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On fuzzy ideals and fuzzy bi-ideals in semigroups
Fuzzy Sets and Systems 5 (1981) 203-215 North-Holland Publishing Company
ON FUZZY IDEALS IN S E M I G R O U P S
AND
FUZZY
BI-IDEALS
Nobuaki K U R O K I Department of Mathematics, College of Science and Technology, Nihon University, Funabashi-shi, Chiba-ken, Japan Received August 1978 Revised December 1979 In this paper we give some properties of fuzzy ideals and fuzzy bi-ideals of semigroups, and characterize semigroups that are (leftj duo. (left) simple and semilattices of subsemigroups in terms of fuzzy ideals and fuzzy bi-ideals.
Keywords: Fuzzy ideal, Fuzzy bi-ideal, Fuzzy duo, lntra-regular, Completely regular. Semilattice of left (right) simple semigroup, Fuzzy simple.
1. Introduction The concept of a fuzzy set, introduced in Zadeh [9], was applied to the elementary theory of groupoids and groups in Rosenfeld [7]. In the present paper we shall apply this to the theory of semigroups. We shall give some properties of fuzzy ideals and fuzzy hi-ideals, and characterize a semigroup which is left (right) simple, left (right) duo and a semilattice of left (right) simple semigroups or another type of semigroups in terms of fuzzy ideals and fuzzy hi-ideals.
2. Definitions and preliminary iemmas Let S be a semigroup. By a subsemigroup of S we mean a non-empty subset A of S such that A~_A 2
and by a left (right) ideal of S we mean a non-empty subset A of S such that
SA c_A
(ASc_A).
By two-sided ideal, or simply ideal, we mean a subset of S which is both a left and a right ideal of S. A semigroup S is called left (right) simple if S itself is the only left (right) ideal of S. S is called simple if it contains no proper ideal. A function 8 from S to the unit interval [0, 1] is called ~1fuzzy set in S. A fuzzy set 8 in S is called a fuzzy subsemigroup of S if ~5(xy) >~min{~(x), ~(y)} 203
204
N, K u r o k i
for all x, y e S, and is called a fuzzy left (right) ideal of S if ~(xy) >~ ~$(y) (~(xy) ~> 6(x)) for all x, y ~ S. A fuzzy set 6 in S is called a fuzzy ideal of S if it is both a fuzzy left and a fuzzy right ideal of S. Then, as is easily seen, 6 is a fuzzy ideal of S if and only if ~(xy) ~>max{6(x), ~(y)} for all x, y e S, and it is clear that any fuzzy left (right, two-sided) ideal of S is a fuzzy subsemigroup of S. Lemma 2.1 [7, Proposition 2.2]. Let A be a non-empty subset of a ~emigroup S
and ~$A the characteristic [unction c,f A. Then (1) A is a st~,bsemigroup of S if and only if ,SA is a fuzzy subsemigroup of S. (2) A is a left" (right) ideal of S if and only if ~A is a fuzzy left (right) ideal of S. (3) A is an sfleal of S if ,,~n¢t only if ~A is a fuzzy ideal of S. A subsemigroup A of a semigroup S is called a bi-ideal of S if
A S A ~_ A. A fuzzy subsemigroup ~i of a semigroup S is called a fuzzy bi-ideal of S if ~5(xyz)-~ min{~i(x), ~$(z)} for all x, y and z ~ S (see [2]). L e m m a 2.2 [2, T h e o r e m 1]. Let A be a non-empty subset of a semigroup S and ~SA
the characteristic [unction of S. Then A is a bi-ideal of S if and only if ~SA is a fuzzy bi-ideal of S. L e m m a 2.3. Every fuzzy left ideal (fuzzy right ideal and fuzzy ideal) is a fuzzy
hi-ideal of S. Proof. Let ~5 be any fuzzy left ideal of S and x, y and z any elements of S. Then ~(xyz) = ~((xy)z) i> ~(z) ~>min{~5(x), ~i(z)}. Thus ~ is a fuzzy bi-ideal of S. Similarly we can see the other cases hold. Remark. Every left (right, two-sided) ideal of a semigroup S is a bi-ideal of S.
3. Regular semigroups
A semigroup S is called regular if, for each element a of S, there exists an element x in S such that d = ~xa.
On fuzzy ideals and fuzzy bi-ideals
205
A semigroup S is called left (right) duo if every left (right) ideal of S is a two-sided ideal of S. A semigroup S is called duo if it is both left and right duo. A semigroup S is called fuzzy left (right) duo if every fuzzy left (right) ideal of S is a fuzzy ideal of S. A semigroup S is called fuzzy duo if it is both fuzzy left and fuzzy right duo. T h e n we have the following.
Theorem 3.1o For a regular semigroup S the following conditions are equivalent. (1) S is left duo. (2) S is fuzzy left duo. Proof. First assume that S is left duo. Let ~5 be any fuzzy left ideal of S, and a, b any elements of S. Then, since the left ideal Sa is a two-sided ideal of S, and since S is regular, we have
a b ~ ( a S a ) b ~_(Sa)S~_Sa. This implies that there exists an element x in S such that
ab = xa. Then, sil, ce c5 is a fuzzy left ideal of S, we have
~(ab)= ~(xa)>~,5(a). This means that ~5 is a fuzzy right ideal of S, and so ~5 is a fuzzy two-sided ideal of S. Thus we obtain that S is fuzzy left duo, and that (1) implies (2). Conversely, assume that S is fuzzy left duo. Let A be any left ideal of S. Then it follows from L e m m a 2.1 that the characteristic function ~SA of A is a fuzzy left ideal of S. Thus by the assumption it is a fuzzy ideal of S. Since A is non-empty, it follows from L e m m a 2.1 that A is an ideal of S. Therefore we obtain that S is left duo, and that (2) implies (1). This completes the proof of the theorem. The left-right dual of Theorem 3.1 reads as follows:
Theorem 3.2. For a regular semigroup S the following conditions are equivalent. (1) S is right duo. (2) S is fuzzy right duo. The combined effect of these two theorems is as follows:
Theorem 3.3. For a iegular semigroup S the following conditions are equivalent. (1) S is duo. (2) S is fuzzy duo. Theorem 3.4. For a regular semigroup S the following conditions are equivalent. (1) Every bi-ideal of S is a right ideal of S. (2) Every fuzzy bi-ideal of S is a fuzzy right ideal of S. Proof. First assume that (1) holds. Let 6 be any fuzzy bi-ideal of S, and a, b any
N. Kuroki
206
elements of S. Then, as is easily seen, aSa is a bi-ideal of S. Then by the assumption aSa is a right ideal of S. Then, since S is regular,
ab ~ ( aSa )S ~_ aSa. This implies that there exists an element x in S such that
ab = axa. Then, since 8 is a fuzzy bi-ideal of S, we have
,5(ab) = ,5(axa)>~min{,5(a), ~(a)} = ~(a). This means that 8 is a fuzzy right ideal of S. Thus we obtain that (1) implies (2). Conversely, assume that (2) holds. Let A be any bi-ideal of S. Then it follows from Lemma 2.2 that the characteristic function ~A of A is a fuzzy bi-ideal of S. Thus by the assumption it is a fuzzy right ideal of S. Then, since A is non-empty, it follows from Lemma 2.1 that A is a right ideal of S. Thus (2) implies (1). This completes the proof. The lefi-right dual of Theorem 3.4 is proved in an analogous way:
Theorem 3.5. For a regular semigroup S the following condihons are equivalent. (1) Every hi-ideal of S is a left ideal of S~ (2) Every fuzzy hi-ideal of S is a fuzzy left ideal of S. The following theorem is the immediate consequence of Theorems 3.4 and 3.5. Theorem 3.6. For a regular semigroup S the following conditions are equivalent. (1) Every hi-ideal of S is a two-sided ideal of S. (2) Every fuzzy hi-ideal of S is a fuzzy two-sided ideal of S. Corollary 3.7. Let S be a regular duo semigroup. Then every jfuzzy bi-ideal of S is a fuzzy right ideal of S. Proof. It follows from [5, Theorem 30] that every bi-ideal of a regular left duo semigroup is a right ideal of it. From this and from Theorem 3.4 it follows that every fuzzy bi-ideal of S is a fuzzy right ideal of S. Corollary 3.8. Let S be a semigroup which is a semilattice of groups. Then every fuzzy hi-ideal of S is a fuzzy two-sided ideal of S. Proof. It follows from [4, Theorem 4] that every bi-ideal of such a semigroup S is a two-sided ideal of S. Therefore it follows from Theorem 3.6 that every fuzzy bi-ideal of S is a fuzzy two-sided ideal of S. We denote by L[a] (J[a]) the principal left (two-sided) ideal of a semigroup S generated by a in S, that is,
L[a]={alUSa, J[a] = {a} U S a U aS U SaS.
On fuzzy ideals and fuzzy hi-ideals
207
As is well-known [1, L e m m a 2.13], if S is a regular semigroup, then L [ a ] = Sa for every a ~ S. A semigroup S is called right (left) zero if xy=y
(xy=x)
for all x, y ~ S. T h e n we have the following. "I]heorem 3.9. For a regular semigroup S lhe following conditions are equivalent. (1) The set of all idempotent elements of S forms a left zero subsernigrov.p of S. (2) For every f u z z y left ideal ,5 of S,
~5(e) = ~5(f) for all idempoten:' elements e and f of S. ProoL First asswrne that the set Es of all idempotents of S forms a left zero subsemigroup of S. Let e and f be any elements of Es, and (5 any fuzzy left ideal of S. Then, since
ef=e
and
fe=f,
we have
,5(e) = 8(ef)>~ cS(f) = ,5(re)>I 8(e), and so 8(e) = cS(f). Thus (1) implies (2). Conversely, assume that (2) holds. Since S is regular, Es is non-empty. Let e and f be any elements of Es. Then i: follows from L e m m a 2.1 that the characteristic function (SLur] of the left ideal L[j'] of S is a fuzzy left ideal of S. Thus we have dSLtn(e) = B ' [ n ( f ) = 1, and so
e ~ L If] = Sf. Then for some x ~ S we have
e=xf=xff=ef. This means that Es is a left zero scmigroup. Thus (2) implies (1). This completes the proof.
Corollary 3.10. For an idempotent semigroup S the following conditions are equivalent. (1) S is left zero. (2) For every f u z z y left ideal ,5 of S, 8(e) = ~(f)
for all e, f ~ S.
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N. Kuroki
The left-right dual of Theorem 3.9 reads as follows: qt~eorem 3.11. For a regular semigroup S the following conditions are equivalent. (1) Es forms a right zero subsemigroup of S. (2) For every fuzzy right ideal 8 of S, ,5(e) = rS(f) for all e, f e Es. Corollary 3.12. For an idempotent semigroup S the following conditions are equivalent. (1) S is right zero. (2) For every fuzzy right ideal 8 of S, 3(e)=8([) for all e, f ~ S. Lemma 3.13 [2, Theorem 2]. For a semigroup S the following conditions are equivalent. (1) S is a group. (2) Every fuzzy bi-ideal of S is a constant function. Theorem 3.14. For a regular semigroup S the following conditions are equivalent. (1) S is a group. (2) For every fuzzy hi-ideal 8 of S,
8(e)= ~5(f) for all idempotents e and f of S. Proof. Assume that (1) holds. Let 8 be any fuzzy bi-ideal of S. Then it follows from Lemma 3.13 that 6 is a constant function. Therefore for any idempotents e and f of S,
,~(e)= ,5(f). Thus (1) implies (2). Conversely, assume that (2) holds. Let e and f be any idempotents of S. We denote by B[x] the principal bi-ideal of S generated by x in S, that is, B[x] = {x} U {x 2} U xSx
[ l, p. 84].
As is well-known, if S is regular, then
B[x]= xSx. Then it follows from Lemma 2.2 that the characteristic function ~SB[r] of the bi-ideal B[f] of S is a fuzzy bi-ideal of S. Sii~ce [ e B[f], we have 8~,m(e) = 8Bm(f)= 1.
On fuzzy ideals and fuzzy bi-ideals
209
T h e n we have e ~ B [ f ] = fSf,
which implies that e = fxf
for s o m e x ~ S. Similarly, we obtain that f = eye for some y ~ S. Then we have e = f x f = f x f f = ef = eeye = eye = f.
This means that, since S is regular, Es is non-empty and S contains exactly one idempotent. Then it follows from [ 1, p. 33 (Ex. 4)] that S is a group. Therefore (2) implies (1). This completes the proof of the theorem.
4. intra-regular semigroups
A semigroup S is called intra-regular if, for each element a of $, there exist elements x and y in S such that a = xa2y. For characterizations of such a semigroup, see [1, T h e o r e m 4.4] and [6, 11.4.5 Theorem]. Now we give a new characterization of intra-regular semigroups. T h e o r e m 4.1. For a semigroup S the f o l l o w i n g conditions are equivalent. (1) S is intra-regular. (2,) For every f u z z y ideal 8 of S, 8(a) = 8(a 2) holds for all a ~ S.
1~oot. First assume that (1) holds. Let 8 be any fuzzy ideal of S and a any element of S. q't~en, since S is intra-regular, there exist elements x and y in S such that a = xa2y.
Then, since 8 is a fuzzy ideal of S, we have 8(a) >I 8(xa2y ) >I t$(xa 2) >t 8( a 2) I> 8(a ), and so we have 8(a) = tr(a2). Thus (1) implies (2). Conversely, assume that (2) ho|ds. Then it follows from L e m m a 2.1 that the characteristic function ~jta2~ of the ideal J [ a 2] of S is a fuzzy
210
N. Kuroki
ideal of S. Since a 2 e j[a2], we have 8jt,,21(a) = 8jt,,21(a 2) = 1. This implies that a e J [ a 2] = {a 2} U Sa 2 U a2S U Sa2S. Then it is easily seen that S is intra-regular. Thus (2) implies (1). This completes the proof.
Theorem 4.2. Let S be an intra-regular semigroup. Then for every f u z z y ideal 8 of s, 8(ab)=8(ba) holds for all a, b ~ S. P r o o | . Let 8 be any fuzzy ideal of S and a and b any elements of S. Then by Theorem 4.1 we have 8 ( a b ) - 8((ab) 2) = 8 ( a ( b a ) b ) ~ 8(ba) = 8((ba)2) = 8(b(ab )a ~>18(ab), and so we have 8(ab) = 6(ba). This completes the proof.
5. Completely regular semigroups A semigroup S is called completely regular if, for each element a of S, there exists an element x in S such that a = axa
and
ax = xa.
A semigroup S is called left (right) regular if, for each element a of S, there exists an element x in S such that a = xa 2 (a = aEx). For characterizations of such a semigroup, see [1, Theorem 4.2].
Theorem 5.1. For a semigroup S the following conditions are equivalent. (1) S is left regular. (2) For every f u z z y left ideal 8 of S, 8(a) = 8(a 2) holds for all a ~ S.
On fuzzy ideals and fuzzy bi-ideals
211
Proof. First assume that (1) holds. Let /5 be any fuzzy left ideal of S and a any element of S. Then, since S is left regular, there exists an elerrent x in S such that a = x a 2.
Then we have /5(a ) = ~5(xa2) ~>/5(a 2) ~>/5(a ), and so we have /5( a ) = /5(a2). Thus (1) implies (2). Conversely, assume that (2) holds. Let a be any element of S. Then it follows from Lemma 2.1 that the characteristic function /sLi,-~l of the left ideal L[a 2] of S is a fuzzy left ideal of S. Since a2~ L[a2], we have ~SLta21(a) = ~Sl~t,21(a2) = 1. This implies that a ~ L[a 2] = {a 2} U S a 2. This means that S is left regular, and so (2) implies (1). This completes the proof. The left-right dual of Theorem 5.1 reads as follows:
Theorem 5.2. For a semigroup S the following conditions are equivalent. (1) S is right regular. (2) For every fuzzy r',.ght ideal ~ of S, tS(a)=~(a 2) holds for all a c S. We recall tiaat a semigroup S is completely regular if and only if it is left and right regular [1, Theorem 4.3]. The equivalence of (1) and (2) in the following theorem is due to the author [2], and the equivalence of (1) and (3) follows fro,,,-a Theorem 5.1 and Theorem 5.2.
Theorem 5.3. For a semigroup S the following conditions are equivalent. ( l ) S is completely regular. (2) For every fuzzy bi-ideal ~ of S, /5(a ) =/5(a 2) holds for all a e S. (3) For every fuzzy left ideal ~ and for every fuzzy right ideal V o[ S, /3(a)=/3(a 2)
hold for all a ~ S.
and
v(a) = v(a 2)
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N. Kuroki
6. Semigroups that are semilattices of left (right) simple semigroups The following characterization of a semigroup that is a semilattice of left simple semigroups is due to Lajos [3] and Salt6 [8].
Lemmm 6.1 For a semigroup S the following conditions are equivalent. (1) S is a semilattice of left simple semigroups. (2) S is left regular and A B = B A for any two left ideals A and B of S. (3) S is left regular and every left ideal of it is a two-sided ideal of S. The following theorem can be proved in a similar way as in the proof of Theorems 3.1 and 3.2. So we omit the proof.
Theorem 6.2. For a left (right) regular semigroup S the following conditions are equivalent. (1} S is left (right) duo. (2) S is fuzzy left (right)duo. The characterization of a semigroup that is a semilattice of left simple semigroups can be found in [6, II.4.9 Theorem]. Now we give a new characterization of such a semigroup.
21heorem 6.3. For a semigroup S the following conditions are equivalent. (1) S is a semilanice ~gf left simple semigroups. (2) For every fuzzy e't ideal ~ of S, 8(a) = ~5(a 2)
:~zd
8(ab) = 8(ba)
hold for all a, b ~ S. ProoL Assume that S is a semilattice of left simple semigroups. Let 8 be any fuzzy left ideal of S. Then, since S is left regular by Lemma 6.1, it follows from Theorem 5.1 that 8(a) = 8(a 2) holds for all a ~ S. It follows from Lemma 6.1 and Theorem 6.2 that ~ is a fuzzy two-sided ideal of S. Let a and b be any elements of S. Then we have
8(ab) = ~((ab) 2) = 8(a(ba)b)>16(ba). Similarly, we have
~5(ba ) >~cS(ab ). "lhus we obtain that
~ ( a b ) - ~(ba) and that (1) implies (2). Conversely, assume that (2) holds. Then it .follows from the first condition of (2) and from Theorem 5.1 that S is left regular. Let A and B
On fuzzy ideals and fuzzy bi-ideals
213
be any left ideal of S and ab (a ~ A , b e B ) be any element of A B . Then the characteristic function `sLtb,~l of the left ideal L[ba] is a fuzzy left ideal of S by Lemma 2.1. And since ba ~ L[ba], we have
`sLtbal( ab ) = `sLtba j( ba ) = 1. This implies that
ab ~ L[ba ] = {ba } O Sba ~ B A LJ S B A ~_ B A , and so we have
A B c_ B A . Similarly, it can be seen that the converse inclusion holds. Thus we obtain that
AB-
BA.
Then it follows from Lemma 6.1 that S is a semilattice of left simple semigroups. Therefore (2) implies (1). This completes the proof of the theorem. The left-right dual of Theorem 6.3 reads as follows:
Theorem 6.4. For a semigroup S the following conditions are equivalent. (1) S is a semilattice of right simple sem~groups. (2) For every f u z z y right ideal ,5 of S, `5(a) =`5(a 2)
and
`5(ab)-`5(ba)
hold ]:or all a, b ~ S.
7. Left (right) simple semigroups A semigroup S is called fuzzy left (right) simple if every fuzzy left (right) ideal of S is a constant function, and is called fuzzy simple if every fuzzy ideal of S is a constant function. Then we have the following.
Theorem 7.1. For a semigroup S the following conditions are equivalent. (1) S is left simple. (2) S is f u z z y left simple.
Proof: First assume that (1) holds. Let `5 be any fuzzy left ideal of S and a and b any elements of S. Then, since S is left simple, it follows from [1, p. 6] that there exist elements x and y in S such that
b=xa
and
a=yb.
Then, since `5 is a fuzzy left ideal of S, we have 6(a) = 6(yb)>I 6 ( b ) = 8(xa)>t 8(a),
N. Kuroki
214 and so we have
6(a)=6(b). Since a and b are any elements of S, this means that 5 is a constant function. Thus we obtain that S is fuzzy left simple, and that (1) implies (2). Conversely, assume that (2) holds, and let A be any left ideal of S. By Lemma 2.1, 6A is a fuzzy left ideal of S, hence a constant function, and since A is non-empty, the constant is 1. Thus every element of S is in A, and so S is left simple. Thus (2) implies (1). The following two theorems can be seen in a similar way as in the proof of Theorem 7.1. Theorem 7.2. For a semigroup S the following conditions are equivalent. (1) S is right simple. (2) S is fuzzy right simple. Theorem 7.3. For a semigroup S the following conditions are equivalent.
(1) S is simple. (2) S is fuzzy simple. As is well-known, a semigroup S is a group if and only if it is left and right simple. From this and from Theorems 7.1 and 7.2, we have the following theorem. Theorem 7.4. For a semigroup S the following conditions are equivalent.
(1) S is a group. (2) S is both fuzzy left and fuzzy right simple. Theorem 7.$. Let S be a left simple semigroup. Then every fuzzy bi-ideal of S is a
fuzzy right ideal of S. Proof. Let 6 be any fuzzy bi-ideal of $, and a and b any elements of $. Then, since S is left simple, there exists an element x in S such that
b=xa. Then, since 6 is a fuzzy bi-ideal of S, we have
6(ab) = 6(axa)>~min{6(a), 6(a)} = 8(a). This means that ~ is a fuzzy right ideal of S. This completes the proof.
Corollary 7.6. Let S be a left simple semigroup. Then every bi-ideal of S is a right ideal of S.
On fuzzy ideals and [uzzy bi-ideals
215
Acknowledgement I express my thanks to the referees for their valuable comments and suggestions.
References [1] A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups Vol. I (Am. Math. Soc., Providence, RI, 1961). [2] N. Kuroki, Fuzzy bi-ideals in semigroups, Comm. Math. Univ. St. Pauli 28 (1979) 17-21. [3] S. Lajos, A note on completely regular semigroups, Acta Sci. Math. (Szeged) 28 (1967) 261-265. [4] S. Lajos, On (m,n)-ideals in regular duo semigroups, Acta Sci. Math. (Szeged) 31 (19701 179-180. [5] S. Lajos, Theorems on (1, 1)-ideals in semigroups, Dept. Math., K. Marx Univ. Economics (1972). [6] M. Petrich, Introduction to Semigroups (Columbus, OH, 1973). [7] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [8] T. SaitS, On semigroups which are semilattices of left simple semigroups, Math. Japon. 18 (1973) 95-97. [9] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-353.
ON FUZZY IDEALS IN S E M I G R O U P S
AND
FUZZY
BI-IDEALS
Nobuaki K U R O K I Department of Mathematics, College of Science and Technology, Nihon University, Funabashi-shi, Chiba-ken, Japan Received August 1978 Revised December 1979 In this paper we give some properties of fuzzy ideals and fuzzy bi-ideals of semigroups, and characterize semigroups that are (leftj duo. (left) simple and semilattices of subsemigroups in terms of fuzzy ideals and fuzzy bi-ideals.
Keywords: Fuzzy ideal, Fuzzy bi-ideal, Fuzzy duo, lntra-regular, Completely regular. Semilattice of left (right) simple semigroup, Fuzzy simple.
1. Introduction The concept of a fuzzy set, introduced in Zadeh [9], was applied to the elementary theory of groupoids and groups in Rosenfeld [7]. In the present paper we shall apply this to the theory of semigroups. We shall give some properties of fuzzy ideals and fuzzy hi-ideals, and characterize a semigroup which is left (right) simple, left (right) duo and a semilattice of left (right) simple semigroups or another type of semigroups in terms of fuzzy ideals and fuzzy hi-ideals.
2. Definitions and preliminary iemmas Let S be a semigroup. By a subsemigroup of S we mean a non-empty subset A of S such that A~_A 2
and by a left (right) ideal of S we mean a non-empty subset A of S such that
SA c_A
(ASc_A).
By two-sided ideal, or simply ideal, we mean a subset of S which is both a left and a right ideal of S. A semigroup S is called left (right) simple if S itself is the only left (right) ideal of S. S is called simple if it contains no proper ideal. A function 8 from S to the unit interval [0, 1] is called ~1fuzzy set in S. A fuzzy set 8 in S is called a fuzzy subsemigroup of S if ~5(xy) >~min{~(x), ~(y)} 203
204
N, K u r o k i
for all x, y e S, and is called a fuzzy left (right) ideal of S if ~(xy) >~ ~$(y) (~(xy) ~> 6(x)) for all x, y ~ S. A fuzzy set 6 in S is called a fuzzy ideal of S if it is both a fuzzy left and a fuzzy right ideal of S. Then, as is easily seen, 6 is a fuzzy ideal of S if and only if ~(xy) ~>max{6(x), ~(y)} for all x, y e S, and it is clear that any fuzzy left (right, two-sided) ideal of S is a fuzzy subsemigroup of S. Lemma 2.1 [7, Proposition 2.2]. Let A be a non-empty subset of a ~emigroup S
and ~$A the characteristic [unction c,f A. Then (1) A is a st~,bsemigroup of S if and only if ,SA is a fuzzy subsemigroup of S. (2) A is a left" (right) ideal of S if and only if ~A is a fuzzy left (right) ideal of S. (3) A is an sfleal of S if ,,~n¢t only if ~A is a fuzzy ideal of S. A subsemigroup A of a semigroup S is called a bi-ideal of S if
A S A ~_ A. A fuzzy subsemigroup ~i of a semigroup S is called a fuzzy bi-ideal of S if ~5(xyz)-~ min{~i(x), ~$(z)} for all x, y and z ~ S (see [2]). L e m m a 2.2 [2, T h e o r e m 1]. Let A be a non-empty subset of a semigroup S and ~SA
the characteristic [unction of S. Then A is a bi-ideal of S if and only if ~SA is a fuzzy bi-ideal of S. L e m m a 2.3. Every fuzzy left ideal (fuzzy right ideal and fuzzy ideal) is a fuzzy
hi-ideal of S. Proof. Let ~5 be any fuzzy left ideal of S and x, y and z any elements of S. Then ~(xyz) = ~((xy)z) i> ~(z) ~>min{~5(x), ~i(z)}. Thus ~ is a fuzzy bi-ideal of S. Similarly we can see the other cases hold. Remark. Every left (right, two-sided) ideal of a semigroup S is a bi-ideal of S.
3. Regular semigroups
A semigroup S is called regular if, for each element a of S, there exists an element x in S such that d = ~xa.
On fuzzy ideals and fuzzy bi-ideals
205
A semigroup S is called left (right) duo if every left (right) ideal of S is a two-sided ideal of S. A semigroup S is called duo if it is both left and right duo. A semigroup S is called fuzzy left (right) duo if every fuzzy left (right) ideal of S is a fuzzy ideal of S. A semigroup S is called fuzzy duo if it is both fuzzy left and fuzzy right duo. T h e n we have the following.
Theorem 3.1o For a regular semigroup S the following conditions are equivalent. (1) S is left duo. (2) S is fuzzy left duo. Proof. First assume that S is left duo. Let ~5 be any fuzzy left ideal of S, and a, b any elements of S. Then, since the left ideal Sa is a two-sided ideal of S, and since S is regular, we have
a b ~ ( a S a ) b ~_(Sa)S~_Sa. This implies that there exists an element x in S such that
ab = xa. Then, sil, ce c5 is a fuzzy left ideal of S, we have
~(ab)= ~(xa)>~,5(a). This means that ~5 is a fuzzy right ideal of S, and so ~5 is a fuzzy two-sided ideal of S. Thus we obtain that S is fuzzy left duo, and that (1) implies (2). Conversely, assume that S is fuzzy left duo. Let A be any left ideal of S. Then it follows from L e m m a 2.1 that the characteristic function ~SA of A is a fuzzy left ideal of S. Thus by the assumption it is a fuzzy ideal of S. Since A is non-empty, it follows from L e m m a 2.1 that A is an ideal of S. Therefore we obtain that S is left duo, and that (2) implies (1). This completes the proof of the theorem. The left-right dual of Theorem 3.1 reads as follows:
Theorem 3.2. For a regular semigroup S the following conditions are equivalent. (1) S is right duo. (2) S is fuzzy right duo. The combined effect of these two theorems is as follows:
Theorem 3.3. For a iegular semigroup S the following conditions are equivalent. (1) S is duo. (2) S is fuzzy duo. Theorem 3.4. For a regular semigroup S the following conditions are equivalent. (1) Every bi-ideal of S is a right ideal of S. (2) Every fuzzy bi-ideal of S is a fuzzy right ideal of S. Proof. First assume that (1) holds. Let 6 be any fuzzy bi-ideal of S, and a, b any
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206
elements of S. Then, as is easily seen, aSa is a bi-ideal of S. Then by the assumption aSa is a right ideal of S. Then, since S is regular,
ab ~ ( aSa )S ~_ aSa. This implies that there exists an element x in S such that
ab = axa. Then, since 8 is a fuzzy bi-ideal of S, we have
,5(ab) = ,5(axa)>~min{,5(a), ~(a)} = ~(a). This means that 8 is a fuzzy right ideal of S. Thus we obtain that (1) implies (2). Conversely, assume that (2) holds. Let A be any bi-ideal of S. Then it follows from Lemma 2.2 that the characteristic function ~A of A is a fuzzy bi-ideal of S. Thus by the assumption it is a fuzzy right ideal of S. Then, since A is non-empty, it follows from Lemma 2.1 that A is a right ideal of S. Thus (2) implies (1). This completes the proof. The lefi-right dual of Theorem 3.4 is proved in an analogous way:
Theorem 3.5. For a regular semigroup S the following condihons are equivalent. (1) Every hi-ideal of S is a left ideal of S~ (2) Every fuzzy hi-ideal of S is a fuzzy left ideal of S. The following theorem is the immediate consequence of Theorems 3.4 and 3.5. Theorem 3.6. For a regular semigroup S the following conditions are equivalent. (1) Every hi-ideal of S is a two-sided ideal of S. (2) Every fuzzy hi-ideal of S is a fuzzy two-sided ideal of S. Corollary 3.7. Let S be a regular duo semigroup. Then every jfuzzy bi-ideal of S is a fuzzy right ideal of S. Proof. It follows from [5, Theorem 30] that every bi-ideal of a regular left duo semigroup is a right ideal of it. From this and from Theorem 3.4 it follows that every fuzzy bi-ideal of S is a fuzzy right ideal of S. Corollary 3.8. Let S be a semigroup which is a semilattice of groups. Then every fuzzy hi-ideal of S is a fuzzy two-sided ideal of S. Proof. It follows from [4, Theorem 4] that every bi-ideal of such a semigroup S is a two-sided ideal of S. Therefore it follows from Theorem 3.6 that every fuzzy bi-ideal of S is a fuzzy two-sided ideal of S. We denote by L[a] (J[a]) the principal left (two-sided) ideal of a semigroup S generated by a in S, that is,
L[a]={alUSa, J[a] = {a} U S a U aS U SaS.
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207
As is well-known [1, L e m m a 2.13], if S is a regular semigroup, then L [ a ] = Sa for every a ~ S. A semigroup S is called right (left) zero if xy=y
(xy=x)
for all x, y ~ S. T h e n we have the following. "I]heorem 3.9. For a regular semigroup S lhe following conditions are equivalent. (1) The set of all idempotent elements of S forms a left zero subsernigrov.p of S. (2) For every f u z z y left ideal ,5 of S,
~5(e) = ~5(f) for all idempoten:' elements e and f of S. ProoL First asswrne that the set Es of all idempotents of S forms a left zero subsemigroup of S. Let e and f be any elements of Es, and (5 any fuzzy left ideal of S. Then, since
ef=e
and
fe=f,
we have
,5(e) = 8(ef)>~ cS(f) = ,5(re)>I 8(e), and so 8(e) = cS(f). Thus (1) implies (2). Conversely, assume that (2) holds. Since S is regular, Es is non-empty. Let e and f be any elements of Es. Then i: follows from L e m m a 2.1 that the characteristic function (SLur] of the left ideal L[j'] of S is a fuzzy left ideal of S. Thus we have dSLtn(e) = B ' [ n ( f ) = 1, and so
e ~ L If] = Sf. Then for some x ~ S we have
e=xf=xff=ef. This means that Es is a left zero scmigroup. Thus (2) implies (1). This completes the proof.
Corollary 3.10. For an idempotent semigroup S the following conditions are equivalent. (1) S is left zero. (2) For every f u z z y left ideal ,5 of S, 8(e) = ~(f)
for all e, f ~ S.
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N. Kuroki
The left-right dual of Theorem 3.9 reads as follows: qt~eorem 3.11. For a regular semigroup S the following conditions are equivalent. (1) Es forms a right zero subsemigroup of S. (2) For every fuzzy right ideal 8 of S, ,5(e) = rS(f) for all e, f e Es. Corollary 3.12. For an idempotent semigroup S the following conditions are equivalent. (1) S is right zero. (2) For every fuzzy right ideal 8 of S, 3(e)=8([) for all e, f ~ S. Lemma 3.13 [2, Theorem 2]. For a semigroup S the following conditions are equivalent. (1) S is a group. (2) Every fuzzy bi-ideal of S is a constant function. Theorem 3.14. For a regular semigroup S the following conditions are equivalent. (1) S is a group. (2) For every fuzzy hi-ideal 8 of S,
8(e)= ~5(f) for all idempotents e and f of S. Proof. Assume that (1) holds. Let 8 be any fuzzy bi-ideal of S. Then it follows from Lemma 3.13 that 6 is a constant function. Therefore for any idempotents e and f of S,
,~(e)= ,5(f). Thus (1) implies (2). Conversely, assume that (2) holds. Let e and f be any idempotents of S. We denote by B[x] the principal bi-ideal of S generated by x in S, that is, B[x] = {x} U {x 2} U xSx
[ l, p. 84].
As is well-known, if S is regular, then
B[x]= xSx. Then it follows from Lemma 2.2 that the characteristic function ~SB[r] of the bi-ideal B[f] of S is a fuzzy bi-ideal of S. Sii~ce [ e B[f], we have 8~,m(e) = 8Bm(f)= 1.
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209
T h e n we have e ~ B [ f ] = fSf,
which implies that e = fxf
for s o m e x ~ S. Similarly, we obtain that f = eye for some y ~ S. Then we have e = f x f = f x f f = ef = eeye = eye = f.
This means that, since S is regular, Es is non-empty and S contains exactly one idempotent. Then it follows from [ 1, p. 33 (Ex. 4)] that S is a group. Therefore (2) implies (1). This completes the proof of the theorem.
4. intra-regular semigroups
A semigroup S is called intra-regular if, for each element a of $, there exist elements x and y in S such that a = xa2y. For characterizations of such a semigroup, see [1, T h e o r e m 4.4] and [6, 11.4.5 Theorem]. Now we give a new characterization of intra-regular semigroups. T h e o r e m 4.1. For a semigroup S the f o l l o w i n g conditions are equivalent. (1) S is intra-regular. (2,) For every f u z z y ideal 8 of S, 8(a) = 8(a 2) holds for all a ~ S.
1~oot. First assume that (1) holds. Let 8 be any fuzzy ideal of S and a any element of S. q't~en, since S is intra-regular, there exist elements x and y in S such that a = xa2y.
Then, since 8 is a fuzzy ideal of S, we have 8(a) >I 8(xa2y ) >I t$(xa 2) >t 8( a 2) I> 8(a ), and so we have 8(a) = tr(a2). Thus (1) implies (2). Conversely, assume that (2) ho|ds. Then it follows from L e m m a 2.1 that the characteristic function ~jta2~ of the ideal J [ a 2] of S is a fuzzy
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N. Kuroki
ideal of S. Since a 2 e j[a2], we have 8jt,,21(a) = 8jt,,21(a 2) = 1. This implies that a e J [ a 2] = {a 2} U Sa 2 U a2S U Sa2S. Then it is easily seen that S is intra-regular. Thus (2) implies (1). This completes the proof.
Theorem 4.2. Let S be an intra-regular semigroup. Then for every f u z z y ideal 8 of s, 8(ab)=8(ba) holds for all a, b ~ S. P r o o | . Let 8 be any fuzzy ideal of S and a and b any elements of S. Then by Theorem 4.1 we have 8 ( a b ) - 8((ab) 2) = 8 ( a ( b a ) b ) ~ 8(ba) = 8((ba)2) = 8(b(ab )a ~>18(ab), and so we have 8(ab) = 6(ba). This completes the proof.
5. Completely regular semigroups A semigroup S is called completely regular if, for each element a of S, there exists an element x in S such that a = axa
and
ax = xa.
A semigroup S is called left (right) regular if, for each element a of S, there exists an element x in S such that a = xa 2 (a = aEx). For characterizations of such a semigroup, see [1, Theorem 4.2].
Theorem 5.1. For a semigroup S the following conditions are equivalent. (1) S is left regular. (2) For every f u z z y left ideal 8 of S, 8(a) = 8(a 2) holds for all a ~ S.
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211
Proof. First assume that (1) holds. Let /5 be any fuzzy left ideal of S and a any element of S. Then, since S is left regular, there exists an elerrent x in S such that a = x a 2.
Then we have /5(a ) = ~5(xa2) ~>/5(a 2) ~>/5(a ), and so we have /5( a ) = /5(a2). Thus (1) implies (2). Conversely, assume that (2) holds. Let a be any element of S. Then it follows from Lemma 2.1 that the characteristic function /sLi,-~l of the left ideal L[a 2] of S is a fuzzy left ideal of S. Since a2~ L[a2], we have ~SLta21(a) = ~Sl~t,21(a2) = 1. This implies that a ~ L[a 2] = {a 2} U S a 2. This means that S is left regular, and so (2) implies (1). This completes the proof. The left-right dual of Theorem 5.1 reads as follows:
Theorem 5.2. For a semigroup S the following conditions are equivalent. (1) S is right regular. (2) For every fuzzy r',.ght ideal ~ of S, tS(a)=~(a 2) holds for all a c S. We recall tiaat a semigroup S is completely regular if and only if it is left and right regular [1, Theorem 4.3]. The equivalence of (1) and (2) in the following theorem is due to the author [2], and the equivalence of (1) and (3) follows fro,,,-a Theorem 5.1 and Theorem 5.2.
Theorem 5.3. For a semigroup S the following conditions are equivalent. ( l ) S is completely regular. (2) For every fuzzy bi-ideal ~ of S, /5(a ) =/5(a 2) holds for all a e S. (3) For every fuzzy left ideal ~ and for every fuzzy right ideal V o[ S, /3(a)=/3(a 2)
hold for all a ~ S.
and
v(a) = v(a 2)
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6. Semigroups that are semilattices of left (right) simple semigroups The following characterization of a semigroup that is a semilattice of left simple semigroups is due to Lajos [3] and Salt6 [8].
Lemmm 6.1 For a semigroup S the following conditions are equivalent. (1) S is a semilattice of left simple semigroups. (2) S is left regular and A B = B A for any two left ideals A and B of S. (3) S is left regular and every left ideal of it is a two-sided ideal of S. The following theorem can be proved in a similar way as in the proof of Theorems 3.1 and 3.2. So we omit the proof.
Theorem 6.2. For a left (right) regular semigroup S the following conditions are equivalent. (1} S is left (right) duo. (2) S is fuzzy left (right)duo. The characterization of a semigroup that is a semilattice of left simple semigroups can be found in [6, II.4.9 Theorem]. Now we give a new characterization of such a semigroup.
21heorem 6.3. For a semigroup S the following conditions are equivalent. (1) S is a semilanice ~gf left simple semigroups. (2) For every fuzzy e't ideal ~ of S, 8(a) = ~5(a 2)
:~zd
8(ab) = 8(ba)
hold for all a, b ~ S. ProoL Assume that S is a semilattice of left simple semigroups. Let 8 be any fuzzy left ideal of S. Then, since S is left regular by Lemma 6.1, it follows from Theorem 5.1 that 8(a) = 8(a 2) holds for all a ~ S. It follows from Lemma 6.1 and Theorem 6.2 that ~ is a fuzzy two-sided ideal of S. Let a and b be any elements of S. Then we have
8(ab) = ~((ab) 2) = 8(a(ba)b)>16(ba). Similarly, we have
~5(ba ) >~cS(ab ). "lhus we obtain that
~ ( a b ) - ~(ba) and that (1) implies (2). Conversely, assume that (2) holds. Then it .follows from the first condition of (2) and from Theorem 5.1 that S is left regular. Let A and B
On fuzzy ideals and fuzzy bi-ideals
213
be any left ideal of S and ab (a ~ A , b e B ) be any element of A B . Then the characteristic function `sLtb,~l of the left ideal L[ba] is a fuzzy left ideal of S by Lemma 2.1. And since ba ~ L[ba], we have
`sLtbal( ab ) = `sLtba j( ba ) = 1. This implies that
ab ~ L[ba ] = {ba } O Sba ~ B A LJ S B A ~_ B A , and so we have
A B c_ B A . Similarly, it can be seen that the converse inclusion holds. Thus we obtain that
AB-
BA.
Then it follows from Lemma 6.1 that S is a semilattice of left simple semigroups. Therefore (2) implies (1). This completes the proof of the theorem. The left-right dual of Theorem 6.3 reads as follows:
Theorem 6.4. For a semigroup S the following conditions are equivalent. (1) S is a semilattice of right simple sem~groups. (2) For every f u z z y right ideal ,5 of S, `5(a) =`5(a 2)
and
`5(ab)-`5(ba)
hold ]:or all a, b ~ S.
7. Left (right) simple semigroups A semigroup S is called fuzzy left (right) simple if every fuzzy left (right) ideal of S is a constant function, and is called fuzzy simple if every fuzzy ideal of S is a constant function. Then we have the following.
Theorem 7.1. For a semigroup S the following conditions are equivalent. (1) S is left simple. (2) S is f u z z y left simple.
Proof: First assume that (1) holds. Let `5 be any fuzzy left ideal of S and a and b any elements of S. Then, since S is left simple, it follows from [1, p. 6] that there exist elements x and y in S such that
b=xa
and
a=yb.
Then, since `5 is a fuzzy left ideal of S, we have 6(a) = 6(yb)>I 6 ( b ) = 8(xa)>t 8(a),
N. Kuroki
214 and so we have
6(a)=6(b). Since a and b are any elements of S, this means that 5 is a constant function. Thus we obtain that S is fuzzy left simple, and that (1) implies (2). Conversely, assume that (2) holds, and let A be any left ideal of S. By Lemma 2.1, 6A is a fuzzy left ideal of S, hence a constant function, and since A is non-empty, the constant is 1. Thus every element of S is in A, and so S is left simple. Thus (2) implies (1). The following two theorems can be seen in a similar way as in the proof of Theorem 7.1. Theorem 7.2. For a semigroup S the following conditions are equivalent. (1) S is right simple. (2) S is fuzzy right simple. Theorem 7.3. For a semigroup S the following conditions are equivalent.
(1) S is simple. (2) S is fuzzy simple. As is well-known, a semigroup S is a group if and only if it is left and right simple. From this and from Theorems 7.1 and 7.2, we have the following theorem. Theorem 7.4. For a semigroup S the following conditions are equivalent.
(1) S is a group. (2) S is both fuzzy left and fuzzy right simple. Theorem 7.$. Let S be a left simple semigroup. Then every fuzzy bi-ideal of S is a
fuzzy right ideal of S. Proof. Let 6 be any fuzzy bi-ideal of $, and a and b any elements of $. Then, since S is left simple, there exists an element x in S such that
b=xa. Then, since 6 is a fuzzy bi-ideal of S, we have
6(ab) = 6(axa)>~min{6(a), 6(a)} = 8(a). This means that ~ is a fuzzy right ideal of S. This completes the proof.
Corollary 7.6. Let S be a left simple semigroup. Then every bi-ideal of S is a right ideal of S.
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215
Acknowledgement I express my thanks to the referees for their valuable comments and suggestions.
References [1] A.H. Clifford and G.B. Preston, The Algebraic Theory of Semigroups Vol. I (Am. Math. Soc., Providence, RI, 1961). [2] N. Kuroki, Fuzzy bi-ideals in semigroups, Comm. Math. Univ. St. Pauli 28 (1979) 17-21. [3] S. Lajos, A note on completely regular semigroups, Acta Sci. Math. (Szeged) 28 (1967) 261-265. [4] S. Lajos, On (m,n)-ideals in regular duo semigroups, Acta Sci. Math. (Szeged) 31 (19701 179-180. [5] S. Lajos, Theorems on (1, 1)-ideals in semigroups, Dept. Math., K. Marx Univ. Economics (1972). [6] M. Petrich, Introduction to Semigroups (Columbus, OH, 1973). [7] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [8] T. SaitS, On semigroups which are semilattices of left simple semigroups, Math. Japon. 18 (1973) 95-97. [9] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-353.