = 1)
= #(ax) A #(z)
= ((ax)O,)(z). (2) If z = a, then
Definition 4.1. A semigroup S is called fuzzy Rees congruence semigroup (abbr. FRC-semigroup) if every fuzzy congruence on S is a fuzzy Rees congruence.
Proposition 4.2. Let S be a FRC-semigroup. Then (1) S has a zero element 0; (2) I f 0 is a f u z z y congruence on S, then O, = O, where #(x) = O(x, O) for all x ~ S. Proof. To prove (1), observe that As is a fuzzy congruence on S. Since S is a FRC-semigroup, there exists a fuzzy ideal # ( # 0 ) of S such that As = 0,. Since p # 0, we have # ( x ) # 0 for some x e S. Then for any y e S such that y # x, we have As(y, x) = O,(x, y) = #(x) A #(y) = O.
(aO.)(a) = 1 = #(a) A #(z) = ((ax)O.)(a).
(3) If z = ax, by (2), we have (aOu)(ax)= ((ax) O.)(ax). It follows that (aO,)= (ax)O, ~_ KO,. By definition of K1, then a e K1. Obviously, KIOu = KOu. It proves that f is onto. The inclusion-preserving o f f is only a routine verification. []
Proposition 3.3. Let S be a semigroup with O. Then
Thus, #(y) = 0. Since # is a fuzzy ideal of S, for all z E S, we have #(zx) >1 p(x), #(xz) ~ #(x), this implies that z x = x z = x. This proves that x is a zero element of S. To prove (2), let 0 be a fuzzy congruence on S. Then, by hypothesis, there exists a fuzzy ideal # ( # 0 ) such that 0 = 0,. Let (Vx ~ S),
v(x) = O(x,O)
(by (1)).
Then for all y e S,
the mapping
v(yx) = O(yx, O) >t O(x, O) = v(x), g : # w-~O.
(# ~ VI(S))
v(yx) = O(yx, 0)/> 0(y, 0) = v(y), is an order-preserving FC(S).
injective from FI(S) into v(O)
= o(o, o ) =
1.
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X.-Y. Xie / Fuzzy Sets and Systems 102 (1999) 353-359
Thus, v is a fuzzy ideal of S. Moreover, we have v = #. In fact, since for any y e S such that y ~ 0,
vCy) = 0(y, 0) = 0,(y, 0) = #(y) Then, we have # = v.
A
< (o//, o o,,2)~(x, y) = (o//, v o//)(x, y)
#(0) = #Cy).
[]
(2) and
Theorem 4.3. Let S be a FRC-semigroup. Let FI(S)
be the lattice of all fuzzy ideals of S and FC(S) the lattice of all fuzzy congruences on S. Then the mapping g:#~-+O.
#2(X) A #2(Y) = O//2(x, Y) ~ CO#, o 0u2)(x' y)
#1(x) A #2(Y) ~< #,(xy) A #2(xy) A #1(X) A #2(Y),
(3) #2(x) A ill(Y) ~< #l(xy) A #2(xy) A #2(x) A #I(Y). (4)
(peEl(S)) We consider (3) in the cases. (1) If xy = x, then
is an isomorphism from FI(S) onto FC(S). Proof. By Propositions 3.3 and 4.2, we have the mapping g is an order-preserving bijective. T o prove that g is lattice-order preserving, we only show that
#1(X) A #2(Y) ~< #1(x) A #2(x) A #2(Y)
(VPl, #2 ~: FI(S))
0.~ A 0.2 = 0//, A//~,
(2) If xy = y, then
0//, V 0//2 = 0., v//~.
#1CX) A #2(y) < #1(Y) A #2(Y) A #1(x)
~< #2(x) A #2@) <~ (0., V O,2)(x, y)
(by (2)).
Since for all x, y e S such that x 4: y, we have
4 #l(x) A #l(y)
(0//, A O//)(x, y) = O.,(x, y) A O//2(x, y)
~< (0//, V 0,2)(x, y) (by C1)).
=
#1(X)
A
#1(Y)
A
#2(X)
A
#2(Y)
= (#1 A m)Cx) A (#1 A m)(Y)
(3) If xy # x, y, then
#1(x) A #2(Y) <~ O//~(x, xy) A O//2(xy, y)
< C0.,o %)(x, y)
= 0//, A /12(X' Y)
and obviously, 0,, A O//~(x, x) = 0//, A ,2(x, x) = 1. This implies that 0//, A 0//2 = 0//, A//~. Since Pl, #2 ~< #1 V #2, by T h e o r e m 4.3, then we have 0//1 ~ 0/A1 V ~/2and 0//2 ~< 0//, v//2" Thus, 0//, V 0//2 <~ 0//, v u~. On the other hand, for all x, y e S such that x va y,
~< ( % ° 0.) ~(x, y) = (0., V O//2)(x,y)
In the same way, we consider (4) in the cases (4) If xy = x, then
#~(x) A #~Cy) < #l(x) A #,Cy) <~ (0//, V O//2)(x, y)
0,, v ,2( x, Y) ----(#1 V #2)(x) A (ill V #2)(Y)
= (#,(x) V #2(x)) A (#1(Y) V #2(Y)) =
(#l(x)
A
V (#ICy)
#I(Y)) A
V
(#1(x)
A
#2(X)) V (#2(x)
#2(Y)) A
#2(Y)).
(by (2)).
(by (1)).
(5) If xy = y, then
#2(X)
A
#I(Y) < #2(X)
A
#2(Y)
<~ CO~~,V O.2)(x, y)
(by (2)).
Furthermore,
(6) If xy # x, y, then
#l(x)
#2(x) A #I(Y) <~ O//,(y, xy) A O.2(xy, x)
A
#iCY) = 0//,(x, y) ~< (0/2, o 0//2)(x. y)
< (%0 %)~(x, y) = (0//, v %)(x, y), (1)
< (0//, v 0//)(y, x) ~< (0//, o % ) ~ ( y , x) = (0//, v O//)tx, y).
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X.-Y. Xie / Fuzzy Sets and Systems 102 (1999) 353-359
In view of all situations, we have
let 0 be a fuzzy congruences on T. Define
Oul v u~(x, Y) <~(0ul V Ou2)(x, y),
O(x, y) = O(f(x),f(y))
thus, 0u, v ~2 ~ (0., V 0~2). Consequently, we prove that
Then we have q~ is a fuzzy congruence on S. In fact, the reflexivity and symmetry of~b are obvious, since
0., v .2 = (0., V 0.2).
~o q~(x, y) = V (q~(x, z) A ¢(z, y))
[]
(Vx, y ~ S).
zeS
Since FI(S) is a distributive lattice, by T h e o r e m 4.3, we have Corollary 4.4. Let S be an FRC-semigrou p. Then
the lattice FC(S) is distributive. Theorem 4.5. Let S be a FRC-semigroup and S 2 ~ {0). Then S is fuzzy congruences free, if and
= V (o(f(x),f(z)) A O(f(z),f(y))) z~iS
V (o(f(x), z) A O(z,f(y))) g~S
= (0o O)(f(x),f(y)) O(f(x), (f(y))) = 49(x, y).
only if S is fuzzy O-simple.
Then it implies that ~b is transitive. On the other hand,
Proof. (=~) Suppose that S is fuzzy congruences
(Vx, y, z, t e S)
free and # ( ¢ 0 ) is a fuzzy ideal of S. Then 0, is a fuzzy congruence on S. By Definition 2.5, then either 0u = x7s or 0, = As. If 0u = Vs, then for all x e S (x ¢ 0),
4)(xz, yt) = O(f (xz),f (yt)) = 0 (f(x)f(z),f(y)f(t))
0.(0, x)= Vs(0, x)= 1 =u(0) A ~(x)=~(x).
>~O(f(x),f(y)) A O(f(z),f(t))
It implies that # = ls. If 0. = As, then, by Proposition 4.2(1), for all x e S (x ¢ 0),
= 4,(x, y) A 4)(y, t).
0.(0, x ) = ~ ( x ) = A s ( O , x)=0. It implies that 0, = 0s. ( ~ ) Suppose that S is fuzzy 0-simple and let 0 be a fuzzy congruence on S. By Definition 2.6, there exists a fuzzy ideal # ( ¢ 0 ) of S such that 0 = 0,. Since S is fuzzy 0-simple, then either p = 0s or kt = ls. If # = ls, then for all x, y e S (x ~ y),
Therefore, q~ is compatible. Since q~ is a fuzzy congruence on S, by hypothesis, then there exist a fuzzy ideal p ( ¢ 0 ) of S such that ~b = ~b,. Let f(p):f(/a)(x) =
V
/t(z),
Vx e S.
zEf-l(X)
Then it is clear that f(~t) is a fuzzy ideal of T. We now claim that 0 = 0z¢u). Since for all x, y e T, if x = y, then O(x,y)=Of(,)(x,y)= 1; if x # y, then there exist a,b E S such that x = f ( a ) , y = f ( b ) . Moreover,
O(x, y) = O.(x, y) = u(x) A u(y) = 1, it implies that 0 = Vs. I f p = 0s, by a routine verification, we have 0 u = As. [] Proposition 4.6. The homomorphic image of a FRC-
O(x, y) = O(f(a),f(b))
(obviously, a # b)
= 49(a, b) = ¢ , ( a , b)
= ,(a) A #(b)
semigroup is a FRC-semigroup, Proof. Let S be a FRC-semigroup and let f : S w-, T
be a h o m o m o r p h i s m of S onto a semigroup T and
=f(/~)(x) Af(/~)(y) = Of(u)(x, y).
X-Y. Xie / Fuzzy Sets and Systems 102 (1999) 353-359
Thus, we have 0 ~< OA.~. On the other hand,
Os(u)(x, y) = f(p)(x) A f(l~)(y) =
V
zef
A
l(x)
V
:
V
w~f-I(y)
(/~(z)
z E f t ( x ) . w e f l(y)
A/~(w))
(obviously, z # w)
V
=
zef-t(x),wef
w)
a(y)
V
=
4)(z,w)
z~ f-l(x), we f - l ( y )
=
~/ zef
O(f(z),f(w))
l ( x ) , w e f l(y)
<~O(x, y). Thus, OI~.) ~< O. Consequently, Oy~.) = O.
[]
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