N -Fuzzy Ideals in Ordered Semigroups

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Volume 2009, Article ID 814861,14pages doi:10.1155/2009/814861

Research Article

N -Fuzzy Ideals in Ordered Semigroups

Asghar Khan,

¹

Young Bae Jun,

²

and Muhammad Shabir

³

1Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan

2Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, South Korea

3Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan

Correspondence should be addressed to Asghar Khan,[email protected] Received 9 July 2009; Accepted 26 October 2009

Recommended by David Dobbs

We introduce the concept ofN-fuzzy leftrightideals in ordered semigroups and characterize ordered semigroups in terms ofN-fuzzy leftrightideals. We characterize left regularright regular and left simpleright simple ordered semigroups in terms ofN-fuzzy leftN-fuzzy rightideals. The semilattice of leftrightsimple semigroups in terms ofN-fuzzy leftright ideals is discussed.

Copyrightq2009 Asghar Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

A fuzzy subset f of a given set S is described as an arbitrary function f : S → 0,1, where0,1is the usual closed interval of real numbers. This fundamental concept was first introduced by Zadeh in his pioneering paper1of 1965, which provides a natural framework for the generalizations of some basic notions of algebra, for example, set theory, group theory, ring theory, groupoids, real analysis, measure theory, topology, and diﬀerential equations, and so forth. Rosenfeldsee2was the first who introduced the concept of a fuzzy set in a group. The concept of a fuzzy ideal in semigroups was first developed by Kurokisee3–8.

He studied fuzzy ideals, fuzzy bi-ideals, fuzzy quasi-ideals, and fuzzy semiprime ideals of semigroups. Fuzzy ideals and Green’s relations in semigroups were studied by McLean and Kummer in9. Dib and Galham in10introduced the definitions of a fuzzy groupoid and a fuzzy semigroup and studied fuzzy ideals and fuzzy bi-ideals of a fuzzy semigroup. Ahsan et al. in 11 characterized semisimple semigroups in terms of fuzzy ideals. A systematic exposition of fuzzy semigroups by Mordeson et al. appeared in 12, where one can find theoretical results on fuzzy semigroups and their use in fuzzy coding, fuzzy finite state machines, and fuzzy languages. The monograph by Mordeson and Malik see13deals with the applications of fuzzy approach to the concept of automata and formal languages.

Fuzzy sets in ordered semigroups/ordered groupoids were first introduced by Kehayopulu

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and Tsingelis in14. They also introduced the concepts of fuzzy bi-ideals and fuzzy quasi- ideals in ordered semigroups in14,15.

In16, Shabir and Khan introduced the concept of a fuzzy generalized bi-ideal of ordered semigroups and characterized diﬀerent classes of ordered semigroups by using fuzzy generalized bi-ideals. They also gave the concept of fuzzy leftresp., bi-filters in ordered semigroups and gave the relations of fuzzy bi-filters and fuzzy bi-ideal subsets of ordered semigroups in17.

In this paper, we introduce the concept of N-fuzzy left resp., right ideals and characterize regular, left, and right simple ordered semigroups and completely regular ordered semigroups in terms ofN-fuzzy leftresp., rightideals. In this respect, we prove that a regular ordered semigroupSis left simple if and only if everyN-fuzzy left idealf of Sis a constant function. We also prove thatSis left regular if and only if for everyN-fuzzy left idealf ofS we havefa fa²for everya∈ S. Next we characterize semilattice of left simple ordered semigroups in terms ofN-fuzzy left ideals. We prove that an ordered semigroupSis a semilattice of left simple semigroups if and only if for everyN-fuzzy left idealfofS,fa fa²andfab fbafor alla, b∈S.

2. Preliminaries

By an ordered semigroupor po-semigroupwe mean a structureS,·,≤in which OS1 S,·is a semigroup,

OS2 S,≤is a poset,

OS3 ∀a, b, x∈S a≤b⇒ax≤bxandxa≤xb.

LetS,·,≤be an ordered semigroup. A nonempty subsetAofSis called a leftresp., rightideal ofSsee14if

iSA⊆Aresp.,AS⊆A,

ii ∀a∈A ∀b∈S b≤a⇒b∈A.

Ais called a two-sided ideal or simply an ideal ifAis both left and right ideal ofS.

ForA⊆S, denoteA:{t∈S|t≤hfor someh∈A}. IfA{a}, then we writea instead of{a}. LetA, B⊆S, thenA⊆A,AB⊆AB, andA A.

LetSbe an ordered semigroup andfa fuzzy subset ofS. Thenfis called a fuzzy left resp., rightideal ofSif

i ∀x, y∈S x≤y⇒fx≥fy,

ii x, y∈S fxy≥fy resp.,fxy≥fx.

A fuzzy left and right ideal ofSis called a fuzzy two-sided ideal or simply a fuzzy ideal ofS.

3. N -Fuzzy Left (Resp., Right) Ideals

LetSbe an ordered semigroup. By a negative fuzzy subsetbrieflyN-fuzzy subsetf ofSwe mean a mappingf :S → −1,0.

Definition 3.1. LetS,·,≤be an ordered semigroups andfanN-fuzzy subset ofS. Thenfis called anN-fuzzy leftresp., rightideal ofSif

1 ∀x, y∈S x≤y⇒fx≤fy,

2 ∀x, y∈S fxy≤fy resp.,fxy≤fx.

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AnN-fuzzy left and right idealfofSis called anN-fuzzy two-sided ideal ofS.

For anyN-fuzzy subsetfofSandt∈−1,0the set

L f;t

:

x∈S|fx≤t

3.1

is called theN-level subset off.

Theorem 3.2. LetS,·,≤be an ordered semigroup. AnN-fuzzy subsetf ofS is anN-fuzzy left (resp., right) ideal ofSif and only if it satisfies

∀t∈−1,0L f;t

/∅is a left ideal iﬀf is anN-fuzzy left ideal

. 3.2

Proof. Suppose thatf is anN-fuzzy left ideal ofS. Letx, y ∈ Sbe such thatx ≤ y. Ify ∈ Lf;t, thenfy≤t. Sincex≤y,we havefx≤fyandfx≤tand we havex∈Lf;t.

Let x, y ∈ S be such thaty ∈ Lf;t. Then fy ≤ t, since fxy ≤ fy. Then one has fxy≤timplies thatxy∈Lf;t. ThusSLf;t⊆Lf;t.

Conversely, assume that for allt ∈ −1,0such thatLf;t/∅, the setLf;tis a left ideal ofS. Letx, y ∈Sbe such thatx ≤y. Iffy 0, then sincefx ≤0 for allx∈ S, we havefy≤fx. Iffy t, theny ∈Lf;tand sincex ≤y ∈Lf;t, andLf;tis a left ideal ofS, we havex ∈Lf;tand sofx ≤t fy. Let x, y ∈S. Iffy 0, then since fxy≤0 for allx, y∈Swe havefxy≤fy. Iffy t, theny∈Lf;tand sinceLf;t is a left ideal ofS, we havexy∈Lf;t. Thenfxy≤tfy.

By left-right dual of the above theorem, we have the following theorem.

Theorem 3.3. LetS,·,≤be an ordered semigroup. AnN-fuzzy subsetf ofS is anN-fuzzy left (resp., right) ideal ofSif and only if it satisfies

∀t∈−1,0L f;t

/∅is a right ideal iﬀf is anN-fuzzy right ideal

. 3.3

Example 3.4. LetS {a, b, c, d, e, f}be the ordered semigroup defined by the multiplication and the order as follows:

· a b c d e f a a a a d a a b a b b d b b c a b c d e e d a a d d d d e a b c d e e f a b c d e f

≤:

a, a,b, b,c, c,d, d,e, e, f, e

, f, f

.

3.4

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Then left ideals ofSare{a},{d},{a, b},{a, d},{a, b, d},{a, b, c, d},{a, b, d, e, f},andS see18. Definef:S → −1,0by

fa −0.8, fb −0.6, fd −0.5, fc −0.4, fe f f

−0.2. 3.5 Then

L f;t

:

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

S if t∈−0.2,0, {a, b, c, d} if t∈−0.4,−0.2, {a, b, d} if t∈−0.5,−0.4, {a, b} if t∈−0.6,−0.5, {a} if t∈−0.8,−0.6,

∅ if t∈−1,−0.8.

3.6

ThenLf;tis a left ideal ofS,and byTheorem 3.2,fis anN-fuzzy left ideal ofS.

LetSbe an ordered semigroup and∅/A ⊆S. The characteristicN-functionκ_A :S → {−1,0}ofAis defined by

κ_A:S−→−1,0, x−→κ_Ax:

⎧⎨

⎩

−1 ifx∈A,

0 ifx /∈A. 3.7

Theorem 3.5. Let S,·,≤ be an ordered semigroup and ∅/A ⊆ S. Then the followings are equivalent.

iAis a left (resp., right) ideal ofS.

iiThe characteristicN-function

κAx

⎧⎨

⎩

−1 ifx∈A,

0 ifx /∈A 3.8

ofAis anN-fuzzy left (resp., right) ideal ofS.

Proof. i⇒iiSuppose thatAis a left ideal ofS. Letx, y∈S,x≤y. Ify∈A, thenκ_Ay −1.

Sincex ≤ y ∈ A andA is a left ideal ofS, we have x ∈ A. Thenκ_Ax −1 and hence κAx≤κAy. Ify /∈A, thenκAy 0. SinceκAis anN-fuzzy subset ofS, we haveκAx≤0 for allx∈S. Henceκ_Ax≤κ_Ay.

Letx∈Sandy∈A. Thenκ_Ay −1. SinceAis a left ideal ofSandy∈A, we have xy ∈ A. ThenκAxy −1, and we haveκAxy ≤ κAy. Ify /∈A. ThenκAy 0. Since κ_Axy≤0 for allx, y∈S. Henceκ_Axy≤κ_Ay. Thusκ_Ais anN-fuzzy left ideal ofS.

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ii⇒iAssume that

κ_Ax

−1 ifx∈A,

0 if x /∈A 3.9

is anN-fuzzy left ideal ofS. Letx, y ∈S,x≤ y. Ify ∈A, thenκ_Ay −1. Sincex ≤y, we haveκ_Ax≤κ_Ay. Thenκ_Ax −1 and we havex∈A.

Letx∈Sandy∈A. ThenκAy −1. SinceκAis anN-fuzzy left ideal ofS, we have κ_Axy≤κ_Ay. Henceκ_Axy −1 andxy∈A. ThusAis a left ideal ofS.

A subsetT of an ordered semigroupSis called semiprimesee15if for everya∈S such that a² ∈ T, we have a ∈ T. Equivalent definition: for each subset Aof S such that A²⊆T, we haveA⊆T.

4. Characterization of Left Simple and Left Regular Ordered Semigroups

Lemma 4.1cf.15. LetS,·,≤be an ordered semigroup. Then the followings are equivalent.

i x_Nis a left simple subsemigroup ofS, for everyx∈S.

iiEvery left ideal ofSis a right ideal ofSand semiprime.

An ordered semigroupSis regular (see [19]) if for everya∈S, there existsx ∈Ssuch that a≤axa.

Equivalent definitions are 1(∀a∈S) (a∈aSa), 2(∀A⊆S) (A⊆ASA).

An ordered semigroupSis left (resp., right ) simple (see [15]) if for every left (resp., right) ideal AofS, one hasAS,andSis called simple if it is left simple and right simple.

Lemma 4.2cf. 15. An ordered semigroupS,·,≤is left (resp., right) simple if and only if for everya∈S,Sa S(resp.,aS S).

Theorem 4.3. For a regular ordered semigroupS, the following conditions are equivalent.

iSis left simple.

iiEveryN-fuzzy left ideal ofSis a constantN-function.

Proof. i⇒iiLetS be a left simple ordered semigroup,f an N-fuzzy left ideal ofS,and a∈S. We consider the set

E_S:

e∈S|e² ≥e

. 4.1

ThenES/∅. In fact, sinceSis regular anda∈S, there existsx∈Ssuch thata≤axa.

It follows fromOS3that

ax² axax≥ax, 4.2 and soax∈E_Sand henceE_S/∅.

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1 f is a constant N-function on ES. Let t ∈ ES. Since S is left simple and t ∈ S we have St S. Since e ∈ S,e ∈ St, so there exists z ∈ S such that e ≤ zt. Hence e²≤ztzt ztzt. Sincefis anN-fuzzy left ideal ofS, we have

f e²

≤fztzt≤ft. 4.3

Sincee ∈ E_S, we havee² ≥ e. Since f is anN-fuzzy left ideal ofS, we havefe ≤ fe². Thusfe ≤ ft. Besides, sinceSis left simple ande ∈ S, we haveSe S. Since t∈S Se, so there existsy∈Ssuch thatt≤ye. Hencet² ≤yeye yeye. Sincefis anN-fuzzy left ideal ofS, we have

f t²

≤f yey

e

≤fe. 4.4

On the other hand,t ∈ ES, we havet² ≥ tand soft ≤ ft² ≤ fyeye ≤ fe. Hence ft fe.

2fis a constantN-function onS. Leta∈S. SinceSis regular, there existsx∈Ssuch thata≤axa. We consider the elementxa∈S. Then it follows byOS3that

xa²xaxa≥xa. 4.5 Hencexa ∈ ESand by1, we havefxa ft. Besides, sincef is anN-fuzzy left ideal ofS, we havefxa≤fa. Thusft≤fa. On the other hand, sinceSis left simple andt ∈S,S St. Sincea∈S, we havea≤ stfor somes ∈ S. Sincef is anN-fuzzy left ideal ofS, we havefa≤fst≤ft. Thusft fa.

ii⇒iLeta ∈ S. Then the setSais a left ideal ofS. In fact,SSa SSa ⊆ SSa⊆Sa. Ifx∈SaandSy≤x, theny∈Sa Sa. SinceSais a left ideal ofS, byTheorem 3.5, the characteristicN-functionκ_SaofSa,

κ_Sa:S−→−1,0, x−→κ_Sax:

⎧⎨

⎩

−1 ifx∈Sa,

0 ifx /∈Sa, 4.6

is a fuzzy left ideal ofS. By hypothesis, κ_Sa is a constantN-function; that is, there exists c∈ {−1,0}such that

κ_Sax c, for everyx∈S. 4.7

LetSa ⊂ Sandt ∈ Sbe such that t /∈Sa. Thenκ_Sat 0. On the other hand, since a² ∈ Sa, we have κ_Saa² 0, a contradiction to the fact thatκ_Sa is a constant N-mapping. HenceS Sa.

From left-right dual ofTheorem 4.3, we have the following.

Theorem 4.4. For a regular ordered semigroup. The following statements are equivalent.

1Sis right simple.

2EveryN-fuzzy right ideal ofSis a constantN-function.

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An ordered semigroupS,·,≤is left (resp., right ) regular [20] if for everya∈Sthere exists x∈Ssuch thata≤xa²(resp.,a≤a²x).

Equivalent definitions are

1(∀a∈S) (a∈Sa²(resp.,a∈a²S)), 2(∀A⊆S) (A⊆SA²(resp.,A⊆A²S)).

An ordered semigroupS,·,≤is intraregular (see [16]) if for everya∈S, there existx, y∈S such thata≤xa²y.

Equivalent definitions are 1(∀a∈S) (a∈Sa²S), 2(∀A⊆S) (A⊆SA²S).

An ordered semigroupSis called completely regular (see [21]) if it is regular, left regular, and right regular.

Lemma 4.5cf.21. An ordered semigroupSis completely regular if and only ifA⊆A²SA²for everyA⊆S. Equivalently,a∈a²Sa²for everya∈S.

Theorem 4.6. An ordered semigroupS,·,≤is left regular if and only if for eachN-fuzzy left ideal fofS, one has

fa f a²

∀a∈S. 4.8

Proof. Suppose thatf is anN-fuzzy left ideal ofSand leta∈S. SinceSis left regular, there existsx∈Ssuch thata≤xa². Sincefis anN-fuzzy left ideal ofS, we have

fa≤f xa²

≤f a²

≤fa. 4.9

Conversely, leta∈S. We consider the left idealLa² a²∪Sa²ofS, generated by a². Then byTheorem 3.5, the characteristicN-function

κ_La²:S−→−1,0|x−→κ_La²x:

⎧⎨

⎩

−1 ifx∈L a²

, 0 ifx /∈L

a² 4.10

is anN-fuzzy left ideal ofS. By hypothesis we have

κ_La²a κ_La² a²

. 4.11

Sincea² ∈La², we haveκ_La²a² −1 andκ_La²a −1. Thena∈ La² a²∪ Sa²anda ≤ yfor somey ∈ a² ∪Sa². Ify a², thena ≤ y a² aa aa² ∈ Sa² and a∈Sa². Ifyxa²for somex∈S, thena≤yxa²∈Sa², anda∈Sa².

From left-right dual ofTheorem 4.6, we have the following theorem.

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Theorem 4.7. An ordered semigroupS,·,≤is right regular if and only if for eachN-fuzzy right idealfofS, one has

fa f a²

∀a∈S. 4.12

From22and by Theorems4.6,4.7, andLemma 4.5, we have the following characterization theorem for completely regular ordered semigroups.

Theorem 4.8. LetS,·,≤be an ordered semigroup. Then the following statements are equivalent.

iSis completely regular.

iiFor eachN-fuzzy bi-idealfofSone has

fa f a²

∀a∈S. 4.13

iiiFor eachN-fuzzy left idealgand eachN-fuzzy right idealhofSwe have

ga g a²

, ha h a²

∀a∈S. 4.14

An ordered semigroupS,·,≤is called left (resp., right ) duo if every left (resp., right) ideal of Sis a two-sided ideal ofS. An ordered semigroupSis called duo if it is both left and right duo.

Definition 4.9. An ordered semigroupS,·,≤is calledN-fuzzy leftresp., rightduo if everyN- fuzzy leftresp., rightideal ofSis anN-fuzzy two-sided ideal ofS. An ordered semigroup Sis calledN-fuzzy duo if it is bothN-fuzzy left andN-fuzzy right duo.

Theorem 4.10. LetS,·,≤be a regular ordered semigroup. Then the followings are equivalent.

iSis left duo.

iiSisN-fuzzy left duo.

Proof. i⇒iiLetSbe left duo andf anN-fuzzy left ideal ofS. Leta, b ∈ S. Then the set Sais a left ideal ofS. In fact,SSa SSa⊆SSa⊆Saand ifx∈SaandSy≤x, then y ∈ Sa Sa. SinceSis left duo, then Sais a two-sided ideal of S. SinceSis regular, there existsx∈Ssuch thata≤axa,

ab≤axab∈aSab⊆SaS⊆SaS⊆Sa. 4.15 Thusab∈Sa Saandab ≤xafor somex∈S. Sincef is anN-fuzzy left ideal ofS, we have

fab≤fxa≤fa. 4.16

Letx, y∈Sbe such thatx≤y. Thenfx≤fy, becausefis anN-fuzzy left ideal of S. Thusfis anN-fuzzy right deal ofSandSisN-fuzzy left duo.

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ii⇒iLetSbeN-fuzzy left duo andAa left ideal ofS. Then the characteristicN- functionκ_AofAis anN-fuzzy left ideal ofS. By hypothesis,κ_Ais anN-fuzzy right ideal of S, and byTheorem 3.5,Ais a right ideal ofS. ThusSis left duo.

By the left-right dual ofTheorem 4.10, we have the following.

iSis right duo.

iiSisN-fuzzy right duo.

iEvery bi-ideal ofSis a right ideal ofS.

iiEveryN-fuzzy bi-ideal ofSis anN-fuzzy right ideal ofS.

Proof. i⇒iiLeta, b ∈Sandf anN-fuzzy bi-ideal ofS. ThenaSais a bi-ideal ofS. In fact,aSa² ⊆aSaaSa⊆aSa,aSaSaSa aSaSaSa⊆aSaand ifx∈aSa andSy≤x∈aSa, theny∈aSa aSa. SinceaSais a bi-ideal ofS, by hypothesis aSais right ideal ofS. Sincea ∈SandSis regular, there existsx ∈Ssuch thata≤ axa, then

ab≤axab∈aSaS⊆aSaS⊆aSa. 4.17

Thenab≤azafor somez∈S. Sincefis anN-fuzzy bi-ideal ofS, we have fab≤faza≤max

fa, fa

fa. 4.18 Letx, y∈Sbe such thatx≤y. Thenfx≤fybecausefis anN-fuzzy bi-ideal of S. Thusfis anN-fuzzy right ideal ofS.

ii⇒iLetAbe a bi-ideal ofS. Then byTheorem 3.5,κ_Ais anN-fuzzy bi-ideal ofS.

By hypothesisκAis anN-fuzzy right ideal ofS. ByTheorem 3.5,Ais a right ideal ofS.

By left-right dual ofTheorem 4.12, we have the following.

iEvery bi-ideal ofSis a left ideal ofS.

iiEveryN-fuzzy bi-ideal ofSis anN-fuzzy left ideal ofS.

5. Characterization of Intraregular Ordered Semigroups in Terms of N -Fuzzy Ideals

Definition 5.1cf.22. LetS,·,≤be an ordered semigroup andfanN-fuzzy subset ofS.

Thenfis called an semiprimeN-fuzzy subset ofSif fa≤f

a²

∀a∈S. 5.1

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Theorem 5.2. LetS,·,≤be an ordered semigroup and∅/A⊆S. Then the followings are equivalent.

iAis semiprime.

iiThe characteristicN-functionκ_AofAis anN-fuzzy semiprime subset.

Proof. i⇒iiSuppose thatAis semiprime subset. Letκ_Aa² 0. Sinceκ_Aa ≤ 0 for all a∈S, thusκAa≤κAa². IfκAa² −1, thena²∈A. SinceAis semiprime, we havea∈A.

Thenκ_Aa −1 and; henceκ_Aa≤κ_Aa².

ii⇒iAssume thatκ_AisN-fuzzy semiprime subset. Leta∈Sbe such thata² ∈A.

ThenκAa² −1. SinceκAa≤κAa², we haveκAa −1, hencea∈A.

Theorem 5.3. LetS,·,≤be an ordered semigroup and letfbe anN-fuzzy subsemigroup ofS. Then fis anN-fuzzy semiprime if and only if for everya∈S, one has

fa f a²

. 5.2

Proof. Suppose thatfis anN-fuzzy subsemigroup ofSsuch thatf is semiprime. Leta∈S.

Then

fa≤f a²

faa≤max

fa, fa

fa. 5.3

The converse is obvious.

Theorem 5.4. An ordered semigroup S is intraregular if and only if every N-fuzzy ideal of S is semiprime.

Proof. Suppose thatS is intraregular andf an N-fuzzy ideal ofS. Leta ∈ S. Thenfa ≤ fa². In fact, sinceSis intraregular, there existx, y∈Ssuch thata≤xa²yxa²y. Then

fa≤f x

a²y

≤f a²y

≤f a²

. 5.4

Assume thatfis anN-fuzzy ideal ofSsuch thatfa≤fa²for alla∈S. Consider the idealIa² a²∪Sa²∪a²S∪Sa²SofSgenerated bya²a∈S. Then byTheorem 3.5, the characteristic N-function κ_Ia² is an N-fuzzy ideal ofS, and by hypothesis, we have κ_Ia²a≤κ_Ia²a². Sincea² ∈Ia², thenκ_Ia²a² −1 andκ_Ia²a −1 ⇒a∈Ia² a²∪Sa² ∪a²S∪Sa²S. Thena ≤ xfor somex ∈ a²∪Sa²∪a²S∪Sa²S. If x a², then a ≤ a² aa ≤ a²a² aa²a ∈ Sa²S and a ∈ Sa²S. If x ya² for some y ∈ S, then a≤ya² yaa≤yya²ayya²a∈Sa²Sanda∈Sa²S. Ifxa²z, thena≤a²z aaz≤ aa²zzaa²zz∈Sa²Sanda∈Sa²S.

6. Some Semilattices of Left Simple Ordered Semigroups in Terms of N-Fuzzy Left Ideals

LetS,·,≤be an ordered semigroup. A subsemigroupFofSis called filtersee15ofSif 1 ∀a, b∈S ab∈F ⇒a∈Fandb∈F,

2 ∀c∈S c≥a∈F⇒c∈F.

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Forx ∈ S, we denote byNxthe filter ofS generated byxx ∈ S i.e., the least filter with respect to inclusion relation containingx.Ndenotes the equivalence relation on Sdefined byN:{x, y∈S×S|Nx Ny}see15.

Definition 6.1cf. 15. LetSbe an ordered semigroup. An equivalence relationσ onS is called congruence if a, b ∈ σ implies ac, bc ∈ σ and ca, cb ∈ σ for every c ∈ S. A congruence σ onS is called semilattice congruence if a², a ∈ σ and ab, ba ∈ σ for each a, b ∈ S. Ifσ is a semilattice congruence onS, then the σ-class x_σ ofS containingxis a subsemigroup ofSfor everyx∈S.

An ordered semigroupSis called a semilattice of left simple semigroups if there exists a semilattice congruenceσonSsuch that theσ-classx_σ ofScontainingxis a left simple subsemigroup ofSfor everyx∈S.

Equivalent definition: there exists a semilatticeY and a family{Sα}_α∈Y of left simple subsemigroups ofSsuch that

1S_α∩S_β∅for allα, β∈Y,α /β, 2S

α∈YSα,

3S_αS_β⊆S_αβfor allα, β∈Y.

In ordered semigroups the semilattice congruences are defined exactly same as in the case of semigroups—without order—so the two definitions are equivalentsee15.

Lemma 6.2cf.22. An ordered semigroupS,·,≤is a semilattice of left simple semigroups if and only if for all left idealsA,BofSone has

A²

A, AB BA. 6.1

Theorem 6.3. An ordered semigroupS,·,≤is a semilattice of left simple semigroups if and only if for everyN-fuzzy left idealfofS, one has

f a²

fa, fab fba ∀a, b∈S. 6.2

Proof. ⇒ALet Sbe a semilattice of left simple semigroups. By hypothesis, there exists a semilatticeY and a family{Sα}_α∈Y of left simple subsemigroups ofSsuch that

iS_α∩S_β∅for allα, β∈Y,α /β, iiS

α∈YSα,

iiiSαSβ⊆Sαβfor allα, β∈Y.

Let f be an N-fuzzy left ideal of S and a ∈ S. Then fa fa². In fact, by Theorem 4.6, it is enough to prove that a ∈ Sa²for everya ∈ S. Let a ∈ S, then there existsα∈Y such thata∈Sα. SinceSαis left simple, we haveSα Sαaand

a≤xa for somex∈S_α. 6.3

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Sincex∈Sα, we havex∈Sαaandx≤yafor somey∈Sα. Thus we have a≤xa≤

ya

aya² 6.4

sincey∈S, we havea∈Sa².

BLeta, b∈S. Then byA, we have fab f

ab²

fabab≥fba. 6.5 By symmetry we can prove thatfba≥fab. Hencefab fba.

⇐Assume that for every fuzzy left idealfofS, we have

f a²

fa, fab fba ∀a, b∈S, 6.6 by condition1andTheorem 4.6, we have thatSis left regular. LetAbe a left ideal ofSand leta∈A. Thena∈S, sinceSis left regular there existsx∈Ssuch that

a≤xa² xaa∈SAA⊆AAA², 6.7

thena ∈ A²andA ⊆ A. On the other hand, sinceAis a left ideal of S, we haveA² ⊆ SA ⊆A, thenA²⊆A A. LetAandBbe left ideals ofSand letx ∈BAthenx≤ba for somea∈ Aandb∈ B. We consider the left idealLabgenerated byab, that is, the set Lab ab∪Sab. Then byTheorem 3.5, the characteristicN-functionf_LabofLabdefined by

f_Lab:S−→0,1|x−→f_Labx:

⎧⎨

⎩

1 ifx∈Lab,

0 ifx /∈Lab 6.8

is a fuzzy left ideal ofS. By hypothesis, we havef_Labab f_Labba. Sinceab∈Lab, we havef_Labab 1 andf_Labba 1 and henceba ∈Lab ab∪Sab. Thenba≤ abor ba ≤ yab for somey ∈ S. Ifba ≤ ab, thenx ≤ ab ∈ ABand x ∈ AB. Ifba ≤ yab, then x≤ yab∈ SAB⊆ ABandx∈AB. ThusBA⊆AB. By symmetry we can prove that AB⊆BA. ThereforeAB BA, and byLemma 6.2, it follows thatSis a semilattice of left simple semigroups.

From left-right dual ofTheorem 6.3, we have the following.

Theorem 6.4. An ordered semigroupS,·,≤is a semilattice of right simple semigroups if and only if for everyN-fuzzy right idealfofS, one has

f a²

fa, fab fba ∀a, b∈S. 6.9 Lemma 6.5. LetS,·,≤be an ordered semigroup andfanN-fuzzy left (resp., right) ideal ofS, and a∈Ssuch thata≤a². Thenfa fa².

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Proof. Sincea≤a²andfis anN-fuzzy left ideal ofS, we have fa≤f

a²

faa≤fa, 6.10

and sofa fa².

7. Conclusion

Here we provided the concept of anN-fuzzy ideal in ordered semigroups and characterized some classes in terms of N-fuzzy leftresp., right ideals of ordered semigroups. In this regard, we provided the characterizations of left resp., right regular, left resp., right simple, and completely regular ordered semigroups in terms of N-fuzzy left resp., right ideals.

In our future work we will consider N-fuzzy prime ideals and N-fuzzy filters in ordered semigroups and will establish the relations between them. We will also try to discuss the quotient structure of ordered semigroup in terms ofN-fuzzy ideals.

Acknowledgments

The authors would like to thank the learned referees for their valuable comments and suggestions which improved the presentation of the paper and for their interest in their work. We would also like to thank Professor Dost Muhammad for his excellent guidance and valuable suggestions during this work.

References

1 L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965.

2 A. Rosenfeld, “Fuzzy groups,” Journal of Mathematical Analysis and Applications, vol. 35, no. 3, pp.

512–517, 1971.

3 N. Kuroki, “Fuzzy bi-ideals in semigroups,” Commentarii Mathematici Universitatis Sancti Pauli, vol.

28, no. 1, pp. 17–21, 1979.

4 N. Kuroki, “On fuzzy ideals and fuzzy bi-ideals in semigroups,” Fuzzy Sets and Systems, vol. 5, no. 2, pp. 203–215, 1981.

5 N. Kuroki, “On fuzzy semiprime ideals in semigroups,” Fuzzy Sets and Systems, vol. 8, no. 1, pp. 71–79, 1982.

6 N. Kuroki, “On fuzzy semigroups,” Information Sciences, vol. 53, no. 3, pp. 203–236, 1991.

7 N. Kuroki, “Fuzzy generalized bi-ideals in semigroups,” Information Sciences, vol. 66, no. 3, pp. 235–

243, 1992.

8 N. Kuroki, “On fuzzy semiprime quasi-ideals in semigroups,” Information Sciences, vol. 75, no. 3, pp.

201–211, 1993.

9 R. G. McLean and H. Kummer, “Fuzzy ideals in semigroups,” Fuzzy Sets and Systems, vol. 48, no. 1, pp. 137–140, 1992.

10 K. A. Dib and N. Galham, “Fuzzy ideals and fuzzy bi-ideals in fuzzy semi-groups,” Fuzzy Sets and System, vol. 92, pp. 203–215, 1997.

11 J. Ahsan, K. Saifullah, and M. F. Khan, “Semigroups characterized by their fuzzy ideals,” Fuzzy Systems and Mathematics, vol. 9, no. 1, pp. 29–32, 1995.

12 J. N. Mordeson, D. S. Malik, and N. Kuroki, Fuzzy Semigroups, vol. 131 of Studies in Fuzziness and Soft Computing, Springer, Berlin, Germany, 2003.

13 J. N. Mordeson and D. S. Malik, Fuzzy Automata and Languages, Theory and Application Computational Mathematics Series, Chapman and Hall/CRC, Boca Raton, Fla, USA, 2002.

(14)

14 N. Kehayopulu and M. Tsingelis, “Fuzzy sets in ordered groupoids,” Semigroup Forum, vol. 65, no. 1, pp. 128–132, 2002.

15 N. Kehayopulu and M. Tsingelis, “Fuzzy bi-ideals in ordered semigroups,” Information Sciences, vol.

171, no. 1–3, pp. 13–28, 2005.

16 M. Shabir and A. Khan, “Characterizations of ordered semigroups by the properties of their fuzzy generalized bi-ideals,” New Mathematics and Natural Computation, vol. 4, no. 2, pp. 237–250, 2008.

17 M. Shabir and A. Khan, “Fuzzy filters in ordered semigroups,” Lobachevskii Journal of Mathematics, vol.

29, no. 2, pp. 82–89, 2008.

18 N. Kehayopulu and M. Tsingelis, “On left regular ordered semigroups,” Semigroups Forum, vol. 25, no. 4, pp. 609–615, 2002.

19 N. Kehayopulu, “On regular duo ordered semigroups,” Mathematica Japonica, vol. 37, no. 3, pp. 535–

540, 1992.

20 N. Kehayopulu and M. Tsingelis, “Regular ordered semigroups in terms of fuzzy subsets,” Information Sciences, vol. 176, no. 24, pp. 3675–3693, 2006.

21 N. Kehayopulu, “On completely regular ordered semigroups,” Scientiae Mathematicae, vol. 1, no. 1, pp. 27–32, 1998.

22 A. Khan, Y. B. Jun, and M. Shabir, “N-fuzzy Bi-ideals in ordered semigroups,” submitted.