Some Characterizations of Strongly Semisimple Ordered Semigroups

TANG JIANAND XIE XIANG-YUN

(1.School of Mathematics and Computational Science,Fuyang Normal College,

Fuyang,Anhui,236037) (2.School of Mathematics and Computational Science,Wuyi University, Jiangmen,Guangdong,529020)

Communicated by Du Xian-kun

Some Characterizations of Strongly Semisimple Ordered Semigroups

TANG JIAN1AND XIE XIANG-YUN2

(1.School of Mathematics and Computational Science,Fuyang Normal College,

Fuyang,Anhui,236037) (2.School of Mathematics and Computational Science,Wuyi University, Jiangmen,Guangdong,529020)

Communicated by Du Xian-kun

In this paper,the concept of quasi-prime fuzzy left ideals of an ordered semigroup S is introduced.Some characterizations of strongly semisimple ordered semigroups are given by quasi-prime fuzzy left ideals of S.In particular,we prove that S is strongly semisimple if and only if each fuzzy left ideal of S is the intersection of all quasi-prime fuzzy left ideals of S containing it.

ordered semigroup,ordered fuzzy point,quasi-prime fuzzy left ideal, strongly semisimple ordered semigroup

1 Introduction

The fundamental concept of a fuzzy set,introduced by Zadeh[1]in 1965,provides a natural framework for generalizing several basic notions of algebra.Following the terminology given by Zadeh[1],Kehayopulu and Tsingelis[2]f i rst considered fuzzy sets in ordered semigroups, and def i ned“fuzzy”analogues for several notations,which have proven to be useful in the theory of ordered semigroups.Moreover,they proved that each ordered groupoid can be embedded into an ordered groupoid having the greatest element(poe-groupoid)in terms of fuzzy sets(see[3]).A theory of fuzzy sets on ordered semigroups has been recently developed(see[4–9]).The concept of ordered fuzzy points of an ordered semigroup S was f i rst introduced by Xie and Tang[7],and prime fuzzy ideals of an ordered semigroupS were studied in[8].Authors also introduced the concepts of weakly prime fuzzy ideals, completely prime fuzzy ideals,completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of an ordered semigroup S,and established the relations among the f i ve types of ideals.Furthermore,Xie and Tang[7]has characterized weakly prime fuzzy ideals, completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of S by their level ideals.

As we know,fuzzy ideals(left,right ideals)with special properties of ordered semigroups always play an important role in the study of ordered semigroups structure.The ordered fuzzy points of an ordered semigroup S are key tools to describe the algebraic subsystems of S.Motivated by the study of prime fuzzy ideals in rings,semigroups and ordered semigroups, and also motivated by Kehayopulu and Tsingelis[10]'s work in ordered semigroups in terms of fuzzy subsets,in this paper we attempt to introduce and give a detailed investigation of quasi-prime fuzzy left ideals of an ordered semigroup S.We characterize quasi-prime fuzzy left ideals of S by ordered fuzzy points of S.Furthermore,we introduce the concept of fuzzy m-systems of an ordered semigroup S,and prove that a fuzzy left ideal f of S is quasi-prime if and only if 1-f is a fuzzy m-system.Finally,we characterize the strongly semisimple ordered semigroups by quasi-prime fuzzy left ideals of S,and prove that S is strongly semisimple if and only if each fuzzy left ideal of S is the intersection of all quasiprime left ideals of S containing it.As an application of the results of this paper,the corresponding results for semigroups(without ordered)are also obtained.

2 Preliminaries and Some Notations

Throughout this paper unless stated otherwise S stands for an ordered semigroup,that is, a semigroup S with an order relation“≤”such that a≤b implies xa≤xb and ax≤bx for any x∈S(for example,see[11]).For convenience we use the notation S1:=S∪{1}, where 1·a=a·1:=a for all a∈S and 1·1=1.A nonempty subset I of S is called a left (resp.right)ideal of S if

(1)SI?I(resp.IS?I);

(2)If a∈I,b≤a with b∈S,then b∈I.

I is called an ideal of S if I is both a left and a right ideal of S(see[11]).Let L be a left ideal of S.L is called quasi-prime if for any two left ideals L1,L2of S,L1L2?L implies L1?L or L2?L;L is called quasi-semiprime if for any left ideal P of S such that P2?L, we have P?L(see[11]).

For H?S,we def i ne

For H={a},we write(a]instead of({a}].We denote by L(a)the left ideal of S generated by a∈S.Then(see[12])

Lemma 2.1[12]Let S be an ordered semigroup.Then the following statements hold:

(1)A?(A]for all A?S;

(2)If A?B?S,then(A]?(B];

(3)(A](B]?(AB]for all A,B∈S;

(4)((A]]=(A]for all A?S;

(5)For every left(resp.right)ideal T of S,one has(T]=T;

(6)If A,B are left ideals of S,then(AB],A∩B,A∪B are left ideals of S;

(7)(SaS],(Sa]are an ideal and a left deal of S for all a∈S,respectively.

Def i nition 2.1[12]Let M be a nonempty subset of an ordered semigroup S.M is called an m-system if for any a,b∈M,there exists an x∈S such that(axb]∩M/=?.

A function f from S to the real closed interval[0,1]is a fuzzy subset of S.The ordered semigroup S itself is a fuzzy subset of S such that

(the fuzzy subset S is also denoted by 1,see[13]).Let f and g be two fuzzy subsets of S. Then the inclusion relation f?g is def i ned by

1-f is a fuzzy subset of S def i ned by

f∩g and f∪g are fuzzy subsets of S def i ned by

respectively.The set of all fuzzy subsets of S is denoted by F(S).One can easily show that (F(S),?,∩,∪)forms a complete lattice with the maximum element S and the minimum element 0,which is a mapping from S into[0,1]def i ned by

Let(S,·,≤)be an ordered semigroup.For x∈S,we def i ne

The product f?g of f and g is def i ned by

It is well known(cf.[3])that this operation“?”is associative.

Let A be a nonempty subset of S.We denote by fAthe characteristic mapping of A, that is,the mapping fAfrom S to[0,1]is def i ned by

Let(S,·,≤)be an ordered semigroup.A fuzzy subset f of S is called a fuzzy left(resp. right)ideal of S if

(1)f(xy)≥f(y)(resp.f(xy)≥f(x))for all x,y∈S;

(2)x≤y implies f(x)≥f(y).Or equivalently:

(1)S?f?f(resp.f?S?f);

(2)x≤y implies f(x)≥f(y)

(see[2],[9]).A fuzzy ideal of S is a fuzzy subset of S which is both a fuzzy left and a fuzzy right ideal of S.

Lemma 2.2[2]Let S be an ordered semigroup and ?/=A?S.Then A is a left(resp. right)ideal of S if and only if the characteristic mapping fAof A is a fuzzy left(resp.right) ideal of S.

We denote by aλan ordered fuzzy point of an ordered semigroup S,where

{

It is easy to see that an ordered fuzzy point of an ordered semigroup S is a fuzzy subset of S.For any fuzzy subset f of S,we also denoteby aλ∈f in the sequel(see[7]).

Def i nition 2.2[8]Let f be a fuzzy subset of S.We def i ne(f]by the rule that

A fuzzy subset of S is called strongly convex if f=(f].

Lemma 2.3[7]Let aλbe an ordered fuzzy point of S.Then the fuzzy left ideal generated by a,denoted by L(a),is

where L(a)is the left ideal of S generated by a.

Let λfAbe a fuzzy subset of S def i ned as follows:

Clearly,λfAis a generalization of the characteristic mapping fAof A.

Lemma 2.4[7]Let A,B be any nonempty subsets of an ordered semigroup S.Then for any λ∈(0,1]the following statements are true:

(1)λfA?λfB=λf(AB].In particular,fA?fB=f(AB];

(2)If A is a left ideal of S,then λfAis a fuzzy left ideal of S.

Lemma 2.5[7]Let aλ,bμ(λ/=0,μ/=0)be ordered fuzzy points of S,and f,g be fuzzy subsets of S.Then the following statements are true:

and S?aλis a fuzzy left ideal of S;

(2)aλ?bμ=(ab)λ∧μ.In particular,aλ?aλ=(a2)λ;

(3)L(ar)=ar∪S?ar;

(4)L(aλ)2?S?aλ;

(5)If f?g and h∈F(S),then f?h?g?h,h?f?h?g.

Def i nition 2.3Let f be any fuzzy subset of an ordered semigroup S.The set

is called a level subset of f.

Lemma 2.6Let S be an ordered semigroup and f be a fuzzy subset of S.Then f is a fuzzy left ideal of S if and only if the level subset ft(t∈(0,1])of f is a left ideal of S for ft/=?.

Proof.Since the proof is similar to the proof of Lemma 2.7 in[7],we omit it.

The reader is referred to[7],[14]–[15]for notations and terminologies not def i ned in this paper.

3 Fuzzy m-systems and Quasi-prime Fuzzy Left Ideals of Ordered Semigroups

Def i nition 3.1Let S be an ordered semigroup.Then a fuzzy left ideal f of S is called quasi-prime if for any two fuzzy left ideals f1and f2of S,f1?f2?f implies that f1?f or f2?f;f is called quasi-semiprime if for any fuzzy left ideal g of S,g2?f implies that g?f.

Theorem 3.1Let S be an ordered semigroup.Then a fuzzy left ideal f of S is quasiprime if and only if for any two ordered fuzzy points xr,yt∈S(rt＞0),xr?S?yt?f implies that xr∈f or yt∈f.

Proof.Let xrand ytbe ordered fuzzy points of S such that xr?S?yt?f.Then

Since f is quasi-prime,S?xrand S?ytare fuzzy left ideals of S,so we have

If S?xr?f,then,by Lemma 2.5(4),

Conversely,let f1,f2be fuzzy left ideals of S such that f1?f2?f.If f1/?f and f2/?f, then there exist x,y∈S such that f1(x)＞f(x)and f2(y)＞f(y).Let r=f1(x),t=f2(y). Then

and so

By hypothesis,xr∈f or yt∈f.If xr∈f,then

It is impossible.

Theorem 3.2Let S be an ordered semigroup.Then a left ideal L of S is quasi-prime if and only if fLis a quasi-prime fuzzy left ideal of S.

Proof.Let L be a quasi-prime left ideal of S.By Lemma 2.2,fLis a fuzzy left ideal of S.For any two fuzzy left ideals f1and f2of S,if f1?f2?fL,then f1?fL,or f2?fL.In fact,if f1/?fL,f2/?fL,then there exist x,y∈S such that f1(x)＞fL(x),f2(y)＞fL(y). Thus

It implies that x,y/∈L.We now show that there exists an s∈S such that(xsy]/?L. Indeed,if(xSy]?L,then(Sx](Sy]?L.Since(Sx]and(Sy]are left ideals of S and L is a quasi-prime left ideal of S,we have

Say(Sx]?L.Then L(x)2?(Sx]?L.Thus x∈L(x)?L,which is impossible.Now if a∈(xsy]such that a/∈L,then fL(a)=0,and

which contradicts the fact that

Therefore fLis a quasi-prime fuzzy left ideal of S.

Conversely,let fLbe a quasi-prime fuzzy left ideal of S,and A,B be left ideals of S such that AB?L.Then,by Lemma 2.1,we have

Thus,by Lemma 2.4,

By hypothesis and Lemma 2.2,since fLis quasi-prime,we have fA?fLor fB?fL,that is,A?L or B?L.

Lemma 3.1If f is a nonconstant quasi-prime fuzzy left ideal of an ordered semigroup S,then|Im(f)|=2.

Proof.The proof is similar to the proof of Lemma 4.3 in[7]with a slight modif i cation.

Theorem 3.3If f is a nonconstant quasi-prime fuzzy left ideal of an ordered semigroup S,then there exists an x0∈S such that f(x0)=1.

Proof.By Lemma 3.1,|Im(f)|=2.If f(x)＜1 for all x∈S,then

Thus there exist x,y∈S and m∈(0,1]such that

For t1,t2∈(0,1)with t＜t1＜s＜t2＜m,we have

Since f is a quasi-prime fuzzy left ideal of S,we have

that is,f(x)≥t1or f(y)≥t2,which is impossible.Thus there exists an x0∈S such that f(x0)=1.

Theorem 3.4If f is a quasi-prime fuzzy left ideal of an ordered semigroup S,then each level subset ft(t∈(0,1])is a quasi-prime left ideal of S for ft/=?.

Proof.Since the proof is similar to the proof of Theorem 4.5 in[7],we omit it.

By Theorems 3.3 and 3.4,we have

Corollary 3.1If f is a quasi-prime fuzzy left ideal of an ordered semigroup S,then f1is a quasi-prime left ideal of S.

Remark 3.1The inverse of Theorem 3.4 is not true.For example,let L be a quasi-prime left ideal of S,and

where 0＜t＜1.Then f is a fuzzy left ideal of S.For any t∈(0,1],if ft/=?,then ft=L, which is a quasi-prime left ideal of S.But f is not a quasi-prime fuzzy left ideal of S since f1=?.

Now quasi-prime fuzzy left ideals of S can be characterized.

Theorem 3.5Let f be a fuzzy subset of an ordered semigroup S.Then f is a quasiprime fuzzy left ideal of S if and only if f satisf i es the following conditions:

(1)|Im(f)|≤2;

(2)f1/=?,and f1is a quasi-prime left ideal of S.

Proof.The proof is similar to the proof of Theorem 4.8 in[7]with suitable modif i cation.

In order to characterize the quasi-prime fuzzy left ideals of ordered semigroups,we need the following concept.

Def i nition 3.2Let S be an ordered semigroup.Then a fuzzy subset f of S is called a fuzzy m-system if for any s,t∈[0,1)and a,b∈S,f(a)＞s,f(b)＞t imply that there exists an x∈S such that f(y)＞s∨t for some y∈(axb].

Theorem 3.6Let M be a nonempty subset of an ordered semigroup S.Then M is an m-system of S if and only if fMis a fuzzy m-system.Proof.For any s,t∈[0,1)and a,b∈S,if

then a,b∈M.Since M is an m-system of S,there exists an x∈S such that(axb]∩M/=?. Let y∈(axb]∩M.Then fM(y)=1.Thus fM(y)＞s∨t for some y∈(axb].

Conversely,let a,b∈M.Then

Thus for any s,t∈[0,1),

which imply that there exists an element x∈S such that fM(y)＞s∨t for some y∈(axb] and that fM(y)=1,that is,y∈M.It thus follows that

Theorem 3.7Let f be a proper fuzzy left ideal of an ordered semigroup S.Then f is quasi-prime if and only if 1-f is a fuzzy m-system.

Proof.For any s,t∈[0,1),a,b∈S,if

then

which imply that

Since f is a quasi-prime fuzzy left ideal of S,by Theorem 3.1,there exists an xr∈S such that

Thus,there exists a y∈(axb]such that

Therefore,

Conversely,let as,bt∈S(ts＞0)such that as?S?bt?f.If as/∈f and bt/∈f,then there exist a1∈(a],b1∈(b]such that

Thus

By hypothesis,there exists an x∈S such that

for some y∈(a1xb1],that is,f(y)＜s∧t.Since S is an ordered semigroup,one gets y∈(axb].It thus follows,by Lemma 2.5(2),that

which is impossible.

4 Strongly Semisimple Ordered Semigroups

A fuzzy left ideal f of an ordered semigroup S is called idempotent if f=f?f,that is, f=f2.Clearly,a fuzzy left ideal f of an ordered semigroup S is idempotent if and only if f?f?f.

Def i nition 4.1An ordered semigroup S is called strongly semisimple if L=(L2]for every left ideal L of S.

Lemma 4.1For an ordered semigroup S the following statements are equivalent:

(1)S is strongly semisimple;

(2)For any a∈S,a∈(SaSa].

The following theorems characterize the strongly semisimple ordered semigroups by means of quasi-prime fuzzy left ideals of S.

Theorem 4.1Let S be an ordered semigroup.Then the following statements are equivalent:

(1)Every fuzzy left ideal of S is idempotent;

(2)For any two fuzzy left ideals f1and f2of S,f1∩f2?f1?f2; (3)For any ordered fuzzy point ar∈S,L(ar)=L(ar)2;

(4)For any ordered fuzzy point ar∈S,ar∈S?ar?S?ar;

(5)Every fuzzy left ideal of S is a quasi-semiprime fuzzy left ideal of S;

(6)Every fuzzy left ideal of S is the intersection of all quasi-prime fuzzy left ideals of S containing it.

Proof.(1)?(2).Let f1and f2be any two fuzzy left ideals of S.Then

Moreover,for any x,y∈S with x≤y,we have

Thus f1∩f2is a fuzzy left ideal of S.By(1)and Lemma 2.5(5)one has

(2)?(3).For any ordered fuzzy point ar∈S,if f1=f2=L(ar),then by(2)we have

On the other hand,By Lemma 2.5(5)we have

Therefore,L(ar)=L(ar)2.

(3)?(4).For any ordered fuzzy point ar∈S,by(3)and Lemma 2.5,we have

Thus

and

Therefore,

which implies that

(4)?(5).Let g be a fuzzy left ideal of S with g2?f.Then,for any ar∈g,by(4),we have

(5)?(1).Let f be any fuzzy left ideal of S.Then f2is also a fuzzy left ideal of S.By (5),since f2?f2,we have f?f2.Clearly,f2?f.It follows that f2=f. (2)?(6).Let f be a fuzzy left ideal of S,and

N={gα|gαis a quasi-prime fuzzy left ideal of S such that f?gα}.

Then,clearly,

Let

B={hβ|hβis a fuzzy left ideal of S such thatClearly,B/=? since f∈B.Thus(B,?)is an ordered set.Let C be a chain in B.Then the setis a fuzzy left ideal of S and Since for any hβ∈C,f(a)=hβ(a), we have

is an upper bound of C in B.By Zorn's Lemma,B has a maximal element.Denote it by hmax.Then ar/∈hmax.We now prove that hmaxis a quasiprime fuzzy left ideal of S.Let f1and f2be two fuzzy left ideals of S with f1?f2?hmax. Then,by(2),

Thus

We claim that

that is,f1?hmaxor f2?hmax.In fact,by hmax=(hmax∪f1)∩(hmax∪f2),we have

This implies

Since hmaxis maximal with respect to the properties f?hmaxand hmax(a)=f(a),we have

(6)?(1).Let f be any fuzzy left ideal of S.Then f2is also a fuzzy left ideal of S.By (6)we have

where M is the set of all quasi-prime fuzzy left ideals of S containing f2.

For any g∈M,clearly,f?f?g.Since g is quasi-prime,f?g holds.Thus

Therefore,

Theorem 4.2An ordered semigroup S is strongly semisimple if and only if every fuzzy left ideal of S is idempotent.

Proof.Similarly to the proof of Proposition 3.7(1)in[7],we can show that

if and only if

Thus the proof is completed by Theorem 4.1 and Lemma 4.1.

Theorem 4.3Let S be a commutative ordered semigroup.Then the fuzzy left ideals of S are quasi-prime if and only if they form a chain and S is strongly semisimple.

Proof.Let g and h be fuzzy left ideals of S.Since g?h is a fuzzy left ideal of S,by

we have

Thus the fuzzy left ideals of S form a chain.Moreover,for any fuzzy left ideal f of S, obviously,f2?f.Since

we have

so that

By Theorem 4.2,S is strongly semisimple.

Conversely,let f,g be two fuzzy left ideals of S,and f?g?h.Since the fuzzy left ideals of S form a chain,i.e.,f?g or g?f,we have

By hypothesis,f?h or g?h holds.

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tion:20M10,06F05

A

1674-5647(2013)02-0155-12

Received date:Sept.15,2011.

The NSF(10961014)of China,the NSF(S2011010003681)of Guangdong Province,the Science and Technology Projects(2010B010600039)of Guangdong Province,the Excellent Youth Talent Foundation (2012SQRL115ZD)of Anhui Province,the University Natural Science Project(KJ2012B133)of Anhui Province and the NSF(2007LZ01)of Fuyang Normal College.

E-mail address:tangjian0901@126.com(Tang J).