Fuzzy Quasi-ideals of Ordered Semigroups

Fuzzy Quasi-ideals of Ordered Semigroups

Muhammad Shabir Department of Mathematics,

Quaid-i-Azam University, Islamabad Pakistan

e-mail: [email protected] Asghar Khan

Departmrnt of Mathematics,

COMSATS Institute of information Technology, Abbottabad, Pakistan

[email protected] June 24, 2009

Abstract

In this paper, we characterize ordered semigroups in terms of fuzzy quasi-ideals. We characterize left simple, right simple and completely reg- ular ordered semigroups in terms of fuzzy quasi-ideals. We de…ne semi- prime fuzzy quasi-ideal of ordered semigroups and characterize completely regular ordered semigroup in terms of semiprime fuzzy quasi-ideals. We also study the decomposition of left and right simple ordered semigroups having the propertya a² for alla2S, by means of fuzzy quasi-ideals.

Keywords. Subsemigroups; left (right) ideals; quasi-(bi-) ideals; left (right) reg- ular; left (right) simple ordered semigroups; completely regular ordered semigroups;

fuzzy sets; fuzzy subsemigroup; fuzzy left (right) ideals; fuzzy quasi-(bi-) ideals; semi- prime (resp. semiprime fuzzy) ideals of ordered semigroups.

2000 MSC:06F05, 06D72, 08A72.

1 Introduction

Worldwide, there has been a rapid growth in the interest of fuzzy set theory and its applications from the past several years. Evidence of this can be found in the increasing number of high-quality research articles on fuzzy sets and related topics that have been published in a variety of international journals, symposia, workshops, and international conferences held every year. It seems

that the fuzzy set theory deals with the applications of fuzzy technology in information processing. The information processing is already important and it will certainly increase in importance in the future. Granular computing refers to the representation of information in the form of aggregates, called granules.

If granules are modeled as fuzzy sets, then fuzzy logics are used. This new computing methodology has been considered by Brageila and Pedrycz in [4].

A presentation of updated trends in fuzzy set theory and its applications has been considered by Pedrycz and Gomide in [23]. A systematic exposition of fuzzy semigroups by Mordeson, Malik and Kuroki appeared in [19], where one can …nd theoritical results on fuzzy semigroups and their use in fuzzy coding, fuzzy …nite state machines and fuzzy languages. The monograph by Mordeson and Malik [20] deals with the applications of fuzzy approach to the concepts of automata and formal languages. The notion of quasi-ideals play an important role in the study of ring theory, semiring theory, semigroup theory and ordered semigroup theory etc. for a detail study of quasi-ideals in rings and semigroups, we refer the reader to [22]. The fuzzy subsets in semigroups were …rst studied by Kuroki [16-18] and Ahsan [1] et al. The fuzzy quasi-ideals in semigroups were studied in [18] and [1], where the basic properties of semigroups in terms of fuzzy quasi-ideals are given. The concept of a quasi-ideal in rings and semigroups was studied by Stienfeld in [22], and Kehayopulu extended the concept of quasi-ideals in ordered semigroupsS as a non-empty subsetQofS such that [8]:

(1)(QS]\(SQ] Qand (2) Ifa2QandS3b athenb2Q. For a detail study of ideal theory and fuzzy ideal theory of ordered semigroups we refer the reader to [10-14] and [7-9] and [15].

In this paper, we characterize regular, left and right simple ordered semi- groups and completely regular ordered semigroups. We prove that an ordered semigroupS is regular, left and right simple if and only if every fuzzy quasi- ideal ofS is a constant function. We also prove thatS is completely regular if and only if for every fuzzy quasi-idealf of S we have f(a) = f(a²) for every a2S. We de…ne semiprime fuzzy quasi-ideal of ordered semigroups and prove that an ordered semigroup S is completely regular if and only if every fuzzy quasi-ideal f of S is semiprime. Next we characterize semilattices of left and right simple ordered semigroups in terms of fuzzy quasi-ideals ofS. We prove that an ordered semigroup S is a semilattice of left and right simple ordered semigroups if and only if for every fuzzy quasi-idealf ofSwe have,f(a) =f(a²) andf(ab) =f(ba), for alla; b2S. In the last of this paper, we discuss ordered semigroups having the propertya a²for alla2S and prove that an ordered semigroupS (having the propertya a² 8a2 S) is a semilattice of left and right simple ordered semigroups if and only if for every fuzzy quasi-idealf ofS we havef(ab) =f(ba), for alla; b2S.

2 Some Basic Results and De…nitions

In this section, we give some basic de…nitions and results, which are necessary for the subsequent sections.

By an ordered semigroup we mean a structure(S; ; )such that:

(OS1) (S; )is a semigroup.

(OS2) (S; )is a poset.

(OS3) (8a; b; x2S)(a b !ax bxandxa xb):

Let(S; ; )be an ordered semigroup and; 6=A S, denote (A] :=ft2Sjt hfor some h2Ag ForA; B S, denote

AB=fabja2A; b2Bg Fora2S, we write(a]instead of(fag].

Let S be an ordered semigroup and A; B S. Then A (A], (A](B]

(AB],((A]] = (A]and((A](B]] (AB](see [8]).

Let (S; ; ) be an ordered semigroup, ; 6= A S. Then A is called a subsemigroup ofS ifA² A(see[9]).

Let (S; ; ) be an ordered semigroup. ; 6=A S is called a right (resp.

lef t)ideal (see[13])of S if: (1)AS A(resp. SA A)and (2) Ifa2Aand S3b a, thenb2A. IfAis both a right and a left ideal ofS;then it is called anideal ofS. A subsemigroup B of S is called a bi-ideal (see [9]) ofS if: (1) BSB B and (2) Ifa2B,S3b a, thenb2B.

Let(S; ; )be an ordered semigroup. By afuzzy subset f of S, we mean a functionf :S ![0;1]:

LetS be an ordered semigroup. A fuzzy subset f of S is called afuzzy left (resp. right) ideal of S if: (1) (8x; y 2 S)(x y ! f(x) f(y)) and (2) (8x; y2S)(f(xy) f(y)(resp. f(xy) f(x)):Iff is both a fuzzy left and a fuzzy right ideal ofS;then it is called afuzzy ideal ofS (see [7]).

Let (S; ; )be an ordered semigroup and ; 6=A S. Thecharacterisitic function fAofAis given by:

fA:S ![0;1]; a7 !fA(a) := 1ifa2A, 0ifa =2A.

Let (S; ; ) be an ordered semigroup and a 2 S, denote Aa := f(y; z) 2 S Sja yzg (see [8]).

For two fuzzy subsetsf andgofS, de…ne

f g:S ![0;1]; a7 !f g(a) 8<

: _

(y;z)2Aa

minff(y); g(z)gifA_a6=;

0 if Aa=;

We denote by F(S) (as given in [8]) the set of all fuzzy subsets of S. We de…ne order relation" " onF(S)as follows:

f gif and only if f(x) g(x) for allx2S

Then(F(S); ; )is an ordered semigroup (see [8]).

For a nonempty family of fuzzy subsetsffigi2I, of an ordered semigroupS, the fuzzy subsets _

i2I

f_i and ^

i2I

f_i ofS are de…ned as follows:

_

i2I

f_i : S ![0;1]; a7 ! _

i2I

f_i

!

(a) :=sup_i2Iff_i(a)gand

^

i2I

fi : S ![0;1]; a7 ! ^

i2I

fi

!

(a) :=infi2Iffi(a)g: IfI is a …nite set, sayI=f1;2; :::; ng, then clearly

_

i2I

f_i(a) = max{f₁(a); f₂(a); :::; f_n(a)gand

^

i2I

f_i(a) = min{f₁(a); f₂(a); :::; f_n(a)g:

For an ordered semigroupS, the fuzzy subsets “0” and “1” ofS are de…ned as follows (see[8]):

0 :S ![0;1]; x7 !0(x) := 0, 1 :S ![0;1]; x7 !1(x) := 1.

Clearly, the fuzzy subset “0”(resp. “1”) ofS is the least (resp. the greatest) element of the ordered set(F(S); ):The fuzzy subset “0” is the zero element of(F(S); ; )(that is,f 0 = 0 f = 0 and0 f for every f 2F(S)):

3 Fuzzy quasi-ideals

In this section we characterize quasi-ideals of ordered semigroups by the prop- erties of their level subsets.

3.1 Proposition (cf. [8]).

If (S; ; )is an ordered semigroup and A; B S. Then (1) A B if and only if fA fB,

(2) fA^fB=fA\B; (3) fA fB =f_(AB]:

3.2 Lemma

Let S be an ordered semigroup. Then every quasi-ideal of S is a subsemigroup of S.

3.3 Lemma (cf. [8]).

An ordered semigroup (S; ; ) is regular if and only if for right ideal A and every left ideal B of S, we have A\B= (AB]:

3.4 De…nition (cf. [8]).

Let(S; ; ) be an ordered semigroup. A fuzzy subset f ofS is called a fuzzy quasi-ideal ofS if:

(1)(f 1)^(1 f) f;

(2)(8x; y2S)(x y !f(x) f(y)).

3.5 De…nition (cf. [9]).

Let(S; ; ) be an ordered semigroup. A fuzzy subset f ofS is called a fuzzy bi-ideal ofS if:

(1)(8x; y2S)(f(xy) minff(x); f(y)g);

(2)(8x; y; z2S)(f(xyz) minff(x); f(z)g).

(3)(8x; y2S)(x y !f(x) f(y)):

3.6 Lemma (cf. [7-9]).

Let (S; ; ) be an ordered semigroups and ; 6=A S. Then A is a left (resp.

right, bi- and quasi-) ideal of S if and only if the characteristic function f_A of Ais a fuzzy left (resp. right, bi- and quasi-) ideal of S.

Let(S; ; )be an ordered semigroup andt2(0;1]then the set U(f;t) :=fx2Sjf(x) tg;

is called a level subset off.

3.7 Theorem

Let (S; ; )be an ordered semigroup and f a fuzzy subset of S. Then (8t2(0;1]U(f;t)6=; is a quasi-ideal if and only if f is a fuzzy quasi-ideal) Proof. =) : Assume that for every t 2 (0;1] such that U(f;t) 6= ; the set U(f;t) is a quasi-ideal of S. Let x; y 2 S, x y be such that f(x) < f(y).

Then there exists t 2 (0;1) such that f(x) < t f(y), theny 2 U(f;t) but x =2U(f;t). This is a contradiction. Hencef(x) f(y)for allx y. Suppose that there existsx2S such that

f(x) ((f 1)^(1 f))(x);

then there existst2(0;1]such that

f(x)< t <((f 1)^(1 f))(x) =min[(f 1)(x);(1 f)(x)]:

and hence(f 1)(x)> tand(1 f)(x)> t. Then _

(p;q)2Ax

minff(p);1(q)g> tand _

(p;q)2Ax

minf1(p); f(q)g> t:

This implies that there exist b; c; d; e2S with (b; c) 2Ax and (d; e)2Ax

such thatf(b)> t andf(e)> t. Then b; e2U(f;t)and so bc2U(f;t)S and de2 SU(f;t): Hence x2 (U(f;t)S] and x2 (SU(f;t)] ! x2 (U(f;t)S]\ (SU(f;t)]:By hypothesis, (U(f;t)S]\(SU(f;t)] U(f;t)and sox2U(f;t):

Thenf(x) t:This is a contradiction. Thusf(x) ((f 1)^(1 f))(x):

(= : Assume that f is a fuzzy quasi-ideal of S and t 2 (0;1] such that U(f;t)6=;. Let x; y2S be such that x y and y 2U(f;t). Then f(y) t.

Sincex y !f(x) f(y)we havef(x) t and sox2U(f;t).

Suppose that x 2 S be such that x 2 (U(f;t)S]\(SU(f;t)]: Then x 2 (U(f;t)S] and x 2 (SU(f;t)] and we have x yz and x y⁰z⁰ for some y; z⁰ 2U(f;t)and z; y⁰ 2S. Then(y; z)2Ax and(y⁰; z⁰)2Ax. SinceAx6=;; by hypothesis

f(x) ((f 1)^(1 f))(x)

= min

2 4 _

(p;q)2Ax

minff(p);1(q)g; _

(p1;q1)2Ax

minf1(p1); f(q1)g 3 5 min[minff(y);1(z)g;minf1(y⁰); f(z⁰)g]

= min[minff(y);1g;minf1; f(z⁰)g]

= min[f(y); f(z⁰)]:

Sincey; z⁰ 2U(f;t)we havef(y) t andf(z⁰) t:Then f(x) min[f(y); f(z⁰)] t;

and sox2U(f;t). Hence (U(f;t)S]\(SU(f;t)] U(f;t). ThusU(f;t)is a quasi-ideal ofS.

3.8 Example

Let S=fa; b; c; d; fg be an ordered semigroup with the following multiplication table

a b c d f

a a a a a a

b a b a d a

c a f c c f

d a b d d b

f a f a c a

We de…ne the order relation " " as follows:

:=f(a; a);(a; b);(a; c);(a; d);(a; f);(b; b);(c; c);(d; d);(f; f)g: Quasi-ideals ofS are:

fag;fa; bg;fa; cg;fa; dg;fa; fg;fa; b; dg; fa; c; dg;fa; b; fg;fa; c; fg andS (see [11]).

De…nef :S ![0;1]by

f(a) = 0:8; f(b) = 0:7; f(d) = 0:6; f(c) =f(f) = 0:5:

Then

U(f;t) :=

8>

>>

><

>>

>>

:

S ift2(0;0:5]

fa; b; dgift2(0:5;0:6]

fa; bg ift2(0:6;0:7]

fagift2(0:7;0:8]

; ift2(0:8;1]

ThenU(f;t)is a quasi-ideal and by Theorem 3.7,f is a fuzzy quasi-ideal of S.

3.9 Lemma

Every quasi-ideal of an ordered semigroup (S; ; )is a bi-ideal of S.

3.10 Lemma

Every fuzzy quasi-ideal of an ordered semigroup S is a fuzzy bi-ideal of S.

Proof. Letf be a fuzzy quasi-ideal ofS. Letx; y2S. Thenxy=x(y)and we have(x; y)2A_xy. SinceA_xy6=;, we have

f(xy) ((f 1)^(1 f))(xy)

= min[(f 1)(xy);(1 f)(xy)]

= min

2 4 _

(p;q)2Axy

minff(p);1(q)g; _

(p1;q1)2Axy

minf1(p1); f(q1)g 3 5 min[minff(x);1(y)g;minf1(x); f(y)g]

= min[minff(x);1g;minf1; f(y)g]

= min[f(x); f(y)]:

Let x; y; z 2 S. Then (xy)z = x(yz) and we have (xy; z);(x; yz) 2 Axyz. SinceAxyz6=;;we have

f(xyz) ((f 1)^(1 f))(xyz)

= min[(f 1)(xyz);(1 f)(xyz)]

= min

2 4 _

(p;q)2Axyz

minff(p);1(q)g; _

(p1;q1)2Axyz

minf1(p1); f(q1)g 3 5 min[minff(x);1(yz)g;minf1(xy); f(z)g]

= min[minff(x);1g;minf1; f(z)g]

= min[f(x); f(z)]:

Let x; y2 S be such that x y. Then f(x) f(y), because f is a fuzzy quasi-ideal ofS. Thusf is a fuzzy bi-ideal ofS.

3.11 Remark

The converse of Lemma 3.10, is not true in general.

3.12 Example

Consider the ordered semigroup S=fa; b; c; dg

a b c d

a a a a a

b a a a a

c a a b a

d a a b b

:=f(a; a);(b; b);(c; c);(d; d);(a; b)g

Thenfa; dgis a bi-ideal ofSbut not a quasi-ideal ofS. De…nef :S ![0;1]

by

f(a) =f(d) = 0:7; f(b) =f(c) = 0:4 Then

U(f;t) :=

8<

:

S ift2(0;0:4]

fa; dgift2(0:4;0:7]

; ift2(0:7;1]

ThenU(f;t)is a bi-ideal ofS and by Theorem 3.7,f is a fuzzy quasi-ideal of S. Furthermore, U(f;t) is a bi-ideal of S for all t 2 (0:4;0:7] but not a quasi-ideal ofS, and hence by Theorem 3.7,f is a fuzzy bi-ideal ofS, but not a fuzzy quasi-ideal ofS.

4 Characterizations of left, right and completely regular ordered semigroup in terms of fuzzy quasi-ideals

In this section, we characterize ordered semigroups in terms of fuzzy quasi-ideals and prove that an ordered semigroup S is regular, left and right simple if and only if every fuzzy quasi-idealfofSis a constant function. We de…ne semiprime fuzzy quasi-ideals of ordered semigroups and prove that an ordered semigroupS is completely regular if and only if every fuzzy quasi-idealf ofSis a semiprime fuzzy quasi-ideal ofS.

An ordered semigroup S is called left (resp. right) simple (see [12]) if for every left (resp. right) idealAofS, we haveA=S.

4.1 Lemma (cf. [9, Lemma 3]).

An ordered semigroup S is left (resp. right) simple if and only if (Sa] = S (resp. (aS] =S) for every a2S.

4.2 Theorem

An ordered semigroup (S; ; ) is regular, left and right simple if and only if every fuzzy quasi-ideal of S is a constant function.

Proof. LetS be regular, left and right simple ordered semigroup. Letf be a fuzzy quasi-ideal ofS anda2S. We consider the set

E :=fe2Sje² eg:

ThenE is non-empty. In fact, sinceSis regular, anda2S, there existsx2S such thata axa. Then it follows by(OS3), that

(ax)²= (axa)x ax;

and soax2E and henceE 6=;:

(1) Lett 2E : Then f(e) =f(t) for every e2E : Indeed, since S is left and right simple, we have(St] = S and (tS] =S. Since e 2S; then e2 (St]

ande2(tS]so there existx; y2S such thate xtande ty. Hence e²=ee (xt)(xt) = (xtx)t;

and we have(xtx; t)2A_e² and ife tythen e²=ee (ty)(ty) =t(yty)

and hence(t; yty)2A_e²: SinceA_e² 6=; andf is a fuzzy quasi-ideals ofS, we have

f(e²) ((f 1)^(1 f))(e²)

= min[(f 1)(e²);(1 f)(e²)]

= min[ _

(y1;z1)2A_e2

minff(y₁);1(z₁)g; _

(y2;z2)2A_e2

minf1(y₂); f(z₂)g] min[minff(t);1(yty)g;minf1(xtx); f(t)g]

= min[minff(t);1g;minf1; f(t)g]

= min[f(t); f(t)] =f(t):

Since e 2 E we have e² e and f is a fuzzy quasi-ideal of S, we have f(e) f(e²): Thusf(e) f(t): SinceS is left and right simple ande2S we have,(Se] =S and (eS] =S. Since t2S, we havet zeand t esfor some z; s2S. Ift zethen

t²=tt (ze)(ze) = (zez)e then(zez; e)2A_t²:Ift esthen

t²=tt (es)(es) =e(ses)

and we have(e; ses)2At²:SinceAt²6=;;we have f(t²) ((f 1)^(1 f))(t²)

= min[(f 1)(t²);(1 f)(t²)]

= min

2

4 _

(p1;q1)2A_t2

minff(p₁);1(q₁)g; _

(p2;q2)2A_t2

minf1(p₂); f(q₂)g 3 5 min[minff(e);1(ses)g;minf1(zez); f(e)g]

= min[minff(e);1g;minf1; f(e)g]

= min[f(e); f(e)] =f(e):

Sincet2E we havet² t and sinef is a fuzzy quasi-ideal ofS, we have f(t) f(t²):Thusf(t) f(e):

(2) Let a 2 S then f(t) = f(a) for every t 2 E : Since a 2 S and S is regular, there existsx2S such thata axa:Then from(OS3)it follows that,

(ax)²= (axa)x axand(xa)²=x(axa) xa:

Then ax; xa 2 E . Then by (1) we have f(ax) = f(t) and f(xa) = f(t): Since (ax)(axa) axa a; and (axa)(xa) axa a; so we have, (ax; axa);(axa; xa)2 Aa: Since Aa 6= ; and f is a fuzzy quasi-ideal of S; we have

f(a) ((f 1)^(1 f))(a)

= min[(f 1)(a);(1 f)(a)]

= min

2 4 _

(y1;z1)2Aa

minff(y1);1(z1)g; _

(y2;z2)2Aa

minf1(y2); f(z2)g 3 5 min[minff(ax);1(axa)g;minf1(axa); f(xa)g]

= min[minff(ax);1g;minf1; f(xa)g]

= min[f(ax); f(xa)]

= min[f(ax); f(ax)] =min[f(t); f(t)] =f(t):

SinceS is left and right simple we have(Sa] =S,(aS] =S:Sincet2S, we have t2(Sa] and t2(aS]. Thent pa andt aqfor some p; q2 S. Then (p; a)2A_tand(a; q)2A_t. SinceA_t6=; andf is a fuzzy ideal ofS;we have

f(t) ((f 1)^(1 f))(t)

= min[(f 1)(t);(1 f)(t)]

= min

2 4 _

(y1;z1)2At

minff(y₁);1(z₁)g; _

(y2;z2)2At

minf1(y₂); f(z₂)g 3 5 min[minff(a);1(q)g;minf1(p); f(a)g]

= min[minff(a);1g;minf1; f(a)g]

= min[f(a); f(a)] =f(a):

SinceSis left and right simple we have(Sa] =S, and(aS] =S. Sincet2S, we havet2(Sa]andt2(aS]. Thent paandt aqfor somep; q2S. Then (p; a)2A_tand(q; a)2A_t. SinceA_t6=;;we have

f(t) ((f 1)^(1 f))(t)

= min[(f 1)(t);(1 f)(t)]

min 2 4 _

(y1;z1)2At

min{f(y₁);1(z₁); _

(y2;z2)

minf1(y₂); f(z₂)g 3 5 min[min{f(a);1(q)g,minf1(p); f(a)g]

= min[min{f(a);1g,minf1; f(a)g]

= min[f(a); f(a)] =f(a):

Conversely, let a 2 S. Then the set (aS] is a quasi-ideal of S. In fact, ((aS]S]\(S(Sa]] (aS]\(SaS] (aS]and ifx2(aS] andS3y x2(aS], then y 2 ((S]] = (aS]. Since (aS] is a quasi-ideal of S, by Theorem 3.6, the characteristic functionf_(aS] of(aS]de…ned by

f_(aS]:S ![0;1]; x7 !f_(aS](x) := 1ifx2(aS]

0ifx =2(aS]

is a fuzzy quasi-ideal ofS. By hypothesis,f_(aS] is a constant function, that is, there existsc2 f0;1g such that

f_(aS](x) =c for everyx2S.

Let(aS] Sand a be an element ofSsuch thata =2(aS], thenf(aS](x) = 0:

On the other hand, sinea² 2 (aS] then f(aS](a²) = 1: A contradiction to the fact that f_(aS] is a constant function. Thus (aS] = S. By symmetry we can prove that(Sa] =S.

Since a 2S and S = (aS] = (Sa], we have a(aS] = (a(Sa]] (aSa], and henceS is regular.

An ordered semigroup S is called left (resp. right) regular (see [10]) if for everya2Sthere existsx2Ssuch thata xa²(resp. a a²x) or, equivalently, if (1) a 2 (Sa²] (resp. a 2 (a²S]) for every a 2 S and (2) A (SA²] (resp.

A (A²S]) for every A S: An ordered semigroup S is called completely regular (see [12]) if it is regular, left regular and right regular.

4.3 Lemma (cf. [12]).

An ordered semigroup(S; ; )is completely regular if and only if A (A²SA²] for every A S or, equivalently, if a2(a²Sa²]for every a2S.

4.4 Lemma

If Sis an ordered semigroup and ; 6=A S;then the set (A[(AS\SA)]is the quasi-ideal of S generated by A. If A=fxg (x2S), we write (x[(xS\Sx)]

instead of (fxg [(fxgS\Sfxg)]:

4.5 Theorem

An ordered semigroup(S; ; )is completely regular if and only if for each quasi- ideal f of S we have,f(a) =f(a²)for all a2S.

Proof. LetS be a completely regular oredered semigroup and f a quasi-ideal of S and let a 2 S. Since S is left and right regular we have a 2 (Sa²] and a2 (a²S]: Then there exists x; y 2 S such that a xa² and a a²y. Then (x; a²),(a²; y)2Aa:SinceAa6=;;we have

f(a) ((f 1)^(1 f))(a)

= min[(f 1)(a);(1 f)(a)]

= min

2 4 _

(y1;z1)

minff(y1);1(z1)g; _

(y2;z2)

minf1(y2); f(z2)g 3 5 min minff(a²);1(y)g;minf1(x); f(a²)g

= min minff(a²);1g;minf1; f(a²)g

= min f(a²); f(a²) =f(a²)

= f(aa) minff(a); f(a)g=f(a).

Thusf(a) =f(a²).

Conversely, leta2S. We consider the quasi-ideal Q(a) of S, generated by a²(a2S). That is the set, Q(a²) = (a²[(a²S\Sa²)]:By Lemma3:6,f_Q(a²₎ is a fuzzy quasi-ideal ofS. By hypothesis, we have

f_Q(a²₎(a) =f_Q(a²₎(a²):

Since a² 2 Q(a²) = (a²[(a²S \Sa²)]; we have fQ(a²)(a²) := 1; then f_Q(a²₎(a) = 1 and we have a 2 Q(a²) = (a²[(a²S\Sa²)]: Then a a² or a a²x and a ya² for some x; y 2 S. If a a² then a a² = aa a²a² = aaa² a²aa² 2 a²Sa² and so a 2 (a²Sa²]: If a a²xand a ya² thena (a²x)(ya²) =a²(yx)a²:Sinceyx2Swe havea²(yx)a²2a²Sa²:Thus a2(a²Sa²]:

4.6 De…nition

Let(S; ; )be an ordered semigroup andf is a fuzzy quasi-ideal ofS. Thenf is calledsemiprime fuzzy quasi-ideal ofS if

f(a) f(a²); for alla2S.

4.7 Theorem

An ordered semigroup (S; ; ) is completely regular if and only if every fuzzy quasi-ideal f of S is semiprime.

Proof. =): LetS be completely regular andf a fuzzy quasi-ideal ofS. Let a2S. Thenf(a²) f(a):In fact, since S is left and right regular, anda2S,

there exist x; y 2 S such that a xa² and a a²y then (x; a²) 2 Aa and (a²; y)2Aa. SinceAa6=;, then

f(a) ((f 1)^(1 f))(a)

= min[(f 1)(a)^(1 f)(a)]

= min

2 4 _

(p;q)2Xa

minff(p);1(q)g; _

(p1;q1)2Xa

minf1(p₁); f(q₁)g 3 5 min min f(a²);1(y) ;min 1(x); f(a²)

= min min f(a²);1 ;min 1; f(a²)

= min f(a²); f(a²)

= f(a²);

(= : Let f be a fuzzy quasi-ideal of S, such that f(a) f(a²) for all a2S. We consider the quasi-idealQ(a²)ofS generated bya². That is, the set Q(a²) = (a²[(a²S\Sa²)]:Then by Lemma 3:6, f_Q(a²₎ is a fuzzy quasi-ideal ofS. By hypothesis,

f_Q(a²₎(a) f_Q(a²₎(a²):

Since a² 2 Q(a²); we have f_Q(a²₎(a²) = 1 and f_Q(a²₎(a) = 1 and we have a2Q(a²) = (a²[(a²S\Sa²)]:Thena a² ora a²xanda za² for some x; z 2S. If a a² then a a² =aa a²a² =aaa² a²aa² 2a²Sa²; then a2 (a²Sa²]: If a a²x and a za², then a (a²x)(za²) = a²(xz)a²; since xz2S;we have a²(xz)a²2a²Sa²and so a2(a²Sa²]:

5 Some semilattices of left and right simple or- dered semigroups in terms of fuzzy quasi-ideals

In this section, we characterize semilattices of left and right simple ordered semi- groups in terms of fuzzy quasi-ideals ofS. We prove that an ordered semigroup S is a semilattice of left and right simple ordered semigroups if and only if for every fuzzy quasi-ideal f of S we have, f(a) =f(a²) and f(ab) = f(ba), for alla; b2 S. We also discuss the semilattice of ordered semigroups having the propertya a²for alla2Sand prove that an ordered semigroupS(having the propertya a² 8a2S)is a semilattice of left and right simple ordered semi- groups if and only if for every fuzzy quasi-idealf of S we havef(ab) =f(ba), for alla; b2S.

A subsetT of an ordered semigroupS is called semiprime if for everya2S such thata²2T we havea2T. Equivalently,A² T we haveA T (see [9]).

5.1 Lemma (cf. [9]).

Let S be an ordered semigroup. Then the following are equivalent:

(i) (x)_N is a left (resp. right) simple subsemigroup of S, for every x2S.

(ii) Every left (resp. right) ideal of S is a right (resp. left) ideal of S and semiprime.

Let (S; ; ) be an ordered semigroup. A subsemigroup F of S is called a f ilter(see [13]) of S if: (1) (8a; b2S)(ab2F !a2F andb2F);and (2) (8a2S)(b2F)(a b !a2F). Forx2S, we denote byN(x)the least …lter ofS generatedx(x2S). ByN we mean the equivalence relation onS de…ned byN :=f(x; y)2S SjN(x) =N(y)g(see[14]). An equivalence relation on Sis called congruence if(a; b)2 implies(ac; bc)2 and(ca; cb)2 for every c2S: A congruence onS is called semilattice congruence onS, if(a; a²)2 and(ab; ba)2 for eacha; b2S (ss[13]). If is a semilattice congruence onS then the -class(x) ofS containingxis a subsemigroup ofS for everyx2S (ss[14]). An ordered semigroupS is called a semilattice of left and right simple semigroups if there exists a semilattice congruence onS such that the -class (x) of S containing xis a left and right simple subsemigroup of S for every x2S.Equivalently, there exists a semilattice Y and a family fS g 2Y of left and right simple subsemigroups ofS such that:

(i)S \S =; 8 ; 2Y; 6= ; (ii)S= [

2Y

S ;

(iii)S S S 8 ; 2Y:

The semilattice congruences in ordered semigroups are de…ned exactly as in semigroups without ordered- so the two de…nitions are equivalent (see[14]).

5.2 Lemma (cf. [9]).

An ordered semigroup(S; ; )is a semilattice of left and right simple semigroups if and only if for all bi-ideals A; B of S, we have

(A] =Aand(AB] = (BA].

5.3 Lemma (cf. [9]).

An ordered semigroup(S; ; )is a semilattice of left and right simple semigroups if and only if for all quasi-ideals A; B of S, we have

(A] =Aand(AB] = (BA].

Proof. Follows from Lemmas 5.1 and 5.2.

5.4 Theorem

An ordered semigroup(S; ; )is a semilattice of left and right simple semigroups if and only if for every fuzzy quasi-ideal f of S, we have

f(a) =f(a²)andf(ab) =f(ba)for alla; b2S.

Proof.=):Suppose thatSis a semilattice of left and right simple semigroups.

Then by hypothesis, there exists a semilatticeY and a familyfS g 2Y of left and right simple subsemigroups ofS such that

(i)S \S =; 8 ; 2Y; 6= ; (ii)S= [

2Y

S ;

(iii)S S S 8 ; 2Y:

(1) Let f be a fuzzy quasi-ideal of S and a 2 S. By Theorem 4.5 and Lemma 4:3, it is enough to prove that a 2 (a²Sa²] for every a 2 S. Since a 2 S = [

2Y

S ; then there exists 2 Y such that a 2 S : Since S is left and right simple we have S = (S a] and S = (aS ]: Then we have (aS ] = (a(S a]] = (aS a]: Since a 2 S we have a 2 (aS a]; then there exists x 2 S such that a axa: Since x 2 (aS a] there exists y 2 S such that, x aya: Thus a axa a(aya)a = a²ya²: Since y 2 S ; we have a²ya²2a²S a² a²Sa² and we havea2(a²Sa²]:

(2)Leta; b2S. By (1), we have

f(ab) =f((ab)²) =f((ab)⁴):

Also we have

(ab)⁴ = (aba)(babab)2Q(aba)Q(babab) (Q(aba)Q(babab)]

= (Q(babab)Q(aba)]( by Lemma5:3)

= ((babab[(bababS\Sbabab)](aba[(abaS\Saba)]]

((babab[(bababS\Sbabab)(aba[(abaS\Saba)]](as ((A](B]] = (AB]) ((babab[bababS)(aba[Saba)]]

= ((babab[bababS)(aba[Saba)](as((A]] = (A]) ((baS)(Sba)] ((baS](Sba]]

= ((baS]\(Sba]](By Lemma 3.3 sinceS is obviously regular)

Then (ab)⁴ (ba)xand (ab)⁴ y(ba) for some x; y 2 S. Then (ba; x) 2 A_(ab)⁴ and (y; ba)2A_(ab)⁴:SinceA_(ab)⁴6=;;we have

f((ab)⁴) = ((f 1)^(1 f))((ab)⁴)

= min[(f 1)((ab)⁴);(1 f)((ab)⁴)]

= min

2

4 _

(y1;z1)2A_(ab)4

minff(y1);1(z1)g; _

(y2;z2)2A_(ab)4

minf1(y2); f(z2)g 3 5 min[minff(ba);1(x)g;minf1(y); f(ba)g]

= min[minff(ba);1g;minf1; f(ba)g]

= min[f(ba); f(ba)]

= f(ba):

Hence f((ab)⁴) f(ab): Sincef(ab) =f((ab)⁴)we havef(ab) f(ba). By symmetry we can prove thatf(ba) f(ab).

(=:SinceN is a semilattice congruence onS, by Lemma5:1;it is enough to prove that every one-sided ideal ofSis a two-sided ideal ofSand semiprime.

LetRbe a right ideal ofS and hence a quasi-ideal ofS. Leta2Rands2S.

SinceR is a quasi-ideal of S; by Lemma3:6; f_R is a fuzzy quasi-idel of S: By hypothesis,

f(as) =f(sa):

Sinceas2RS R;thenf(as) = 1 =)f(sa) = 1:Thussa2R=)SR Rand if a2R,S 3b athenb2R:ThusR is a left ideal ofS. HenceR is an ideal ofS. Letx2S such that x²2R. Thenx2R. In fact: SinceRis a quasi-ideal ofS by Lemma 3.6,fRis a fuzzy quasi-idel of S:By hypothesis,

fR(x²) =fR(x):

Sincex²2R;we havefR(x²) = 1thenfR(x) = 1;and we havex2R:Thus Ris semiprime. Similarly, we can prove that every left ideal ofS is an ideal and semiprime.

5.5 Lemma

Let (S; ; )be an ordered semigroup such that a a² for all a2S. Then for every fuzzy quasi-ideal f of S we have,

f(a) =f(a²)for every a2S.

Proof. Let a 2 S such that a a²: Let f be a fuzzy quasi-ideal of S: By Proposition2:9,f is a fuzzy subsemigroup ofS, then we have,

f(a) f(a²) minff(a); f(a)g=f(a):

5.6 Theorem

Let (S; ; )be an ordered semigroup and a2S such that a a² for all a2S.

Then the following are equivalent:

(1) ab2(baS]\(Sba] for each a; b2S.

(2) For every fuzzy quasi-ideal f of S, we have, f(ab) =f(ba)for every a; b2S:

Proof. (1)=)(2). Letf be a fuzzy quasi-ideal of S:Since ab2(baS]\(Sba], then ab 2 (baS] and we have ab (ba)x for some x 2 S. By (i), we have (ba)x2(xbaS]\(Sxba]:Then(ba)x2(Sxba]and we have(ba)x (yx)(ba)and so,ab (yx)(ba) =)(yx; ba)2Aab:Again, since ab2(Sba];then ab z(ba) for some z 2 S and by (1) we have z(ba)2 (bazS], then z(ba) (ba)(zt)for

some t 2 S. So we haveab (ba)(zt) =) (ba; zt)2 Aab: Since f is a fuzzy quasi-ideal ofS andAab6=;;we have

f(ab) ((f 1)^(1 f))(ab)

= min[(f 1)(ab);(1 f)(ab)]

= min[ _

(p₁;q₁)2Aab

minff(p₁);1(q₁)g; _

(p₂;q₂)2Xab

minf1(p₂); f(q₂)g] min[minff(ba);1(zt)g;minf1(yx); f(ba)g]

= min[minff(ba);1g;minf1; f(ba)g]

= min[f(ba); f(ba)] =f(ba):

By symmetry we can prove thatf(ba) f(ab):

(2)=)(1). Let f be a fuzzy quasi-ideal ofS. Sincea a²for alla2S, by Lemma5:5, we havef(a) = f(a²)for every a; b2S: By (2), we havef(ba) = f(ab)for every a; b2S: By Theorem 5:4, it follows that S; is a semilattice of left and right simple semigroups. Thus by hypothesis, there exist a semilattice Y and a familyfS g 2Y of left and right simple subsemigroups such that

(1)S \S =;for all ; 2Y and 6= ; (2)S= [

2Y

S ;and

(3)S S S for all ; 2Y:

Leta; b2S, then there exist ; 2Y such that a2S andb2S . Then ab2S S S andba2S S S =S :SinceS is left simple we have S = (S c] andS is right simple, by Lemma 4.1, we haveS = (cS ]for eachc2S :Sinceab; ba2S ;we haveab2(baS ]\(S ba] (baS]\(Sba]:

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