On Ordered Quasi-Gamma-Ideals of Regular Ordered Gamma-Semigroups

Hindawi Publishing CorporationAlgebraVolume 2013 Article ID 565848 7 pageshttpdxdoiorg1011552013565848

Research ArticleOn Ordered Quasi-Gamma-Ideals of RegularOrdered Gamma-Semigroups

M Y Abbasi and Abul Basar

Department of Mathematics Jamia Millia Islamia New Delhi 110025 India

Correspondence should be addressed to Abul Basar basarjmigmailcom

Received 31 March 2013 Accepted 7 October 2013

Academic Editor Sorin Dascalescu

Copyright copy 2013 M Y Abbasi and A Basar This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We introduce the notion of ordered quasi-Γ-ideals of regular ordered Γ-semigroups and study the basic properties of orderedquasi-Γ-ideals of ordered Γ-semigroups We also characterize regular ordered Γ-semigroups in terms of their ordered quasi-Γ-ideals ordered right Γ-ideals and left Γ-ideals Finally we have shown that (i) a partially ordered Γ-semigroup 119878 is regular if andonly if for every ordered bi-Γ-ideal 119861 every ordered Γ-ideal 119868 and every ordered quasi-Γ-ideal 119876 we have 119861 cap 119868 cap 119876 sube (119861Γ119868Γ119876]

and (ii) a partially ordered Γ-semigroup 119878 is regular if and only if for every ordered quasi-Γ-ideal 119876 every ordered left Γ-ideal 119871and every ordered right-Γ-ideal 119877 we have that 119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871]

1 Introduction

Steinfeld [1ndash3] introduced the notion of a quasi-ideal forsemigroups and rings Since then this notion has been thesubject of great attention of many researchers and conse-quently a series of interesting results have been published byextending the notion of quasi-ideals to Γ-semigroupsordered semigroups ternary semigroups semirings Γ-semi-rings regular rings near-rings and many other differentalgebraic structures [4ndash15]

It is a widely known fact that the notion of a one-sidedideal of rings and semigroups is a generalization of the notionof an ideal of rings and semigroups and the notion of a quasi-ideal of semigroups and rings is a generalization of a one-sided ideal of semigroups and rings In fact the concept ofordered semigroups and Γ-semigroups is a generalization ofsemigroups Also the ordered Γ-semigroup is a generalizationof Γ-semigroups So the concept of ordered quasi-ideals ofordered semigroups is a generalization of the concept ofquasi-ideals of semigroups In the same way the notion ofan ordered quasi-ideal of ordered semigroups is a general-ization of a one-sided ordered ideal of ordered semigroupsDue to these motivating facts it is naturally significant togeneralize the results of semigroups to Γ-semigroups and ofΓ-semigroups to ordered Γ-semigroups

In 1998 the concept of an ordered quasi-ideal in orderedsemigroups was introduced by Kehayopulu [16] He stud-ied theory of ordered semigroups based on ordered idealsanalogous to the theory of semigroups based on idealsThe concept of po-Γ-semigroup was introduced by Kwonand Lee in 1996 [17] and since then it has been studiedby several authors [18ndash22]Our purpose in this paper is toexamine many important classical results of ordered quasi-Γ-ideals in ordered Γ-semigroups and then to characterizethe regular ordered Γ-semigroups through ordered quasi-Γ-ideals ordered bi-Γ-ideals and ordered one-sided Γ-ideals

2 Preliminaries

We note here some basic definitions and results that arerelevant for our subsequent results

Let 119878 and Γ be two nonempty sets Then 119878 is called a Γ-semigroup if 119878 satisfies (119886120574119887)120583119888 = 119886120574(119887120583119888) for all 119886 119887 119888 isin 119878

and 120574 120583 isin Γ A nonempty subset 119873 of a Γ-semigroup 119878 iscalled a sub-Γ-semigroup of 119878 if 119886120572119887 isin 119873 for all 119886 119887 isin 119873 and120572 isin Γ For any nonempty subsets 119860 119861 of 119878 119860Γ119861 = 119886120572119887

119886 isin 119860 119887 isin 119861 and 120572 isin Γ We also denote 119886Γ119861 119860Γ119887 and119886Γ119887 respectively by 119886Γ119861 119860Γ119887 and 119886Γ119887 Many classicalresults of semigroups have been generalized and extendedto Γ-semigroups [23ndash25] By an ordered Γ-semigroup 119878

2 Algebra

(also called po-Γ-semigroups) wemean an ordered set (119878 le)at the same time a Γ-semigroup satisfying the following con-ditions

119886 le 119887 997904rArr 119886120574119888 le 119887120574119888 119888120574119886 le 119888120574119887 forall119886 119887 119888 isin 119878 120574 isin Γ (1)

Throughout this paper 119878 will stand for an ordered Γ-semigroup unless otherwise stated An ordered Γ-semigroup119878 is called regular if for each 119904 isin 119878 and for each 120572 120573 isin Γ thereexists 119886 isin 119878 such that 119904 le 119904120572119886120573119904 Equivalent definitions ofregular ordered Γ-semigroup are as follows (i) 119860 sube (119860Γ119878Γ119860]

for each119860 sube 119878 and (ii) 119904 isin (119904Γ119878Γ119904] for each 119904 isin 119878 Let (119878 le) bean ordered Γ-semigroup and119873 a sub-Γ-semigroup of 119878 then(119873 le) is an ordered Γ-semigroup Let119860 be a nonempty subsetof119873 Then similarly to [26] we write (119860]

119873= 119899 isin 119873 119899 le 119886

for some 119886 isin 119860 and 119860 cup 119886 = 119860 cup 119886 We also write (119860]119873

by simply (119860] if 119873 = 119878 (see [27]) A nonempty subset 119868 ofan ordered Γ-semigroup 119878 is called an ordered right-Γ-ideal(left-Γ-ideal) of 119878 if 119868Γ119878 sube 119868(119878Γ119868 sube 119868) and for any 119909 isin 119868(119909] sube 119868 119868 is called an ordered Γ-ideal of 119878 if it is both a leftand a right Γ-ideals of 119878 Also for any 119904 isin 119878 we have that (119878Γ119904]is an ordered left Γ-ideal of 119878 and (119904Γ119878] is an ordered right Γ-ideal of 119878 [18] A nonempty subset 119876 of 119878 is called an orderedquasi-Γ-ideal of 119878 if (i) (119876Γ119878] cap (119878Γ119876] sube 119876 and (ii) (119876] sube 119876A sub-Γ-semigroup 119861 of an ordered Γ-semigroup 119878 is calledan ordered bi-Γ-ideal of 119878 if 119861Γ119878Γ119861 sube 119861 and for any 119909 isin 119861(119909] sube 119861

Let119883 be a nonempty subset of 119878Then the least right (left)ordered Γ-ideal of 119878 containing 119883 is given by 119877(119883) = (119883 cup

119883Γ119878](119871(119883) = (119878Γ119883 cup 119883]) If 119883 = 119904 119904 isin 119878 we write 119877119904and 119871119904 respectively by 119877(119904) and 119871(119904) and 119877(119904) = (119904 cup 119904Γ119878]119871(119904) = (119878Γ119904 cup 119904] and the ideal generated by 119904 isin 119878 is given by119868(119904) = (119904 cup 119878Γ119904 cup 119904Γ119878 cup 119878Γ119904Γ119878] Also the least quasi-Γ-ideal of119878 containing 119883 is denoted by 119876(119883) Moreover we willl needsome notations as follows (i)119873

119876= 119876 119876 = 0 where 119876 sube 119878

and (119876] sube 119876 (ii) 119877119868is a set of ordered right Γ-ideals of 119878

(iii) 119871119868is a set of ordered left Γ-ideals of 119878 and (iv) 119868

119879is a

two-sided Γ-ideal of 119878Now for any two elements 119876

1 1198762isin 119873119876 we define an

operation lowast in119873119876as follows

1198761lowast Γ lowast 119876

2= (1198761Γ1198762] (2)

Further let 119873 be a sub-Γ-semigroup of 119878 Then we caneasily observe here the following (see [16 18 21 28ndash30])

(i) 119860 sube (119860]119873sube (119860] = ((119860]] for 119860 sube 119873

(ii) for 119860 sube 119873 and 119861 sube 119873 we have (119860 cup 119861] = (119860] cup (119861](iii) for 119860 sube 119873 and 119861 sube 119873 we have (119860 cap 119861] sube (119860] cap (119861](iv) for 119886 and 119887 isin 119873 with 119886 le 119887 we have (119886Γ119873] sube (119887Γ119873]

and (119873Γ119886] sube (119873Γ119887](v) (119860]Γ(119861] sube (119860Γ119861](vi) for every left (right two-sided) ideal 119871 of 119878 (119871] = 119871(vii) if 119860 and 119861 are ordered Γ-ideals of 119878 then (119860Γ119861] and

119860 cup 119861 are also ideals of 119878(viii) for any 119904 isin 119878 (119878Γ119904Γ119878] is an ideal of 119878

3 Ordered Γ-Semigroups and OrderedQuasi-Γ-Ideals

In this section we study some classical properties of theordered Γ-semigroup 119878 We start with the following lemma

Lemma 1 Let 119878 be an ordered Γ-semigroup Then

(i) (119873119876 lowast sube) is an ordered Γ-semigroup

(ii) (119871119868 lowast sube) (119877

119868 lowast sube) and (119868

119879 lowast sube) are sub-Γ-semi-

groups of (119873119876 lowast sube)

Proof (i) Suppose 119875 119876 119877 isin 119873119876 Since 119875Γ119876 isin (119875Γ119876] we

obtain ((119875Γ119876)Γ119877] sube ((119875Γ119876]Γ119877] Next we have (119875 lowast Γ lowast 119876) lowastΓlowast119877 = (119875Γ119876Γ119877] by using (119875lowastΓlowast119876)lowastΓlowast119877 = (119875Γ119876]lowastΓlowast119877 =

((119875Γ119876]Γ119877] sube ((119875Γ119876)Γ119877] = (119875Γ119876Γ119877] In a similar way wecan show that 119875 lowast Γ lowast (119876 lowast Γ lowast 119877) = (119875Γ119876Γ119877] and therefore(119875 lowast Γ lowast 119876) lowast Γ lowast 119877 = 119875 lowast Γ lowast (119876 lowast Γ lowast 119877) Hence (119873

119876 lowast) is

a Γ-semigroup Suppose 119875 sube 119876 Then 119875 lowast Γ lowast 119877 = (119875Γ119876] sube

(119876Γ119877] = 119876lowastΓlowast119877 and119877lowastΓlowast119875 = (119877Γ119875] sube (119877Γ119876] = 119877lowastΓlowast119876Hence (119873

119876 lowast sube) is an ordered Γ-semigroup

(ii) We have that 119871119868 119877119868 and 119868

119879are nonempty subsets of

119873119876 Suppose 119871

1 1198712isin 119871119868 Then obviously we have (119871

1lowast Γ lowast

1198712] = ((119871

1Γ1198712]] = (119871

1Γ1198712] Moreover using

119878Γ (1198711lowast Γ lowast 119871

2) = 119878Γ (119871

1Γ1198712]

sube (119878Γ (1198711Γ1198712]]

sube ((119878Γ1198711) Γ1198712]

sube (1198711Γ1198712]

= 1198711lowast Γ lowast 119871

2

(3)

we infer that1198711lowastΓlowast119871

2is a left Γ-ideal of 119878 that is119871

1lowastΓlowast119871

2isin

119871119868 Thus (119871

119868 lowast sube) is a sub-Γ-semigroup of (119873

119876 lowast sube)

Dually we can prove that (119877119868 lowast sube) is a sub-Γ-semigroup

of (119873119876 lowast sube) Since 119868

119879= 119871119868cap 119877119868 it follows that (119868

119879 lowast sube) is a

sub-Γ-semigroup of (119873119876 lowast sube)

Let 119876119868= 119876 119876 is an ordered quasi-Γ-ideal of 119878 Then

obviously we have 119871119868cup 119877119868sube 119876119868sube 119873119876 This implies that

every one-sided Γ-ideal of an ordered Γ-semigroup is a quasi-Γ-ideal of 119878 Thus the class of ordered quasi-Γ-ideals of 119878 is ageneralization of the class of one-sided ordered Γ-ideals of 119878

Lemma 2 Each ordered quasi-Γ-ideal 119876 of an ordered Γ-semigroup 119878 is a sub-Γ-semigroup of 119878

Proof Proof is straightforward In fact we have119876Γ119876 sube 119876Γ119878cap

119878Γ119876 sube (119876Γ119878] cap (119878Γ119876] sube 119876

Lemma 3 For every ordered right Γ-ideal 119877 and an orderedleft Γ-ideal 119871 of an ordered Γ-semigroup 119878 119877 cap 119871 is an orderedquasi-Γ-ideal of 119878

Proof As 119877Γ119871 sube 119878Γ119871 sube 119871 and 119877Γ119871 sube 119877Γ119878 sube 119877 we obtain119877Γ119871 sube 119877 cap 119871 so 119877 cap 119871 = 0 Now the fact that 119877 cap 119871 is anordered quasi-Γ-ideal of 119878 follows from the following

(i) (119877 cap 119871] sube (119877] cap (119871] sube 119877 cap 119871

Algebra 3

(ii) ((119877cap119871)Γ119878]cap(119878Γ(119877cap119871)] sube (119877Γ119878]cap(119878Γ119871] sube (119877]cap(119871] sube119877 cap 119871

Lemma 4 Let 119876 be an ordered quasi-Γ-ideal of 119878 then oneobtains 119876 = 119871(119876) cap 119877(119876) = (119878Γ119876 cup 119876] cap (119876 cup 119876Γ119878]

Proof The following relation

119876 sube (119878Γ119876 cup 119876] cap (119876 cup 119876Γ119878] is obvious (4)

Conversely suppose 119886 isin (119878Γ119876 cup 119876] cap (119876 cup 119876Γ119878] Then119886 le 119887 or 119886 le 119909120572119906 and 119886 le V120573119910 for some 119887 119906 V isin 119876 119909 119910 isin 119878and 120572 120573 isin Γ As119876 is an ordered quasi-Γ-ideal of 119878 the formercase implies that 119886 isin (119876] sube 119876 and the latter case implies that119886 isin (119878Γ119876] cap (119876Γ119878] sube 119876 Therefore (119878Γ119876 cup 119876] cap (119876 cup 119876Γ119878] =

119876

We recall here that if 119883 is a nonempty subset of anordered Γ-semigroup 119878 then we write the least quasi-ideal of119878 containing119883 by 119876(119883) If119883 = 119886 we write 119876(119886) by 119876(119886)

Theorem 5 Suppose 119878 is an ordered Γ-semigroup Then onehas the following

(i) for every 119904 isin 119878 119876(119904) = 119871(119904) cap 119877(119904) = (119878Γ119904 cup 119904] cap (119904 cup

119904Γ119878](ii) let 0 =119883 sube 119878 119876(119883) = 119871(119883) cap 119877(119883) = (119878Γ119883 cup 119883] cap

(119883 cup 119883Γ119878]

Proof of (i) Suppose 119904 isin 119878 Using Lemma 3 119871(119904) cap 119877(119904) is aquasi-Γ-ideal of 119878 containing 119904 therefore 119876(119904) sube 119871(119904) cap 119877(119904)and by Lemma 4 we obtain

119871 (119904) cap 119877 (119904) = (119878Γ119904 cup 119904] cap (119904 cup 119904Γ119878]

sube (119878Γ119876 (119904) cup 119876 (119904)] cap (119876 (119904) cup 119876 (119904) Γ119878]

= 119876 (119904)

(5)

Hence 119876(119904) = 119871(119904) cap 119877(119904)

Proof of (ii) Its proof can be given as (i)

The notion of a bi-Γ-ideal of Γ-semigroups is a gener-alization of the notion of a quasi-Γ-ideal of Γ-semigroupsSimilarly the class of ordered quasi-Γ-ideals of ordered Γ-semigroups is a particular case of the class of ordered bi-Γ-ideals of ordered Γ-semigroups This is what we have shownin the following result

Theorem 6 Suppose 119868 is a two-sided ordered Γ-ideal of anordered Γ-semigroup 119878 and 119876 is a quasi-Γ-ideal of 119868 then 119876is an ordered bi-Γ-ideal of 119878

Proof Since 119876 is an ordered quasi-Γ-ideal of 119868 and 119876 sube 119868 weobtain

119876Γ119876 sube 119876Γ119878Γ119868

= 119876Γ (119878Γ119868) sube 119876Γ119868

sube (119876Γ119868] sube (119878Γ119868] sube (119868] sube 119868

119876Γ119878Γ119876 sube 119868Γ119876Γ119878

= (119868Γ119878) Γ119876 sube 119868Γ119876 sube (119868Γ119876]

sube (119868Γ119878] sube (119868] sube 119868

(6)

and 119902 isin (119876] rArrThere exists 1199021isin 119876 sube 119868 such that 119902 le 119902

1rArr

119902 isin (119868] = 119868 and 119902 isin (119876] rArr 119902 isin 119868 cap (119876] = (119876]119868sube 119876

Therefore

119876Γ119878Γ119876 sube (119868 cap (119868Γ119876]) cap (119868 cap (119876Γ119868])

= (119868Γ119876]119868 cap (119876Γ119868]119868 sube 119876 (119876] sube 119876

(7)

Hence applying these facts together with Lemma 2 we haveshown that 119876 is an ordered bi-Γ-ideal of 119878

4 Regular Ordered Γ-Semigroups andOrdered Quasi-Γ-Ideals

In this section we use the concept of ordered quasi-Γ-idealsto characterize regular ordered Γ-semigroups

Lemma 7 Let 119878 be an ordered Γ-semigroup Then the orderedsub-Γ-semigroup of (119873

119876 lowast) generated by (119871

119868 lowast) and (119877

119868 lowast) is

in the following form

⟨119871119868cup 119877119868⟩ = 119871

119868cup 119877119868cup (119877119868lowast Γ lowast 119871

119868) (8)

Proof One can easily see that

⟨119871119868cup 119877119868⟩ =119884

1lowast Γ1lowast 1198842lowast sdot sdot sdot lowast 119884

119899minus1lowast Γ119899minus1

lowast 119884119899| 119884119895isin 119871119868

or 119884119895isin 119877119868 119895 = 1 sdot sdot sdot 119899 119899 isin 119885

+ Γ119895isin Γ

(9)

Suppose 119884119895 119884119895+1

isin 119871119868cup119877119868 Then the conditions that arise

are as follows (i) 119884119895 119884119895+1

isin 119871119868 in this condition by Lemma 1

we obtain 119884119895lowast Γ lowast 119884

119895+1isin 119871119868 (ii) 119884

119895 119884119895+1

isin 119877119868 in this

condition 119884119895lowast Γ lowast 119884

119895+1isin 119877119868by also Lemma 1 (iii) 119884

119895isin 119871119868

119884119895+1

isin 119877119868 in this condition 119884


119895+1= (119884119895Γ119884119895+1] is an

ordered Γ-ideal of 119878 so119884119895lowastΓlowast119884119895+1

isin 119868119879= 119871119868cap119877119868 (iv)119884

119895isin 119877119868

119884119895+1



119895+1isin 119877119868lowast Γ lowast 119871

119868in

(119873119876 lowast)Therefore for any119884

1 119884

119899isin 119871119868cup119877119868 where 119899 isin 119885+

using (i)ndash(iv) there arise three conditions as follows

(i)1015840 If1198841isin 119871119868 then119884

1lowastΓ1lowast1198842lowastsdot sdot sdotlowast119884

119899minus1lowastΓ119899minus1

lowast119884119899isin 119871119868

(ii)1015840 If119884119899isin 119877119868 then119884



lowast119884119899isin 119877119868

(iii)1015840 If 1198841isin 119877119868and 119884

119899isin 119871119868 where 119899 ge 2 then 119884

1lowast Γ1lowast

1198842lowast sdot sdot sdot lowast 119884


lowast 119884119899isin 119877119868lowast Γ lowast 119871

119868 Hence the

lemma holds

Theorem 8 Let 119878 be an ordered Γ-semigroupThen the follow-ing assertions on 119878 are equivalent

4 Algebra

(i) 119878 is a regular ordered Γ-semigroup(ii) For every ordered left Γ-ideal 119871 and every ordered right

Γ-ideal 119877 one has

(119877Γ119871] = 119877 cap 119871 (10)

(iii) For every ordered right Γ-ideal 119877 and ordered left Γ-ideal 119871 of 119878

(1) (119877Γ119877] = 119877(2) (119871Γ119871] = 119871(3) (119877Γ119871] is an ordered quasi-Γ-ideal of 119878

(iv) (119871119868 lowast) and (119877

119868 lowast) are ordered idempotent Γ-semi-

groups and (119876119868 lowast) is the sub-Γ-semigroup of (119873

119876 lowast)

generated by (119871119868 lowast) and (119877

119868 lowast)

(v) (119876119868 lowast) is a regular ordered sub-Γ-semigroup of the Γ-

semigroup (119873119876 lowast)

(vi) Every ordered quasi-Γ-ideal 119876 of 119878 is given by 119876 =

(119876Γ119878Γ119876](vii) (119876

119868 lowast sube) is a regular sub-Γ-semigroup of the ordered

Γ-semigroup of (119873119876 lowast sube)

Proof (i) rArr (ii) Suppose 119877 and 119871 are ordered right and leftΓ-ideals of 119878 respectively then we have

(119877Γ119871] sube 119877 cap 119871 (11)

Let 119878 be regular we need to prove only that 119877 cap 119871 sube (119877Γ119871]Suppose 119886 isin 119877cap119871 Since 119878 is regular we obtain 119886 le 119886120572119909120573119886 forsome 119909 isin 119878 and 120572 120573 isin Γ and so 119886 isin 119877 and 119909120572119886 isin 119871 therefore119886120572119909120573119886 isin 119877Γ119871 Therefore 119886 isin (119877Γ119871] and thus 119877cap119871 sube (119877Γ119871]

(ii) rArr (iii) (119877Γ119871] is an ordered quasi-Γ-ideal of 119878 thatfollows directly from Lemma 3 and the condition (ii) As theordered two-sided Γ-ideal of 119878 is generated by 119877 = (119877cup 119878Γ119877]the condition (ii) implies that

119877 = 119877 cap (119877 cup 119878Γ119877] = (119877Γ (119877 cup 119878Γ119877]]

therefore (119877Γ119877] sube (119877Γ (119877 cup 119878Γ119877]] = 119877(12)

Conversely suppose 119886 isin (119877Γ(119877 cup 119878Γ119877]] Then 119886 le 119903120572119887 for119903 isin 119877 and 119887 isin (119877 cup 119878Γ119877] From 119887 isin (119877 cup 119878Γ119877] we have119887 le 119888 where 119888 = 119903

1015840isin 119877 or 119888 = 119904120572119903

10158401015840 for some 119904 isin 119878 and11990310158401015840isin 119877 Therefore 119886 le 119903120572119888 = 119903120572119903

1015840isin 119877Γ119877 or 119886 le 119903120572119888 =

119903120572(11990412057311990310158401015840) = (119903120572119904)120574119903

10158401015840isin 119877Γ119877 for 120572 120573 120574 isin Γ thus 119886 isin (119877Γ119877]

Thus 119877 sube (119877Γ119877] so that (119877Γ119877] = 119877 Similarly we can provethat (119871Γ119871] = 119871 dually

(iii) rArr (iv)The conditions (1) (2) in (iii) and Lemma 7show that (119871

119868 lowast) and (119877

119868 lowast) are idempotent Γ-semigroups

respectively Applying (iii) (3) we obtain 119877119868lowast Γ lowast 119871

119868sube 119876119868

therefore ⟨119871119868cup 119877119868⟩ sube 119876

119868in (119873119876 lowast)

Conversely suppose 119876 isin 119876119868 Then (119876 cup 119878Γ119876] is the

ordered left Γ-ideal of 119878 generated by 119876 The condition (iii)(2) implies that

119876 sube (119876 cup 119878Γ119876] = ((119876 cup 119878Γ119876] Γ (119876 cup 119878Γ119876]]

sube (119876Γ119876 cup 119878Γ119876Γ119876 cup 119876Γ119878Γ119876

cup (119878Γ119876) Γ (119878Γ119876)] sube (119878Γ119876]

(13)

We can dually prove that 119876 sube (119876Γ119878] Therefore usingthese facts and Lemma 4 it follows that

(a) 119876 sube (119878Γ119876] cap (119876Γ119878] sube (119878Γ119876 cup 119876] cap (119876 cup 119876Γ119878] = 119876

(14)

Therefore for 119876 isin 119876119868 we have 119876 = (119878Γ119876] cap (119876Γ119878] and the

condition (iii) (3) together with (a) implies that

(b) (119877Γ119871] = (119878Γ (119877Γ119871]] cap ((119877Γ119871] Γ119878]

(15)

Moreover by the assertion (iii) (2) we have 119878 = (119878Γ119878] and

(119878Γ119876] = ((119878Γ119876]2]

= ((119878Γ119876] Γ (119878Γ119876]] = ((119878Γ119876] Γ ((119878Γ119878] Γ119876]]

sube (119878Γ119876Γ119878Γ119878Γ119876] sube (119878Γ (119876Γ119878] Γ (119878Γ119876]]

sube (119878Γ ((119876Γ119878] Γ (119878Γ119876]]] sube (119878Γ (119876Γ119878Γ119878Γ119876)]

sube (119878Γ119876]

(16)

Therefore (119878Γ119876] = (119878Γ((119876Γ119878]Γ(119878Γ119876]]] Dually we canprove that

(119876Γ119878] = (((119876Γ119878] Γ (119878Γ119876]] Γ119878] (17)

From these facts (a) and (b) we obtain

(c) 119876 = (119876Γ119878] cap (119878Γ119876]

= (((119876Γ119878] Γ (119878Γ119876]] Γ119878] cap (119878Γ ((119876Γ119878] Γ (119878Γ119876]]]

= ((119876Γ119878] Γ (119878Γ119876]]

= (119876Γ119878] lowast Γ lowast (119878Γ119876] isin 119877119868 lowast Γ lowast 119871119868

sube ⟨119871119868cup 119877119868⟩

(18)

by Lemma 7Therefore119876119868sube ⟨119871119868cup119877119868⟩ Hence119876

119868= ⟨119871119868cup119877119868⟩

in (119873119876 lowast)

(iv) rArr (iii) It is a consequence of Lemma 7(iii) rArr (v) By (iii) rArr (iv) we have (b) and (c) Sup-

pose 1198761 1198762are two ordered quasi-Γ-ideals of 119878 Then

(119878Γ(1198761Γ1198762] cup (119876

1Γ1198762]] is the least ordered left Γ-ideal of 119878

containing (1198761Γ1198762] Then the condition (iii) (2) implies that

(1198761Γ1198762] sube (119878Γ (119876

1Γ1198762] cup (119876

1Γ1198762]]

= ((119878Γ (1198761Γ1198762] cup (119876

1Γ1198762]]2]

sube (119878Γ (1198761Γ1198762]] = ((119878Γ119878] Γ (1198761Γ1198762]]

sube (119878Γ (119878Γ (1198761Γ1198762]]]

(19)

Algebra 5

Dually one can prove that (1198761Γ1198762] sube ((119876

1Γ1198762] cup

(1198761Γ1198762]Γ119878] sube (((119876

1Γ1198762]Γ119878]Γ119878]These facts togetherwith (b)

show that

(1198761Γ1198762] sube (119878Γ (119876

1Γ1198762] cup (119876

1Γ1198762]]

cap ((1198761Γ1198762] cup (119876

1Γ1198762] Γ119878]

sube (119878Γ (119878Γ (1198761Γ1198762]]]

cap (((1198761Γ1198762] Γ119878] Γ119878]

= (((1198761Γ1198762] Γ119878] Γ (119878Γ (119876

1Γ1198762]]]

sube ((1198761Γ (1198762Γ119878Γ119878) Γ119876

1) Γ1198762] sube (119876

1Γ1198762]

(20)

By Theorem 5 (ii) (1198761Γ1198762] = (119878Γ(119876

1Γ1198762] cup (119876

1Γ1198762]] cap

((1198761Γ1198762] cup (119876

1Γ1198762]Γ119878] is an ordered quasi-Γ-ideal of 119878

therefore 1198761lowast Γ lowast 119876

2isin 119876119868 Hence (119876

119868 lowast) is a sub-Γ-

semigroup of (119873119876 lowast) For every 119876 isin 119876

119868 by (c) we obtain

119876 = ((119876Γ119878]Γ(119878Γ119876]] sube (119876Γ119878Γ119878Γ119876] sube (119876Γ119878Γ119876] sube 119876 and so119876 = (119876Γ119878Γ119876] = 119876lowastΓlowast119878lowastΓlowast119876 where 119878 isin 119876

119868 Thus (119876

119868 lowast)

is a regular sub Γ-semigroup of (119873119876 lowast)

(v) rArr (vi) Suppose 119876 is an ordered quasi-Γ-ideal of 119878Applying the condition (iv) there is an ordered quasi-Γ-ideal1198761of 119878 so that by Lemma 4

119876 = 119876 lowast Γ lowast 1198761lowast Γ lowast 119876 = (119876Γ119876

1Γ119876]

sube (119876Γ119878Γ119876] sube (119878Γ119876] cap (119876Γ119878]

sube (119878Γ119876 cup 119876] cap (119876 cup 119876Γ119878] = 119876

(21)

and therefore 119876 = (119876Γ119878Γ119876](vi) rArr (vii) It is straightforward(vii) rArr (i) For every 119904 isin 119878 usingTheorem 5 119877(119904) cap 119871(119904)

is an ordered quasi-Γ-ideal of 119878 containing 119904 By (vii) thereexists 119876 isin 119876

119878so that

119904 isin 119877 (119904) cap 119871 (119904) sube (119877 (119904) cap 119871 (119904)) lowast Γ lowast 119876 lowast Γ lowast (119877 (119904) cap 119871 (119904))

= ((119877 (119904) cap 119871 (119904)) Γ119876Γ (119877 (119904) cap 119871 (119904))]

sube (119877 (119904) Γ119878Γ119871 (119904)]

= ((119904 cup 119904Γ119878] Γ119878Γ (119878Γ119904 cup 119904]] sube (119904Γ119878Γ119904]

(22)

Hence 119878 is a regular ordered Γ-semigroup

Lemma 9 Every two-sided ordered Γ-ideal 119868 of a regularordered Γ-semigroup 119878 is a regular sub-Γ-semigroup of 119878

Proof Suppose 119894 isin 119868 As 119878 is regular there exists 119904 isin 119878 so thatfor 120572 120573 120574 120575 isin Γ we have

119894 le 119894120572119904120573119894 le 119894120572119904120573119894120574119904120575119894 = 119894120572 (119904120573119894120574119904) 120575119894 (23)

As 119904120572119894120573119904 isin 119878Γ119868Γ119878 sube 119868 we observe that 119894 isin (119894Γ119868Γ119894]119868

Theorem10 Suppose 119878 is a regular ordered Γ-semigroupThenthe following statements are true

(i) Every ordered quasi-Γ-ideal of 119878 can be expressed asfollows

119876 = 119877 cap 119871 = (119877Γ119871] (24)

where 119877 and 119871 are respectively the ordered right andleft Γ-ideals of 119878 generated by 119876

(ii) Let 119876 be an ordered quasi-Γ-ideal of 119878 then (119876Γ119876] =(119876Γ119876Γ119876]

(iii) Every ordered bi-Γ-ideal of 119878 is an ordered quasi-Γ-ideal of 119878

(iv) Every ordered bi-Γ-ideal of any ordered two sided-Γ-ideal of 119878 is a quasi-Γ-ideal of 119878

(v) For every 1198711 1198712isin 119871119868and 119877

1 1198772isin 119877119868 one obtains

1198711cap 1198712sube (1198711Γ1198712]

1198771cap 1198772sube (1198771Γ1198772]

(25)

Proof Because 119878 is a regular ordered Γ-semigroup then byLemma 4 and Theorem 8 the statement (i) is done Since(119876Γ119876Γ119876] sube (119876Γ119876] is always true we need to show that(119876Γ119876] sube (119876Γ119876Γ119876] We have that (119876Γ119876] is also an orderedquasi-Γ-ideal of 119878 by Theorem 8 Moreover we have thefollowing equation

(119876Γ119876] = (119876Γ119876Γ119878Γ119876Γ119876]

= (119876Γ (119876Γ119878Γ119876) Γ119876] sube (119876Γ119876Γ119876]

(26)

Suppose 1198761is an ordered bi-Γ-ideal of 119878 Then (119878Γ119876

1] is

an ordered left Γ-ideal and (1198761Γ119878] is an ordered right Γ-ideal

of 119878 ApplyingTheorem 8 we obtain

(119878Γ1198761] cap (119876

1Γ119878] = ((119876

1Γ119878] Γ (119878Γ119876

1]]

sube (1198761Γ119878Γ1198761] sube (119876

1] sube 1198761

(27)

Therefore 1198761is an ordered quasi-Γ-ideal of 119878

Suppose 119868 is a two-sided ordered Γ-ideal of 119878 and 119861 is anordered bi-Γ-ideal of 119868 By the relation (iii) and Lemma 9 119861 isan ordered quasi-Γ-ideal of 119868 therefore usingTheorem 6 119861 isan ordered bi-Γ-ideal of 119878 Also from the relation (iii) againwe obtain 119861 as an ordered quasi-Γ-ideal of 119878

Lastly suppose 1198711 1198712isin 119871119868 Because 119878 is regular and

1198711cap 1198712is an ordered quasi-Γ-ideal of 119878 using Theorem 8

we obtain

1198711cap 1198712= ((119871

1cap 1198712) Γ119878Γ (119871

1cap 1198712)]

sube (1198711Γ (119878Γ119871

2)] sube (119871

1Γ1198712]

(28)

Dually we can prove that 1198771cap 1198772sube (1198771Γ1198772] for all 119877

1

1198772isin 119877119868

Theorem 11 A partially ordered Γ-semigroup 119878 is regular ifand only if for every ordered bi-Γ-ideal 119861 every ordered Γ-ideal119868 and every ordered quasi-Γ-ideal 119876 one has

119861 cap 119868 cap 119876 sube (119861Γ119868Γ119876] (29)

6 Algebra

Proof Let 119878 be regularThen for any 119886 isin 119861cap119868cap119876 there exists119904 isin 119878 such that

119886 le 119886120572119904120573119886 le (119886120572119904120573119886) 120574119904120575 (119886120572119904120573119886)

= (119886120572119904120573119886) 120574 (119904120572119886120573119904) 120575119886 isin (119861Γ119861) Γ (119878Γ119868Γ119878) Γ119876

sube 119861Γ119868Γ119876

(30)

Hence 119886 isin (119861Γ119868Γ119876] where 120572 120573 120574 120575 isin ΓConversely let 119861cap 119868 cap119876 sube (119861Γ119868Γ119876] for every ordered bi-

Γ-ideal 119861 every ordered Γ-ideal 119868 and every ordered quasi-Γ-ideal 119876 of 119878 Suppose 119904 isin 119878 Let 119861(119904) and 119876(119904) be theordered bi-Γ-ideal and ordered quasi-Γ-ideal of 119878 generatedby 119904 respectively So we have the following

119904 isin 119861 (119904) cap 119868 (119904) cap 119876 (119904)

sube (119861 (119904) Γ119868 (119904) Γ119876 (119904)]

sube ((119904 cup 119904Γ119878Γ119904] Γ119878Γ (119904 cup (119878Γ119904 cap 119904Γ119878)]]

sube ((119904 cup 119904Γ119878Γ119904] Γ119878Γ (119904 cup 119878Γ119904]] sube (119904Γ119878Γ119904]

(31)

Hence 119878 is regular

Next consider 119877 in place of119876 inTheorem 11 to obtain thefollowing

Corollary 12 An ordered Γ-semigroup 119878 is regular if and onlyif for every ordered bi-Γ-ideal 119861 every ordered Γ-ideal 119868 andevery right Γ-ideal 119877 of 119878

119861 cap 119868 cap 119877 sube (119861Γ119868Γ119877] (32)

Theorem 13 A partially ordered Γ-semigroup 119878 is regular ifand only if for every ordered quasi-Γ-ideal119876 every ordered leftΓ-ideal 119871 and every ordered right-Γ-ideal 119877 one has

119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (33)

Proof Let 119878 be regular then for any 119886 isin 119877 cap 119876 cap 119871 thereexists 119904 isin 119878 such that 119886 le 119886120572119904120573119886 le (119886120572119904120573119886)120574119904120575(119886120579119904120582119886) =

(119886120572119904)120573119886120574(119904120575119886120579119904120582119886) isin (119877Γ119878)Γ119876Γ(119878Γ119871Γ119878Γ119871) sube 119877Γ119876Γ119871 for120572 120573 120574 120579 120575 120582 isin Γ Hence 119886 isin (119877Γ119876Γ119871]

Conversely let

119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (34)

for every ordered right Γ-ideal 119877 every ordered quasi-Γ-ideal119876 and every ordered left Γ-ideal 119871 of 119878 Suppose 119904 isin 119878 So wehave

119904 isin 119877 (119904) cap 119876 (119904) cap 119871 (119904)

sube (119877 (119904) Γ119876 (119904) Γ119871 (119904)] sube (119877 (119904) Γ119878Γ119871 (119904)]

sube (119877 (119904) Γ119871 (119904)] sube ((119904 cup 119904Γ119878] Γ (119904 cup 119878Γ119904]]

sube ((119904Γ119904 cup 119904Γ119878Γ119904]]

(35)

So for 120572 120573 120574 isin Γ 119904 le 119904120572119904 or 119904 le 119904120572119909120573119904 for some 119909 isin 119878 If119904 le 119904120572119904 then 119904 le 119904120572119904 le (119904120572119904)120573(119904120574119904) = 119904120572(119904120573119904)

2120574119904 isin 119904Γ119878Γ119904

If 119904 le 119904120572119909120573119904 for some 119909 isin 119878 then 119904 isin 119904Γ119878Γ119904 So finally weobtain 119904 isin (119904Γ119878Γ119904] Hence 119878 is regular

Corollary 14 If one considers an ordered left Γ-ideal 119871 (or anordered right Γ-ideal 119877) in place of the ordered quasi-Γ-ideal119876in Theorem 13 one obtains

119871 cap 119877 sube (119877Γ119871] (36)

Acknowledgment

The authors are grateful to the referee for the useful com-ments and valuable suggestions

References

[1] O Steinfeld ldquoOn ideal-quotients and prime idealsrdquoActaMathe-matica Academiae Scientiarum Hungaricae vol 4 pp 289ndash2981953

[2] O Steinfeld ldquoUber die Quasiideale von Halbgruppenrdquo Publica-tiones Mathematicae Debrecen vol 4 pp 262ndash275 1956

[3] O Steinfeld Quasi-Ideals in Rings and Semigroups vol 10of Disquisitiones Mathematicae Hungaricae Akademiai KiadoBudapest Hungary 1978

[4] R Chinram ldquoA note on Quasi-ideals in Γ-semiringsrdquo Interna-tional Mathematical Forum vol 3 no 25ndash28 pp 1253ndash12592008

[5] R Chinram ldquoOn quasi gamma-ideals in Γ-semigroupsrdquo Sci-enceAsia vol 32 pp 351ndash353 2006

[6] C Donges ldquoOn Quasi-ideals of semiringsrdquo International Jour-nal of Mathematics and Mathematical Sciences vol 17 no 1 pp47ndash58 1994

[7] A H Clifford ldquoRemarks on o-minimal Quasi-ideals in semi-groupsrdquo Semigroup Forum vol 16 no 2 pp 183ndash196 1978

[8] P Choosuwan and R Chinram ldquoA study on Quasi-ideals internary semigroupsrdquo International Journal of Pure and AppliedMathematics vol 77 no 5 pp 39ndash647 2012

[9] V N Dixit and S Dewan ldquoMinimal Quasi-ideals in ternarysemigrouprdquo Indian Journal of Pure and Applied Mathematicsvol 28 no 5 pp 625ndash632 1997

[10] V N Dixit and S Dewan ldquoA note on quasi and bi-ideals internary semigroupsrdquo International Journal of Mathematics andMathematical Sciences vol 18 no 3 pp 501ndash508 1995

[11] K Iseki ldquoQuasi-ideals in semirings without zerordquo Proceedingsof the Japan Academy vol 34 pp 79ndash81 1958

[12] R D Jagatap and Y S Pawar ldquoQuasi-ideals and minimal quasi-ideals in Γ-semiringsrdquoNovi Sad Journal of Mathematics vol 39no 2 pp 79ndash87 2009

[13] N Kehayopulu S Lajos and G Lepouras ldquoA note on bi-and Quasi-ideals of semigroups ordered semigroupsrdquo PureMathematics and Applications vol 8 no 1 pp 75ndash81 1997

[14] S Lajos ldquoOn quasiideals of regular ringrdquo Proceedings of theJapan Academy vol 38 pp 210ndash211 1962

[15] I Yakabe ldquoQuasi-ideals in near-ringsrdquoMathematical Reports ofCollege of General Education Kyushu University vol 14 no 1pp 41ndash46 1983

[16] N Kehayopulu ldquoOn completely regular ordered semigroupsrdquoScientiae Mathematicae vol 1 no 1 pp 27ndash32 1998

[17] Y I Kwon and S K Lee ldquoSome special elements in orderedΓ-semigroupsrdquo Kyungpook Mathematical Journal vol 35 no 3pp 679ndash685 1996

[18] A Iampan and M Siripitukdet ldquoOn minimal and maximalordered left ideals in PO-Γ-semigroupsrdquoThai Journal of Math-ematics vol 2 no 2 pp 275ndash282 2004

Algebra 7

[19] A Iampan ldquoCharacterizing ordered bi-ideals in ordered Γ-semigroupsrdquo Iranian Journal of Mathematical Sciences amp Infor-matics vol 4 no 1 pp 17ndash25 2009

[20] A Iampan ldquoCharacterizing ordered Quasi-ideals of ordered Γ-semigroupsrdquo Kragujevac Journal of Mathematics vol 35 no 1pp 13ndash23 2011

[21] Y I Kwon and S K Lee ldquoTheweakly semi-prime ideals of po-Γ-semigroupsrdquo Kangweon-Kyungki Mathematical Journal vol 5no 2 pp 135ndash139 1997

[22] M Siripitukdet and A Iampan ldquoOn the least (ordered) semi-lattice congruence in ordered Γ-semigroupsrdquo Thai Journal ofMathematics vol 4 no 2 pp 403ndash415 2006

[23] N K Saha ldquoOn Γ-semigroup IIrdquo Bulletin of the CalcuttaMathematical Society vol 79 no 6 pp 331ndash335 1987

[24] M K Sen and N K Saha ldquoOn Γ-semigroup Irdquo Bulletin of theCalcutta Mathematical Society vol 78 no 3 pp 180ndash186 1986

[25] M K Sen ldquoOn Γ-semigroupsrdquo in Proceedings of the Interna-tional Confernec on Algebra and Itrsquos Applications pp 301ndash308Decker New York NY USA 1981

[26] N Kehayopulu ldquoOn prime weakly prime ideals in orderedsemigroupsrdquo Semigroup Forum vol 44 no 3 pp 341ndash346 1992

[27] Y Cao and X Xinzhai ldquoNil-extensions of simple po-semigroupsrdquo Communications in Algebra vol 28 no 5pp 2477ndash2496 2000

[28] N Kehayopulu ldquoNote on Greenrsquos relations in ordered semi-groupsrdquoMathematica Japonica vol 36 no 2 pp 211ndash214 1991

[29] N Kehayopulu ldquoOn regular ordered semigroupsrdquoMathematicaJaponica vol 45 no 3 pp 549ndash553 1997

[30] N Kehayopulu ldquoOn weakly prime ideals of ordered semi-groupsrdquo Mathematica Japonica vol 35 no 6 pp 1051ndash10561990

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013


Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics


Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics


Journal of Function Spaces

International Journal of Mathematics and Mathematical Sciences


ISRN Geometry


Discrete Dynamics in Nature and Society



Volume 2013

Advances in

Mathematical Physics

ISRN Algebra


ProbabilityandStatistics

Journal of


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Journal ofApplied Mathematics


Advances in

DecisionSciences



Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal


ISRN Discrete Mathematics


Differential EquationsInternational Journal of

Volume 2013

2 Algebra

(also called po-Γ-semigroups) wemean an ordered set (119878 le)at the same time a Γ-semigroup satisfying the following con-ditions

119886 le 119887 997904rArr 119886120574119888 le 119887120574119888 119888120574119886 le 119888120574119887 forall119886 119887 119888 isin 119878 120574 isin Γ (1)

Throughout this paper 119878 will stand for an ordered Γ-semigroup unless otherwise stated An ordered Γ-semigroup119878 is called regular if for each 119904 isin 119878 and for each 120572 120573 isin Γ thereexists 119886 isin 119878 such that 119904 le 119904120572119886120573119904 Equivalent definitions ofregular ordered Γ-semigroup are as follows (i) 119860 sube (119860Γ119878Γ119860]

for each119860 sube 119878 and (ii) 119904 isin (119904Γ119878Γ119904] for each 119904 isin 119878 Let (119878 le) bean ordered Γ-semigroup and119873 a sub-Γ-semigroup of 119878 then(119873 le) is an ordered Γ-semigroup Let119860 be a nonempty subsetof119873 Then similarly to [26] we write (119860]

119873= 119899 isin 119873 119899 le 119886

for some 119886 isin 119860 and 119860 cup 119886 = 119860 cup 119886 We also write (119860]119873

by simply (119860] if 119873 = 119878 (see [27]) A nonempty subset 119868 ofan ordered Γ-semigroup 119878 is called an ordered right-Γ-ideal(left-Γ-ideal) of 119878 if 119868Γ119878 sube 119868(119878Γ119868 sube 119868) and for any 119909 isin 119868(119909] sube 119868 119868 is called an ordered Γ-ideal of 119878 if it is both a leftand a right Γ-ideals of 119878 Also for any 119904 isin 119878 we have that (119878Γ119904]is an ordered left Γ-ideal of 119878 and (119904Γ119878] is an ordered right Γ-ideal of 119878 [18] A nonempty subset 119876 of 119878 is called an orderedquasi-Γ-ideal of 119878 if (i) (119876Γ119878] cap (119878Γ119876] sube 119876 and (ii) (119876] sube 119876A sub-Γ-semigroup 119861 of an ordered Γ-semigroup 119878 is calledan ordered bi-Γ-ideal of 119878 if 119861Γ119878Γ119861 sube 119861 and for any 119909 isin 119861(119909] sube 119861

Let119883 be a nonempty subset of 119878Then the least right (left)ordered Γ-ideal of 119878 containing 119883 is given by 119877(119883) = (119883 cup

119883Γ119878](119871(119883) = (119878Γ119883 cup 119883]) If 119883 = 119904 119904 isin 119878 we write 119877119904and 119871119904 respectively by 119877(119904) and 119871(119904) and 119877(119904) = (119904 cup 119904Γ119878]119871(119904) = (119878Γ119904 cup 119904] and the ideal generated by 119904 isin 119878 is given by119868(119904) = (119904 cup 119878Γ119904 cup 119904Γ119878 cup 119878Γ119904Γ119878] Also the least quasi-Γ-ideal of119878 containing 119883 is denoted by 119876(119883) Moreover we willl needsome notations as follows (i)119873

119876= 119876 119876 = 0 where 119876 sube 119878

and (119876] sube 119876 (ii) 119877119868is a set of ordered right Γ-ideals of 119878

(iii) 119871119868is a set of ordered left Γ-ideals of 119878 and (iv) 119868

119879is a

two-sided Γ-ideal of 119878Now for any two elements 119876

1 1198762isin 119873119876 we define an

operation lowast in119873119876as follows


2= (1198761Γ1198762] (2)

Further let 119873 be a sub-Γ-semigroup of 119878 Then we caneasily observe here the following (see [16 18 21 28ndash30])

(i) 119860 sube (119860]119873sube (119860] = ((119860]] for 119860 sube 119873

(ii) for 119860 sube 119873 and 119861 sube 119873 we have (119860 cup 119861] = (119860] cup (119861](iii) for 119860 sube 119873 and 119861 sube 119873 we have (119860 cap 119861] sube (119860] cap (119861](iv) for 119886 and 119887 isin 119873 with 119886 le 119887 we have (119886Γ119873] sube (119887Γ119873]

and (119873Γ119886] sube (119873Γ119887](v) (119860]Γ(119861] sube (119860Γ119861](vi) for every left (right two-sided) ideal 119871 of 119878 (119871] = 119871(vii) if 119860 and 119861 are ordered Γ-ideals of 119878 then (119860Γ119861] and

119860 cup 119861 are also ideals of 119878(viii) for any 119904 isin 119878 (119878Γ119904Γ119878] is an ideal of 119878

3 Ordered Γ-Semigroups and OrderedQuasi-Γ-Ideals

In this section we study some classical properties of theordered Γ-semigroup 119878 We start with the following lemma

Lemma 1 Let 119878 be an ordered Γ-semigroup Then

(i) (119873119876 lowast sube) is an ordered Γ-semigroup

(ii) (119871119868 lowast sube) (119877

119868 lowast sube) and (119868

119879 lowast sube) are sub-Γ-semi-

groups of (119873119876 lowast sube)

Proof (i) Suppose 119875 119876 119877 isin 119873119876 Since 119875Γ119876 isin (119875Γ119876] we

obtain ((119875Γ119876)Γ119877] sube ((119875Γ119876]Γ119877] Next we have (119875 lowast Γ lowast 119876) lowastΓlowast119877 = (119875Γ119876Γ119877] by using (119875lowastΓlowast119876)lowastΓlowast119877 = (119875Γ119876]lowastΓlowast119877 =

((119875Γ119876]Γ119877] sube ((119875Γ119876)Γ119877] = (119875Γ119876Γ119877] In a similar way wecan show that 119875 lowast Γ lowast (119876 lowast Γ lowast 119877) = (119875Γ119876Γ119877] and therefore(119875 lowast Γ lowast 119876) lowast Γ lowast 119877 = 119875 lowast Γ lowast (119876 lowast Γ lowast 119877) Hence (119873

119876 lowast) is

a Γ-semigroup Suppose 119875 sube 119876 Then 119875 lowast Γ lowast 119877 = (119875Γ119876] sube

(119876Γ119877] = 119876lowastΓlowast119877 and119877lowastΓlowast119875 = (119877Γ119875] sube (119877Γ119876] = 119877lowastΓlowast119876Hence (119873

119876 lowast sube) is an ordered Γ-semigroup

(ii) We have that 119871119868 119877119868 and 119868

119879are nonempty subsets of

119873119876 Suppose 119871

1 1198712isin 119871119868 Then obviously we have (119871

1lowast Γ lowast

1198712] = ((119871

1Γ1198712]] = (119871

1Γ1198712] Moreover using

119878Γ (1198711lowast Γ lowast 119871

2) = 119878Γ (119871

1Γ1198712]

sube (119878Γ (1198711Γ1198712]]

sube ((119878Γ1198711) Γ1198712]

sube (1198711Γ1198712]

= 1198711lowast Γ lowast 119871

2

(3)

we infer that1198711lowastΓlowast119871

2is a left Γ-ideal of 119878 that is119871

1lowastΓlowast119871

2isin

119871119868 Thus (119871

119868 lowast sube) is a sub-Γ-semigroup of (119873

119876 lowast sube)

Dually we can prove that (119877119868 lowast sube) is a sub-Γ-semigroup

of (119873119876 lowast sube) Since 119868

119879= 119871119868cap 119877119868 it follows that (119868

119879 lowast sube) is a

sub-Γ-semigroup of (119873119876 lowast sube)

Let 119876119868= 119876 119876 is an ordered quasi-Γ-ideal of 119878 Then

obviously we have 119871119868cup 119877119868sube 119876119868sube 119873119876 This implies that

every one-sided Γ-ideal of an ordered Γ-semigroup is a quasi-Γ-ideal of 119878 Thus the class of ordered quasi-Γ-ideals of 119878 is ageneralization of the class of one-sided ordered Γ-ideals of 119878

Lemma 2 Each ordered quasi-Γ-ideal 119876 of an ordered Γ-semigroup 119878 is a sub-Γ-semigroup of 119878

Proof Proof is straightforward In fact we have119876Γ119876 sube 119876Γ119878cap

119878Γ119876 sube (119876Γ119878] cap (119878Γ119876] sube 119876

Lemma 3 For every ordered right Γ-ideal 119877 and an orderedleft Γ-ideal 119871 of an ordered Γ-semigroup 119878 119877 cap 119871 is an orderedquasi-Γ-ideal of 119878

Proof As 119877Γ119871 sube 119878Γ119871 sube 119871 and 119877Γ119871 sube 119877Γ119878 sube 119877 we obtain119877Γ119871 sube 119877 cap 119871 so 119877 cap 119871 = 0 Now the fact that 119877 cap 119871 is anordered quasi-Γ-ideal of 119878 follows from the following

(i) (119877 cap 119871] sube (119877] cap (119871] sube 119877 cap 119871

Algebra 3






119876





(119883 cup 119883Γ119878]


119871 (119904) cap 119877 (119904) = (119878Γ119904 cup 119904] cap (119904 cup 119904Γ119878]


= 119876 (119904)

(5)

Hence 119876(119904) = 119871(119904) cap 119877(119904)





119876Γ119876 sube 119876Γ119878Γ119868

= 119876Γ (119878Γ119868) sube 119876Γ119868


119876Γ119878Γ119876 sube 119868Γ119876Γ119878

= (119868Γ119878) Γ119876 sube 119868Γ119876 sube (119868Γ119876]

sube (119868Γ119878] sube (119868] sube 119868

(6)


1rArr


Therefore


= (119868Γ119876]119868 cap (119876Γ119868]119868 sube 119876 (119876] sube 119876

(7)






119868 lowast) and (119877

119868 lowast) is


⟨119871119868cup 119877119868⟩ = 119871


119868) (8)


⟨119871119868cup 119877119868⟩ =119884



lowast 119884119899| 119884119895isin 119871119868


+ Γ119895isin Γ

(9)

Suppose 119884119895 119884119895+1


are as follows (i) 119884119895 119884119895+1



119895+1isin 119871119868 (ii) 119884

119895 119884119895+1




119895isin 119871119868

119884119895+1



119895+1= (119884119895Γ119884119895+1] is an


isin 119868119879= 119871119868cap119877119868 (iv)119884

119895isin 119877119868

119884119895+1




119868in


1 119884



(i)1015840 If1198841isin 119871119868 then119884



lowast119884119899isin 119871119868

(ii)1015840 If119884119899isin 119877119868 then119884



lowast119884119899isin 119877119868

(iii)1015840 If 1198841isin 119877119868and 119884


1lowast Γ1lowast




119868 Hence the

lemma holds


4 Algebra



(119877Γ119871] = 119877 cap 119871 (10)



(iv) (119871119868 lowast) and (119877



119876 lowast)


119868 lowast)




(119876Γ119878Γ119876](vii) (119876




(119877Γ119871] sube 119877 cap 119871 (11)



119877 = 119877 cap (119877 cup 119878Γ119877] = (119877Γ (119877 cup 119878Γ119877]]



1015840isin 119877 or 119888 = 119904120572119903


1015840isin 119877Γ119877 or 119886 le 119903120572119888 =

119903120572(11990412057311990310158401015840) = (119903120572119904)120574119903




119868 lowast) and (119877



119868sube 119876119868


119868in (119873119876 lowast)





cup (119878Γ119876) Γ (119878Γ119876)] sube (119878Γ119876]

(13)



(14)



(b) (119877Γ119871] = (119878Γ (119877Γ119871]] cap ((119877Γ119871] Γ119878]

(15)


(119878Γ119876] = ((119878Γ119876]2]

= ((119878Γ119876] Γ (119878Γ119876]] = ((119878Γ119876] Γ ((119878Γ119878] Γ119876]]



sube (119878Γ119876]

(16)


(119876Γ119878] = (((119876Γ119878] Γ (119878Γ119876]] Γ119878] (17)


(c) 119876 = (119876Γ119878] cap (119878Γ119876]

= (((119876Γ119878] Γ (119878Γ119876]] Γ119878] cap (119878Γ ((119876Γ119878] Γ (119878Γ119876]]]

= ((119876Γ119878] Γ (119878Γ119876]]


sube ⟨119871119868cup 119877119868⟩

(18)


119868= ⟨119871119868cup119877119868⟩

in (119873119876 lowast)



(119878Γ(1198761Γ1198762] cup (119876



(1198761Γ1198762] sube (119878Γ (119876

1Γ1198762] cup (119876

1Γ1198762]]

= ((119878Γ (1198761Γ1198762] cup (119876

1Γ1198762]]2]

sube (119878Γ (1198761Γ1198762]] = ((119878Γ119878] Γ (1198761Γ1198762]]

sube (119878Γ (119878Γ (1198761Γ1198762]]]

(19)

Algebra 5


1Γ1198762] cup

(1198761Γ1198762]Γ119878] sube (((119876


show that

(1198761Γ1198762] sube (119878Γ (119876

1Γ1198762] cup (119876

1Γ1198762]]

cap ((1198761Γ1198762] cup (119876

1Γ1198762] Γ119878]

sube (119878Γ (119878Γ (1198761Γ1198762]]]

cap (((1198761Γ1198762] Γ119878] Γ119878]

= (((1198761Γ1198762] Γ119878] Γ (119878Γ (119876

1Γ1198762]]]

sube ((1198761Γ (1198762Γ119878Γ119878) Γ119876

1) Γ1198762] sube (119876

1Γ1198762]

(20)

By Theorem 5 (ii) (1198761Γ1198762] = (119878Γ(119876

1Γ1198762] cup (119876

1Γ1198762]] cap

((1198761Γ1198762] cup (119876



2isin 119876119868 Hence (119876





119868 Thus (119876

119868 lowast)




1Γ119876]



(21)



119878so that


= ((119877 (119904) cap 119871 (119904)) Γ119876Γ (119877 (119904) cap 119871 (119904))]

sube (119877 (119904) Γ119878Γ119871 (119904)]


(22)




119894 le 119894120572119904120573119894 le 119894120572119904120573119894120574119904120575119894 = 119894120572 (119904120573119894120574119904) 120575119894 (23)




119876 = 119877 cap 119871 = (119877Γ119871] (24)






1 1198772isin 119877119868 one obtains

1198711cap 1198712sube (1198711Γ1198712]

1198771cap 1198772sube (1198771Γ1198772]

(25)


(119876Γ119876] = (119876Γ119876Γ119878Γ119876Γ119876]

= (119876Γ (119876Γ119878Γ119876) Γ119876] sube (119876Γ119876Γ119876]

(26)


1] is



(119878Γ1198761] cap (119876

1Γ119878] = ((119876

1Γ119878] Γ (119878Γ119876

1]]

sube (1198761Γ119878Γ1198761] sube (119876

1] sube 1198761

(27)





we obtain

1198711cap 1198712= ((119871

1cap 1198712) Γ119878Γ (119871

1cap 1198712)]

sube (1198711Γ (119878Γ119871

2)] sube (119871

1Γ1198712]

(28)


1

1198772isin 119877119868


119861 cap 119868 cap 119876 sube (119861Γ119868Γ119876] (29)

6 Algebra


119886 le 119886120572119904120573119886 le (119886120572119904120573119886) 120574119904120575 (119886120572119904120573119886)

= (119886120572119904120573119886) 120574 (119904120572119886120573119904) 120575119886 isin (119861Γ119861) Γ (119878Γ119868Γ119878) Γ119876

sube 119861Γ119868Γ119876

(30)



119904 isin 119861 (119904) cap 119868 (119904) cap 119876 (119904)

sube (119861 (119904) Γ119868 (119904) Γ119876 (119904)]



(31)




119861 cap 119868 cap 119877 sube (119861Γ119868Γ119877] (32)


119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (33)



Conversely let

119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (34)


119904 isin 119877 (119904) cap 119876 (119904) cap 119871 (119904)

sube (119877 (119904) Γ119876 (119904) Γ119871 (119904)] sube (119877 (119904) Γ119878Γ119871 (119904)]


sube ((119904Γ119904 cup 119904Γ119878Γ119904]]

(35)


2120574119904 isin 119904Γ119878Γ119904



119871 cap 119877 sube (119877Γ119871] (36)

Acknowledgment


References



















Algebra 7














OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013

Algebra 3






119876





(119883 cup 119883Γ119878]


119871 (119904) cap 119877 (119904) = (119878Γ119904 cup 119904] cap (119904 cup 119904Γ119878]


= 119876 (119904)

(5)

Hence 119876(119904) = 119871(119904) cap 119877(119904)





119876Γ119876 sube 119876Γ119878Γ119868

= 119876Γ (119878Γ119868) sube 119876Γ119868


119876Γ119878Γ119876 sube 119868Γ119876Γ119878

= (119868Γ119878) Γ119876 sube 119868Γ119876 sube (119868Γ119876]

sube (119868Γ119878] sube (119868] sube 119868

(6)


1rArr


Therefore


= (119868Γ119876]119868 cap (119876Γ119868]119868 sube 119876 (119876] sube 119876

(7)






119868 lowast) and (119877

119868 lowast) is


⟨119871119868cup 119877119868⟩ = 119871


119868) (8)


⟨119871119868cup 119877119868⟩ =119884



lowast 119884119899| 119884119895isin 119871119868


+ Γ119895isin Γ

(9)

Suppose 119884119895 119884119895+1


are as follows (i) 119884119895 119884119895+1



119895+1isin 119871119868 (ii) 119884

119895 119884119895+1




119895isin 119871119868

119884119895+1



119895+1= (119884119895Γ119884119895+1] is an


isin 119868119879= 119871119868cap119877119868 (iv)119884

119895isin 119877119868

119884119895+1




119868in


1 119884



(i)1015840 If1198841isin 119871119868 then119884



lowast119884119899isin 119871119868

(ii)1015840 If119884119899isin 119877119868 then119884



lowast119884119899isin 119877119868

(iii)1015840 If 1198841isin 119877119868and 119884


1lowast Γ1lowast




119868 Hence the

lemma holds


4 Algebra



(119877Γ119871] = 119877 cap 119871 (10)



(iv) (119871119868 lowast) and (119877



119876 lowast)


119868 lowast)




(119876Γ119878Γ119876](vii) (119876




(119877Γ119871] sube 119877 cap 119871 (11)



119877 = 119877 cap (119877 cup 119878Γ119877] = (119877Γ (119877 cup 119878Γ119877]]



1015840isin 119877 or 119888 = 119904120572119903


1015840isin 119877Γ119877 or 119886 le 119903120572119888 =

119903120572(11990412057311990310158401015840) = (119903120572119904)120574119903




119868 lowast) and (119877



119868sube 119876119868


119868in (119873119876 lowast)





cup (119878Γ119876) Γ (119878Γ119876)] sube (119878Γ119876]

(13)



(14)



(b) (119877Γ119871] = (119878Γ (119877Γ119871]] cap ((119877Γ119871] Γ119878]

(15)


(119878Γ119876] = ((119878Γ119876]2]

= ((119878Γ119876] Γ (119878Γ119876]] = ((119878Γ119876] Γ ((119878Γ119878] Γ119876]]



sube (119878Γ119876]

(16)


(119876Γ119878] = (((119876Γ119878] Γ (119878Γ119876]] Γ119878] (17)


(c) 119876 = (119876Γ119878] cap (119878Γ119876]

= (((119876Γ119878] Γ (119878Γ119876]] Γ119878] cap (119878Γ ((119876Γ119878] Γ (119878Γ119876]]]

= ((119876Γ119878] Γ (119878Γ119876]]


sube ⟨119871119868cup 119877119868⟩

(18)


119868= ⟨119871119868cup119877119868⟩

in (119873119876 lowast)



(119878Γ(1198761Γ1198762] cup (119876



(1198761Γ1198762] sube (119878Γ (119876

1Γ1198762] cup (119876

1Γ1198762]]

= ((119878Γ (1198761Γ1198762] cup (119876

1Γ1198762]]2]

sube (119878Γ (1198761Γ1198762]] = ((119878Γ119878] Γ (1198761Γ1198762]]

sube (119878Γ (119878Γ (1198761Γ1198762]]]

(19)

Algebra 5


1Γ1198762] cup

(1198761Γ1198762]Γ119878] sube (((119876


show that

(1198761Γ1198762] sube (119878Γ (119876

1Γ1198762] cup (119876

1Γ1198762]]

cap ((1198761Γ1198762] cup (119876

1Γ1198762] Γ119878]

sube (119878Γ (119878Γ (1198761Γ1198762]]]

cap (((1198761Γ1198762] Γ119878] Γ119878]

= (((1198761Γ1198762] Γ119878] Γ (119878Γ (119876

1Γ1198762]]]

sube ((1198761Γ (1198762Γ119878Γ119878) Γ119876

1) Γ1198762] sube (119876

1Γ1198762]

(20)

By Theorem 5 (ii) (1198761Γ1198762] = (119878Γ(119876

1Γ1198762] cup (119876

1Γ1198762]] cap

((1198761Γ1198762] cup (119876



2isin 119876119868 Hence (119876





119868 Thus (119876

119868 lowast)




1Γ119876]



(21)



119878so that


= ((119877 (119904) cap 119871 (119904)) Γ119876Γ (119877 (119904) cap 119871 (119904))]

sube (119877 (119904) Γ119878Γ119871 (119904)]


(22)




119894 le 119894120572119904120573119894 le 119894120572119904120573119894120574119904120575119894 = 119894120572 (119904120573119894120574119904) 120575119894 (23)




119876 = 119877 cap 119871 = (119877Γ119871] (24)






1 1198772isin 119877119868 one obtains

1198711cap 1198712sube (1198711Γ1198712]

1198771cap 1198772sube (1198771Γ1198772]

(25)


(119876Γ119876] = (119876Γ119876Γ119878Γ119876Γ119876]

= (119876Γ (119876Γ119878Γ119876) Γ119876] sube (119876Γ119876Γ119876]

(26)


1] is



(119878Γ1198761] cap (119876

1Γ119878] = ((119876

1Γ119878] Γ (119878Γ119876

1]]

sube (1198761Γ119878Γ1198761] sube (119876

1] sube 1198761

(27)





we obtain

1198711cap 1198712= ((119871

1cap 1198712) Γ119878Γ (119871

1cap 1198712)]

sube (1198711Γ (119878Γ119871

2)] sube (119871

1Γ1198712]

(28)


1

1198772isin 119877119868


119861 cap 119868 cap 119876 sube (119861Γ119868Γ119876] (29)

6 Algebra


119886 le 119886120572119904120573119886 le (119886120572119904120573119886) 120574119904120575 (119886120572119904120573119886)

= (119886120572119904120573119886) 120574 (119904120572119886120573119904) 120575119886 isin (119861Γ119861) Γ (119878Γ119868Γ119878) Γ119876

sube 119861Γ119868Γ119876

(30)



119904 isin 119861 (119904) cap 119868 (119904) cap 119876 (119904)

sube (119861 (119904) Γ119868 (119904) Γ119876 (119904)]



(31)




119861 cap 119868 cap 119877 sube (119861Γ119868Γ119877] (32)


119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (33)



Conversely let

119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (34)


119904 isin 119877 (119904) cap 119876 (119904) cap 119871 (119904)

sube (119877 (119904) Γ119876 (119904) Γ119871 (119904)] sube (119877 (119904) Γ119878Γ119871 (119904)]


sube ((119904Γ119904 cup 119904Γ119878Γ119904]]

(35)


2120574119904 isin 119904Γ119878Γ119904



119871 cap 119877 sube (119877Γ119871] (36)

Acknowledgment


References



















Algebra 7














OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013

4 Algebra



(119877Γ119871] = 119877 cap 119871 (10)



(iv) (119871119868 lowast) and (119877



119876 lowast)


119868 lowast)




(119876Γ119878Γ119876](vii) (119876




(119877Γ119871] sube 119877 cap 119871 (11)



119877 = 119877 cap (119877 cup 119878Γ119877] = (119877Γ (119877 cup 119878Γ119877]]



1015840isin 119877 or 119888 = 119904120572119903


1015840isin 119877Γ119877 or 119886 le 119903120572119888 =

119903120572(11990412057311990310158401015840) = (119903120572119904)120574119903




119868 lowast) and (119877



119868sube 119876119868


119868in (119873119876 lowast)





cup (119878Γ119876) Γ (119878Γ119876)] sube (119878Γ119876]

(13)



(14)



(b) (119877Γ119871] = (119878Γ (119877Γ119871]] cap ((119877Γ119871] Γ119878]

(15)


(119878Γ119876] = ((119878Γ119876]2]

= ((119878Γ119876] Γ (119878Γ119876]] = ((119878Γ119876] Γ ((119878Γ119878] Γ119876]]



sube (119878Γ119876]

(16)


(119876Γ119878] = (((119876Γ119878] Γ (119878Γ119876]] Γ119878] (17)


(c) 119876 = (119876Γ119878] cap (119878Γ119876]

= (((119876Γ119878] Γ (119878Γ119876]] Γ119878] cap (119878Γ ((119876Γ119878] Γ (119878Γ119876]]]

= ((119876Γ119878] Γ (119878Γ119876]]


sube ⟨119871119868cup 119877119868⟩

(18)


119868= ⟨119871119868cup119877119868⟩

in (119873119876 lowast)



(119878Γ(1198761Γ1198762] cup (119876



(1198761Γ1198762] sube (119878Γ (119876

1Γ1198762] cup (119876

1Γ1198762]]

= ((119878Γ (1198761Γ1198762] cup (119876

1Γ1198762]]2]

sube (119878Γ (1198761Γ1198762]] = ((119878Γ119878] Γ (1198761Γ1198762]]

sube (119878Γ (119878Γ (1198761Γ1198762]]]

(19)

Algebra 5


1Γ1198762] cup

(1198761Γ1198762]Γ119878] sube (((119876


show that

(1198761Γ1198762] sube (119878Γ (119876

1Γ1198762] cup (119876

1Γ1198762]]

cap ((1198761Γ1198762] cup (119876

1Γ1198762] Γ119878]

sube (119878Γ (119878Γ (1198761Γ1198762]]]

cap (((1198761Γ1198762] Γ119878] Γ119878]

= (((1198761Γ1198762] Γ119878] Γ (119878Γ (119876

1Γ1198762]]]

sube ((1198761Γ (1198762Γ119878Γ119878) Γ119876

1) Γ1198762] sube (119876

1Γ1198762]

(20)

By Theorem 5 (ii) (1198761Γ1198762] = (119878Γ(119876

1Γ1198762] cup (119876

1Γ1198762]] cap

((1198761Γ1198762] cup (119876



2isin 119876119868 Hence (119876





119868 Thus (119876

119868 lowast)




1Γ119876]



(21)



119878so that


= ((119877 (119904) cap 119871 (119904)) Γ119876Γ (119877 (119904) cap 119871 (119904))]

sube (119877 (119904) Γ119878Γ119871 (119904)]


(22)




119894 le 119894120572119904120573119894 le 119894120572119904120573119894120574119904120575119894 = 119894120572 (119904120573119894120574119904) 120575119894 (23)




119876 = 119877 cap 119871 = (119877Γ119871] (24)






1 1198772isin 119877119868 one obtains

1198711cap 1198712sube (1198711Γ1198712]

1198771cap 1198772sube (1198771Γ1198772]

(25)


(119876Γ119876] = (119876Γ119876Γ119878Γ119876Γ119876]

= (119876Γ (119876Γ119878Γ119876) Γ119876] sube (119876Γ119876Γ119876]

(26)


1] is



(119878Γ1198761] cap (119876

1Γ119878] = ((119876

1Γ119878] Γ (119878Γ119876

1]]

sube (1198761Γ119878Γ1198761] sube (119876

1] sube 1198761

(27)





we obtain

1198711cap 1198712= ((119871

1cap 1198712) Γ119878Γ (119871

1cap 1198712)]

sube (1198711Γ (119878Γ119871

2)] sube (119871

1Γ1198712]

(28)


1

1198772isin 119877119868


119861 cap 119868 cap 119876 sube (119861Γ119868Γ119876] (29)

6 Algebra


119886 le 119886120572119904120573119886 le (119886120572119904120573119886) 120574119904120575 (119886120572119904120573119886)

= (119886120572119904120573119886) 120574 (119904120572119886120573119904) 120575119886 isin (119861Γ119861) Γ (119878Γ119868Γ119878) Γ119876

sube 119861Γ119868Γ119876

(30)



119904 isin 119861 (119904) cap 119868 (119904) cap 119876 (119904)

sube (119861 (119904) Γ119868 (119904) Γ119876 (119904)]



(31)




119861 cap 119868 cap 119877 sube (119861Γ119868Γ119877] (32)


119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (33)



Conversely let

119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (34)


119904 isin 119877 (119904) cap 119876 (119904) cap 119871 (119904)

sube (119877 (119904) Γ119876 (119904) Γ119871 (119904)] sube (119877 (119904) Γ119878Γ119871 (119904)]


sube ((119904Γ119904 cup 119904Γ119878Γ119904]]

(35)


2120574119904 isin 119904Γ119878Γ119904



119871 cap 119877 sube (119877Γ119871] (36)

Acknowledgment


References



















Algebra 7














OperationsResearch

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Algebra 5


1Γ1198762] cup

(1198761Γ1198762]Γ119878] sube (((119876


show that

(1198761Γ1198762] sube (119878Γ (119876

1Γ1198762] cup (119876

1Γ1198762]]

cap ((1198761Γ1198762] cup (119876

1Γ1198762] Γ119878]

sube (119878Γ (119878Γ (1198761Γ1198762]]]

cap (((1198761Γ1198762] Γ119878] Γ119878]

= (((1198761Γ1198762] Γ119878] Γ (119878Γ (119876

1Γ1198762]]]

sube ((1198761Γ (1198762Γ119878Γ119878) Γ119876

1) Γ1198762] sube (119876

1Γ1198762]

(20)

By Theorem 5 (ii) (1198761Γ1198762] = (119878Γ(119876

1Γ1198762] cup (119876

1Γ1198762]] cap

((1198761Γ1198762] cup (119876



2isin 119876119868 Hence (119876





119868 Thus (119876

119868 lowast)




1Γ119876]



(21)



119878so that


= ((119877 (119904) cap 119871 (119904)) Γ119876Γ (119877 (119904) cap 119871 (119904))]

sube (119877 (119904) Γ119878Γ119871 (119904)]


(22)




119894 le 119894120572119904120573119894 le 119894120572119904120573119894120574119904120575119894 = 119894120572 (119904120573119894120574119904) 120575119894 (23)




119876 = 119877 cap 119871 = (119877Γ119871] (24)






1 1198772isin 119877119868 one obtains

1198711cap 1198712sube (1198711Γ1198712]

1198771cap 1198772sube (1198771Γ1198772]

(25)


(119876Γ119876] = (119876Γ119876Γ119878Γ119876Γ119876]

= (119876Γ (119876Γ119878Γ119876) Γ119876] sube (119876Γ119876Γ119876]

(26)


1] is



(119878Γ1198761] cap (119876

1Γ119878] = ((119876

1Γ119878] Γ (119878Γ119876

1]]

sube (1198761Γ119878Γ1198761] sube (119876

1] sube 1198761

(27)





we obtain

1198711cap 1198712= ((119871

1cap 1198712) Γ119878Γ (119871

1cap 1198712)]

sube (1198711Γ (119878Γ119871

2)] sube (119871

1Γ1198712]

(28)


1

1198772isin 119877119868


119861 cap 119868 cap 119876 sube (119861Γ119868Γ119876] (29)

6 Algebra


119886 le 119886120572119904120573119886 le (119886120572119904120573119886) 120574119904120575 (119886120572119904120573119886)

= (119886120572119904120573119886) 120574 (119904120572119886120573119904) 120575119886 isin (119861Γ119861) Γ (119878Γ119868Γ119878) Γ119876

sube 119861Γ119868Γ119876

(30)



119904 isin 119861 (119904) cap 119868 (119904) cap 119876 (119904)

sube (119861 (119904) Γ119868 (119904) Γ119876 (119904)]



(31)




119861 cap 119868 cap 119877 sube (119861Γ119868Γ119877] (32)


119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (33)



Conversely let

119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (34)


119904 isin 119877 (119904) cap 119876 (119904) cap 119871 (119904)

sube (119877 (119904) Γ119876 (119904) Γ119871 (119904)] sube (119877 (119904) Γ119878Γ119871 (119904)]


sube ((119904Γ119904 cup 119904Γ119878Γ119904]]

(35)


2120574119904 isin 119904Γ119878Γ119904



119871 cap 119877 sube (119877Γ119871] (36)

Acknowledgment


References



















Algebra 7














OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013

6 Algebra


119886 le 119886120572119904120573119886 le (119886120572119904120573119886) 120574119904120575 (119886120572119904120573119886)

= (119886120572119904120573119886) 120574 (119904120572119886120573119904) 120575119886 isin (119861Γ119861) Γ (119878Γ119868Γ119878) Γ119876

sube 119861Γ119868Γ119876

(30)



119904 isin 119861 (119904) cap 119868 (119904) cap 119876 (119904)

sube (119861 (119904) Γ119868 (119904) Γ119876 (119904)]



(31)




119861 cap 119868 cap 119877 sube (119861Γ119868Γ119877] (32)


119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (33)



Conversely let

119877 cap 119876 cap 119871 sube (119877Γ119876Γ119871] (34)


119904 isin 119877 (119904) cap 119876 (119904) cap 119871 (119904)

sube (119877 (119904) Γ119876 (119904) Γ119871 (119904)] sube (119877 (119904) Γ119878Γ119871 (119904)]


sube ((119904Γ119904 cup 119904Γ119878Γ119904]]

(35)


2120574119904 isin 119904Γ119878Γ119904



119871 cap 119877 sube (119877Γ119871] (36)

Acknowledgment


References



















Algebra 7














OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013

Algebra 7














OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013


OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013

On Ordered Quasi-Gamma-Ideals of Regular Ordered Gamma-Semigroups

Documents

Transcript of On Ordered Quasi-Gamma-Ideals of Regular Ordered Gamma-Semigroups