Ordered Quasi(BI)-Γ-Ideals in Ordered...

Research ArticleOrdered Quasi(BI)-Γ-Ideals in Ordered Γ-Semirings

Ali N A Koam AzeemHaider and Moin A Ansari

Department of Mathematics College of Science Jazan University Jazan Saudi Arabia

Correspondence should be addressed to Ali N A Koam akoumjazanuedusa

Received 19 January 2019 Accepted 28 February 2019 Published 20 March 2019

Academic Editor Andrei V Kelarev

Copyright copy 2019 Ali N A Koam et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we have defined ordered quasi-Γ-ideals and ordered bi-Γ-ideals in ordered Γ-semirings by defining the relation ldquolerdquoin ordered Γ semiring 119878 as 119886 le 119887 if 119886 + 119909 = 119887 for any 119886 119887 119909 isin 119878 By using this relation we have shown that ordered quasi-Γ-idealsand ordered bi-Γ-ideals in ordered Γ-semirings are generalization of quasi-ideals and bi-ideals in ordered semirings Propertiesof many types of ordered Γ-ideals including (semi)prime (strongly) irreducible and maximal ordered quasi-Γ-ideals and orderedbi-Γ-ideals in ordered Γ-semirings 119878 have been studied

1 Introduction

The notion of semiring was introduced by Vandiver [1] in1934 whereas Γ-semiring was introduced and studied by Rao[2] in 1995 as a generalization of the notion of Γ-rings aswell as of semirings It is well known that semiring is ageneralization of a ring and Γ-semiring is a generalization ofsemiring but interestingly ideals of semirings do not coincidewith ideals of rings Applications of semirings are not limitedto theoretical computer science graph theory optimizationtheory automata coding theory and formal languages but inother branches of sciences and technologies as well

Quasi-ideal was first introduced by O Steinfeld for semi-groups [3] and then for rings by the same author O Steinfeld[4] where it is seen that quasi-ideal is a generalization of leftand right ideals Kiyoshi Iseki [5] introduced quasi-ideals forsemirings without zero and showed results based on themDonges studied quasi-ideals in semirings in [6] whereasShabir et al [7] characterized semirings by using quasi-idealsMinimal quasi-ideals for Γ-semiring was studied by Iseki [5]

The concept of bi-ideal for semigroups was given byGood and Hughes [8] in 1952 Generalized bi-ideal wasthen introduced for rings in 1970 by Szasz [9 10] and thenfor semigroups by Lajos [11] Many types of ideals on thealgebraic structures were characterized by several authorssuch as the following In 2000 Dutta and Sardar studiedthe characterization of semiprime ideals and irreducibleideals of Γ-semirings [12] In 2004 Sardar and Dasgupta[13] introduced the notions of primitive Γ-semirings and

primitive ideals of Γ-semirings In 2008 Kaushik et al [14]introduced and studied bi-Γ-ideals in Γ-semirings In 2008Pianskool et al [15] introduced and studied valuation Γ-semirings and valuation Γ-ideals of a Γ-semiring

In 2008 Chinram studied properties of quasi-ideals in Γ-semirings [16] whereas in 2009 quasi-ideals and minimalquasi-ideals in Γ-semiring were studied by Jagatap andPawar [17] In 2014 Jagatap [18] studied prime 119896-Bi-idealsin Γ-semirings In 2015 Lavanya et al [19] studied primebiternary Γ-ideals in ternary Γ-semirings In 2016 Jagatap[20] defined the concept of bi-ideals in a Γ-semiring andstudied properties of bi-ideals in a regular Γ-semiring In2018 Rao introduced ideals in ordered Γ-semirings [21] Itis natural to define and study properties of ordered quasi-Γ-ideals and ordered bi-Γ-ideals in ordered Γ-semiringsTherefore we have extended the concept of ordered idealsin ordered Γ-semirings to that of ordered quasi-Γ-ideals andordered bi-Γ-ideals in ordered Γ-semiring 119878 by consideringdifferent order than the order defined in [21]

2 Preliminaries and Basic Definitions

In this section we will define important terminologies basedon those concepts which are being used in this paper

Definition 1 Let (119878 +) and (Γ +) be commutative semi-groupsThen we call 119878 a Γ-semiring if there exists a mapping119878 times Γ times 119878 997888rarr 119878 is written (119909 120572 119910) as 119909120572119910 such that it satisfiesthe following axioms for all 119909 119910 119911 isin 119878 and 120572 120573 isin Γ

HindawiJournal of MathematicsVolume 2019 Article ID 9213536 8 pageshttpsdoiorg10115520199213536

2 Journal of Mathematics

(i) 119909120572(119910 + 119911) = 119909120572119910 + 119909120572119911

(ii) (119909 + 119910)120572119911 = 119909120572119911 + 119910120572119911

(iii) 119909(120572 + 120573)119910 = 119909120572119910 + 119909120573119910

(iv) 119909120572(119910120573119911) = (119909120572119910)120573119911

Example 2 (see [21]) Let 119878 be a semiring and119872119901119902(119878) denotethe additive abelian semigroup of all 119901 times 119902 matrices withidentity element whose entries are from 119878 Then 119872119901119902(119878) isa Γ-semiring with Γ = 119872119901119902(119878) ternary operation is definedby 119909120572119911 = 119909(120572119905)119911 as the usual matrix multiplication where120572119905 denotes the transpose of the matrix 120572 for all 119909 119911 and120572 isin 119872119901119902(119878)

Definition 3 A Γ-semiring 119878 is called an ordered Γ-semiringif it admits a compatible relation le ie le is a partial orderingon 119878 satisfying the following conditions

If 119886 le 119887 and 119888 le 119889 then

(i) 119886 + 119888 le 119887 + 119889

(ii) 119886120574119888 le 119887120574119889

(iii) 119888120574119886 le 119889120574119887 forall119886 119887 119888 119889 isin 119878 and 120574 isin Γ

Definition 4 An ordered Γ-semiring 119878 is said to be commu-tative ordered Γ-semiring if 119909120574119910 = 119910120574119909 for all 119909 119910 isin 119878 and120574 isin Γ

Definition 5 (see [21]) A nonempty subset 119860 of an orderedΓ-semiring 119878 is called a sub-Γ-semiring of 119878 if (119860 +) is asubsemigroup of (119878 +) and 119886120574119887 isin 119860 for all 119886 119887 isin 119860 and 120574 isin Γ

Definition 6 (see [21]) A nonempty subset 119860 of an orderedΓ-semiring 119878 is called a left (right) ideal if 119860 is closed underaddition 119878Γ119860 sube 119860(119860Γ119878 sube 119860) and if for any isin 119878 119887 isin 119860 119886 le 119887then 119886 isin 119860

Definition 7 If 119860 is both a left and a right ideal of orderedΓ-semirings 119878 then 119860 is called an ideal of 119878

Definition 8 A right ideal 119860 of an ordered Γ-semiring 119878 iscalled a right 119896 ideal if 119886 isin 119860 and 119909 isin 119878 such that 119886 + 119909 isin 119860and then 119909 isin 119860

A left 119896 ideal of 119878 is defined in a similar way

Definition 9 (see [21]) A nonempty subset119870 of an ordered Γ-semiring 119878 is called a 119896 ideal if119870 is an ideal and 119909 isin 119878 119909+119910 isin119870 119910 isin 119870 then 119909 isin 119870

Definition 10 An ordered Γ-semiring 119878 is called totallyordered if any two elements of 119878 are comparable

Definition 11 Let119870 and119871 be ordered Γ-semirings Amapping119891 119870 997888rarr 119871 is called a homomorphism if

(i) 119891(119886 + 119887) = 119891(119886) + 119891(119887)

(ii) 119891(119886120572119887) = 119891(119886)120572119891(119887) for all 119886 119887 isin 119870 120572 isin Γ

3 Ordered Quasi-Γ-Ideals inOrdered-Γ-Semirings

In this section we define ordered quasi-Γ-ideal ordered 119896quasi-Γ-ideal ordered 119898 minus 119896 quasi-Γ-ideal ordered primequasi-Γ-ideal ordered semiprime quasi-Γ-ideal orderedmaximal quasi-Γ-ideal ordered irreducible and orderedstrongly irreducible quasi-Γ-ideal of an ordered Γ-semiring119878We define the relation ldquolerdquo in 119887 as 119886 le 119887 if 119886 + 119909 = 119887 for any119886 119887 119909 isin 119878 For a nonempty subset 119883 of 119878 and 119886 119887 isin 119878 we saythat 119886 le 119887 in 119883 if 119886 + 119909 = 119887 for any 119909 isin 119883

Definition 12 An ordered sub-Γ-semiring 119876 of an ordered Γ-semiring 119878 is called an ordered quasi-Γ-ideal of 119878 if 119878Γ119876 cap119876Γ119878 sube 119876 and if for any isin 119878 119902 isin 119876 119886 le 119902 then 119886 isin 119876 in119876

Definition 13 An ordered quasi-Γ-ideals 119876 is called ordered119896 quasi-Γ-ideal of 119878 if 119876 is an ordered sub-Γ-semiring of119878 and if 119902 isin 119876 and 119909 isin 119878 such that 119909 + 119902 isin 119876 then119909 isin 119876

Clearly every ordered 119896 quasi-Γ-ideal is an ordered quasi-Γ-ideal but converse is not true in general as shown in thefollowing example

Example 14 Let 119878 be the set of nonnegative integers andΓ = 119873 be additive abelian semigroups Tennary operation isdefined as (119909 120574 119910) 997888rarr 119909120574119910 usual multiplication of integersThen 119878 is an ordered Γ-semirings A subset 119876 = 5119878 5 of119878 is an ordered quasi-Γ-ideal of 119878 but it is not an ordered 119896quasi-Γ-ideal of 119878

Definition 15 An ordered quasi-Γ-ideal 119876 of an ordered Γ-semiring 119878 is called ordered 119898 minus 119896 quasi-Γ-ideal of 119878 if 119902 isin119876 119909 isin 119878 119902120574119909 isin 119876 for 120574 isin Γ then 119909 isin 119876

Theorem 16 Every ordered119898minus 119896 quasi-Γ-ideal of an orderedΓ-semiring 119878 is an ordered 119896 quasi-Γ-ideal of 119878

Proof Let119876 be an ordered119898minus 119896 quasi-Γ-ideal of an orderedΓ-semiring 119878 Consider 119909 + 119902 isin 119876 119902 isin 119876 119909 isin 119878 and 120574 isin Γthen (119909 + 119902)120574119909 isin 119876Then 119909 isin 119876 since 119876 is an ordered119898 minus 119896quasi-Γ-ideal Hence119876 is an ordered 119896 quasi-Γ-ideal of 119878

The following example shows that the converse of Theo-rem 16 is not true in general

Example 17 Let 119878 be the set of all natural numbers Then 119878with usual ordering is an ordered semiring If Γ = 119878 then 119878is an ordered Γ-semiring If 119876 = 2 4 6 then 119876 forms anordered 119896 quasi-Γ-ideal but not ordered119898minus 119896 quasi-Γ-idealsof ordered Γ-semiring 119878

Definition 18 An ordered Γ-semiring 119878 is called band if everyelement of 119878 is an idempotent

Theorem19 Let119876 be an ordered sub-Γ-semiring of an orderedΓ-semiring 119878 in which semigroup (119878 +) is a band 13en119876 is anordered quasi-Γ-ideal of 119878 if and only if119876 is an ordered 119896 quasi-Γ-ideals of 119878

Journal of Mathematics 3

Proof Let 119876 be an ordered quasi-Γ-ideal of an ordered Γ-semiring 119878 and 119909 + 119902 isin 119876 119902 isin 119876 and then 119909 + (119909 + 119902) =(119909+119909)+119902 = 119909+119902 997904rArr 119909 le 119909+119902 997904rArr 119909 isin 119876 as119876 is an orderedquasi-Γ-ideal of 119878Hence119876 is a an ordered 119896 quasi-Γ-ideal of119878

Conversely suppose that 119876 is an ordered 119896 quasi-Γ-idealof an ordered Γ-semiring 119878 Let 119909 isin 119878 119902 isin 119876 and 119909 le 119902 997904rArr119909 + 1199021015840 = 119902 997904rArr 119909 + 1199021015840 isin 119876 997904rArr 119909 isin 119876 since 119876 is an ordered119896 quasi-Γ-ideal of an ordered Γ-semiring 119878 Hence 119876 is anordered quasi-Γ-ideal of 119878

Definition 20 An ordered quasi-Γ-ideal 119876 of 119878 is called anirreducible ordered quasi-Γ-ideal if1198761cap1198762 = 119876 implies1198761 =119876 or 1198762 = 119876 for any ordered quasi-Γ-ideal 1198761 and 1198762 of 119878

Definition 21 An ordered quasi-Γ-ideal 119876 of 119878 is calledstrongly irreducible ordered quasi-Γ-ideal if for any orderedquasi-Γ-ideal 1198761 and 1198762 of 1198781198761 cap1198762 sube 119876 implies 1198761 sube 119876 or1198762 sube 119876 It is obvious that every strongly irreducible orderedquasi-Γ-ideal is an irreducible ordered quasi-Γ-ideal

Definition 22 An ordered quasi-Γ-ideal 119876 of 119878 is called aprime ordered quasi-Γ-ideal if 1198761Γ1198762 sube 119876 implies 1198761 sube 119876or 1198762 sube 119876 for any ordered quasi-Γ-ideals 1198761 1198762 of 119878

Definition 23 An ordered 119896 quasi-Γ-ideal 119876 of 119878 is called aprime ordered 119896 quasi-Γ-ideal if 1198761Γ1198762 sube 119876 implies 1198761 sube 119876or 1198762 sube 119876 for any ordered 119896 quasi-Γ-ideals 1198761 1198762 of 119878

Definition 24 An ordered quasi-Γ-ideal 119876 of 119878 is called astrongly prime ordered quasi-Γ-ideal if (1198761Γ1198762)cap (1198762Γ1198761) sube119876 implies that 1198761 sube 119876 or 1198762 sube 119876 for any ordered quasi-Γ-ideals 1198761 1198762 of 119878

Definition 25 An ordered quasi-Γ-ideal 119876 of 119878 is called asemiprime ordered quasi-Γ-ideal if for any ordered quasi-Γ-ideals 1198761 of 119878 119876

21 = 1198761Γ1198761 sube 119876 implies 1198761 sube 119876

It is clear that every strongly prime ordered quasi-Γ-idealin 119878 is a prime ordered quasi-Γ-ideal and every prime orderedquasi-Γ-ideal in 119878 is a semiprime ordered quasi-Γ-ideal

Definition 26 An ordered quasi-Γ-ideal 119876 of an ordered Γ-semiring 119878 is said to be maximal ordered quasi-Γ-ideal if119876 =119878 and for every ordered quasi-Γ-ideal 119862 of 119878 with 119876 sube 119862 sube 119878then either 119876 = 119862 or 119876 = 119878

Theorem 27 In an ordered Γ-semiring 119878 every maximalordered quasi-Γ-ideal of 119878 is irreducible ordered quasi-Γ-idealof 119878

Proof Let 119876119898 be a maximal ordered quasi-Γ-ideal of anordered Γ-semiring 119878 Suppose 119876119898 is not irreducible and119876119898 = 119880 cap 119881 997904rArr 119876119898 = 119880 and 119876119898 = 119881 997904rArr 119876119898 sub119880 sub 119878 and 119876119898 sub 119881 sub 119878 a contradiction Hence 119876119898 isirreducible ordered quasi-Γ-deal of an ordered Γ-semiring119878

Theorem 28 Let 119876 be an ordered quasi-Γ-ideal of an orderedΓ-semiring 119878

(i) If 119876 is a prime ordered quasi-Γ-ideal then 119876 is astrongly irreducible ordered quasi-Γ-ideal

(ii) If119876 is a strongly irreducible ordered quasi-Γ-ideal then119876 is an irreducible ordered quasi-Γ-ideal

Proof Let 119876 be an ordered quasi-Γ-ideal of an ordered Γ-semiring 119878

(i) Suppose119876 is a prime ordered quasi-Γ-ideal J andK areordered quasi-Γ-ideal of an ordered Γ-semiring 119878 such that119869 cap 119870 sube 119861 Then 119869Γ119870 sube 119876 997904rArr 119869 sube 119876 or 119870 sube 119876 Hence 119876 is astrongly irreducible ordered quasi-Γ-ideal

(ii) Suppose 119876 is a strongly irreducible ordered quasi-Γ-ideal 119869 and 119870 are ordered quasi-Γ-ideal of an ordered Γ-semiring 119878 such that 119869cap119870 = 119876Then certainly 119869cap119870 sube 119876 997904rArr119869 sube 119876 or 119870 sube 119876Hence 119869 = 119876 or 119870 = 119876

Therefore 119876 is an irreducible ordered quasi-Γ-ideal of 119878

Corollary 29 Let 119878 be an ordered Γ-semiring If 119876 is a primeordered quasi-Γ-ideal of 119878 then 119876 is an irreducible orderedquasi-Γ-ideal of 119878

Theorem 30 Let 119891 119870 997888rarr 119871 be a homomorphism of orderedΓ-semirings If 119869 is an ordered quasi-Γ-ideal of 119871 then 119891minus1(119869)is an ordered quasi-Γ-ideal of an ordered Γ-semiring 119870

Proof Suppose 119869 is an ordered quasi-Γ-ideal of 119871 119891 119870 997888rarr119871 is a homomorphism of ordered Γ-semirings and 119909 119910 isin119891minus1(119869) 997904rArr 119891(119909) 119891(119910) isin 119869 997904rArr 119891(119909) + 119891(119910) = 119891(119909 + 119910) isin119869 997904rArr 119909 + 119910 isin 119891minus1(119869) If 119909 119910 isin 119891minus1(119869) 997904rArr 119891(119909) 119891(119910) isin119869 997904rArr 119891(119909)120574119891(119910) isin 119869 997904rArr 119891(119909120574119910) isin 119869 997904rArr 119909120574119910 isin 119891minus1(119869) Weneed to show here that 119891minus1(119869) is an ordered quasi-Γ-ideal of119870 ie 119870Γ119891minus1(119869) cap 119891minus1(119869)Γ119870 sube 119891minus1(119869) For that we suppose119895 isin 119870Γ119891minus1(119869) cap 119891minus1(119869)Γ119870 997904rArr 119895 = 119904120572119909 and 119895 = 1199101205731199041015840 for119904 1199041015840 isin 119870 119909 119910 isin 119891minus1(119869) and 120572 120573 isin Γ 997904rArr 119891(119909) 119891(119910) isin 119869which is a quasi-ideal Now 119891(119895) = 119891(119904120572119909) = 119891(119904)120572119891(119909) and119891(119895) = 119891(119910)120573119891(1199041015840) 997904rArr 119891(119895) isin 119871Γ119869 cap 119869Γ119871 as 119891(119904) 119891(1199041015840) isin119871 997904rArr 119891(119895) isin 119869 997904rArr 119895 isin 119891minus1(119869)

Let 119909 isin 119870 119910 isin 119891minus1(119869) such that 119909 le 119910 997904rArr 119909 + 119910 =119910 997904rArr 119891(119909 + 119910) = 119891(119910) 997904rArr 119891(119909) + 119891(119910) = 119891(119910) isin 119869 997904rArr119891(119909) le 119891(119910) 997904rArr 119891(119909) isin 119869 997904rArr 119909 isin 119891minus1(119869) Hence 119891minus1(119869) isan ordered quasi-Γ-ideal of an ordered Γ-semiring 119870

Theorem31 Let 119878 be an ordered Γ-semiring If119876 is an ordered119898 minus 119896 quasi-Γ-ideal of 119878 then 119876 is a maximal ordered quasi-Γ-ideal of 119878

Proof Let119876 be an ordered119898minus 119896 quasi-Γ-ideal of an orderedΓ-semiring 119878 Suppose 119869 is an ordered quasi-Γ-ideal of 119878 suchthat 119876 sube 119869 119909 isin 119869 119910 isin 119876 and 120574 isin Γ We have 119910120574119909 isin 119876 997904rArr 119909 isin119876 since 119876 is an ordered 119898 minus 119896 quasi-Γ-ideal of 119878 Therefore119876 = 119869 Hence ordered 119898 minus 119896 quasi-Γ-ideal 119876 of 119878 is maximalordered quasi-Γ-ideal of 119878

Theorem 32 Let 119878 be an ordered Γ-semiring in which (119878 +)is cancellative semigroup and 119867 = 119909 isin 119878 | 119909 + 119909 = 119909 If119867 = 0 then119867 is an ordered 119898minus 119896 quasi-Γ-ideal of an orderedΓ-semiring 119878


Proof Let 119909 isin 119867 119910 isin 119878 and 120573 isin Γ Then 119909 = 119909 + 119909 997904rArr119909120573119910 = (119909 + 119909)120573119910 = 119909120573119910 + 119909120573119910 997904rArr 119909120573119910 isin 119867 Similarly119910120573119909 isin 119867 Suppose 119909 119910 isin 119867 Then 119909 + 119909 = 119909 119910 + 119910 = 119910 997904rArr(119909 + 119910) + (119909 + 119910) = (119909 + 119909) + (119910 + 119910) = 119909 + 119910 997904rArr 119909 + 119910 isin 119867Suppose 119909 le 119910 119910 isin 119867 Then 119909 + 119910 = 119910 997904rArr 119909 + 119909 + 119910 =119909 + 119910 997904rArr 119909 + 119909 = 119909 Let 119904120572ℎ isin 119878Γ119867 for 119904 isin 119878 120572 isin Γ andℎ isin 119867 Next 119904120572ℎ + 119904120572ℎ = 119904120572(ℎ + ℎ) = 119904120572ℎ 997904rArr 119904120572ℎ isin 119867 997904rArr119878Γ119867 sube 119867 In a similar way we can show that 119867Γ119878 sube 119867Therefore 119878Γ119867 cap 119867Γ119878 sube 119867Hence119867 is a quasi-ideal

Therefore119867 is an ordered quasi-Γ-ideal of an ordered Γ-semiring Suppose 119909120574119910 isin 119867 119909 isin 119867 and 120574 isin Γ Then 119909120574119910 +119909120574119910 = 119909120574119910 997904rArr 119909120574(119910 + 119910) = 119909120574119910 997904rArr 119910 + 119910 = 119910 997904rArr 119910 isin 119867Hence119867 is an ordered 119898 minus 119896 quasi-Γ-ideal of 119878

Theorem 33 13e intersection of family of prime (or semi-prime) ordered quasi-Γ-ideal of 119878 is a semiprime ordered quasi-Γ-ideal

Proof Let 119875119894 | 119894 isin Λ be the family of prime ordered quasi-Γ-ideal of S and 119875 = ⋂119896119894=1 119875119894 Since 119878Γ119875 = 119878Γ(⋂

119896119894=1 119875119894) sube

⋂119896119894=1(119878Γ119875119894) and 119875Γ119878 = (⋂119896119894=1 119875119894)Γ119878 sube ⋂119896119894=1(119875119894Γ119878) 997904rArr

119878Γ119875⋂119875Γ119878 sube ((⋂119896119894=1(119878Γ119875119894)) cap (⋂119896119894=1(119875119894)Γ119878) = ⋂

119896119894=1(119878Γ119875119894 cap

119875119894Γ119878) sube ⋂119896119894=1 119875119894 = 119875Hence 119875 is a quasi-Γ-ideal

Now for any 119910 isin 119875 such that 119909 le 119910 997904rArr 119909 isin 119875119894 forall119894 997904rArr 119909 isin119875 Hence 119875 is an ordered quasi-Γ-ideal of 119878 For any orderedquasi-Γ-ideal 119876 of 119878119876Γ119876 sube 119875 implies119876Γ119876 sube 119875119894 for all 119894 isin ΛAs 119875119894 are semiprime ordered quasi-Γ-ideals 119876 sube 119875119894 for all119894 isin Λ Hence 119876 sube 119875

Theorem 34 Every strongly irreducible semiprime orderedquasi-Γ-ideal of 119878 is a strongly prime ordered quasi-Γ-ideal

Proof Let119876 be a strongly irreducible and semiprime orderedquasi-Γ-ideal of 119878 For any ordered quasi-Γ-ideal 1198761 and 1198762of 119878 let (1198761Γ1198762) cap (1198762Γ1198761) sube 119876 1198761 cap 1198762 is a quasi-Γ-idealof 119878 since (1198761 cap 1198762)

2 = (1198761 cap 1198762)Γ(1198761 cap 1198762) sube 1198761Γ1198762Similarly we get (1198761 cap1198762)

2 = (1198761 cap1198762)Γ(1198761 cap1198762) sube 1198762Γ1198761Therefore (1198761 cap 1198762)

2 sube 1198761Γ1198762 cap 1198762Γ1198761 sube 119876 As 119876 is asemiprime ordered quasi-Γ-ideal 1198761 cap 1198762 sube 119876 But 119876 is astrongly irreducible ordered quasi-Γ-idealTherefore1198761 sube 119876or 1198762 sube 119876 Thus 119876 is a strongly prime ordered quasi-Γ-idealof 119878

Theorem 35 If 119876 is an ordered 119896 quasi-Γ-ideal of 119878 and 119902 isin119878 such that 119902 notin 119876 then there exists an irreducible ordered 119896quasi-Γ-ideal1198761 of 119878 such that 119876 sube 1198761 and 119902 notin 1198761

Proof Let 119865 be the family of ordered 119896 quasi-Γ-ideals of 119878which contain 119876 but do not contain element 119902 Then 119865 isnonempty as119876 isin 119865This family of ordered 119896 quasi-Γ-ideals ofS forms a partially ordered set under the set inclusion Henceby Zornrsquos lemma there exists a maximal ordered 119896 quasi-Γ-ideal say 1198761 in 119865 Therefore 119876 sube 1198761 and 119902 notin 1198761 Now toshow that 1198761 is an irreducible ordered 119896 quasi-Γ-ideal of 119878let119870 and 119871 be any two ordered 119896 quasi-Γ-ideals of 119878 such that119870 cap 119871 = 1198761 Suppose that 119870 and 119871 both contain 1198761 properlyBut 1198761 is a maximal ordered 119896 quasi-Γ-ideal in 119865 Hence weget 119902 isin 119870 and 119902 isin 119871 Therefore 119902 isin 119870 cap 119871 = 1198761 which is

absurd Thus either 119870 = 1198761 or 119871 = 1198761 Therefore 1198761 is anirreducible ordered 119896 quasi-Γ-ideal of 119878

Theorem 36 13e following statements are equivalent in 119878

(1) 13e set of ordered 119896 quasi-Γ-ideals of 119878 is totally orderedset under inclusion of sets

(2) Each ordered 119896 quasi-Γ-ideal of 119878 is strongly irreducible(3) Each ordered 119896 quasi-Γ-ideal of 119878 is irreducible

Proof (1) 997904rArr (2) Suppose that the set of ordered 119896 quasi-Γ-ideals of 119878 is a totally ordered set under inclusion of sets Let119861 be any ordered 119896 quasi-Γ-ideal of 119878 Then 119861 is a stronglyirreducible ordered 119896 quasi-Γ-ideal of 119878 for that let1198761 and 1198762be any two ordered 119896 quasi-Γ-ideal of 119878 such that1198761cap1198762 sube 119876But by the hypothesis we have either 1198761 sube 1198762 or 1198762 sube 1198761Therefore 1198761 cap 1198762 = 1198761 or 1198761 cap 1198762 = 1198762 Hence 1198761 sube 119876 or1198762 sube 119876 Thus 119876 is a strongly irreducible ordered 119896 quasi-Γ-ideal of 119878(2) 997904rArr (3) Suppose that each ordered 119896 quasi-Γ-ideal of

119878 is strongly irreducible Let119876 be any ordered 119896 quasi-Γ-idealof 119878 such that 119876 = 1198761 cap 1198762 for any ordered 119896 quasi-Γ-ideal1198761 and 1198762 of 119878 But by hypothesis 1198761 sube 119876 or 1198762 sube 119876 As119876 sube 1198761 and 119876 sube 1198762 we get 1198761 = 119876 or 1198762 = 119876 Hence 119876 isan irreducible ordered 119896 quasi-Γ-ideal of 119878(3) 997904rArr (1) Suppose that each ordered 119896 quasi-Γ-ideal of

119878 is an irreducible ordered 119896 quasi-Γ-ideal Let 1198761 and 1198762 beany two ordered 119896 quasi-Γ-ideal of 119878 Then 1198761 cap 1198762 is alsoordered 119896 quasi-Γ-ideal of 119878 Hence 1198761 cap 1198762 = 1198761 cap 1198762 willimply1198761cap1198762 = 1198761 or1198761cap1198762 = 1198762 by assumptionThereforeeither 1198761 sube 1198762 or1198762 sube 1198761This shows that the set of ordered119896 quasi-Γ-ideal of 119878 is totally ordered set under inclusion ofsets

Theorem37 Aprime ordered 119896 quasi-Γ-ideal119876of 119878 is a primeone sided ordered 119896 ideal of 119878

Proof Let 119876 be a prime 119896 quasi-Γ-ideal of 119878 Suppose 119876is not a one sided 119896 ideal of 119878 Therefore 119876Γ119878 sube 119876 and119878Γ119876 sube 119876 As119876 is a prime 119896 quasi-Γ-ideal (119876Γ119878)Γ(119878Γ119876) sube 119876(119876Γ119878)Γ(119878Γ119876) sube (119876Γ119878) cap (119878Γ119876) sube 119876 which is a contradictionTherefore119876Γ119878 sube 119876 or 119878Γ119876 sube 119876 Thus119876 is a prime one sided119896 ideal of 119878

Theorem38 An ordered 119896 quasi-Γ-ideal119876 of 119878 is prime if andonly if for a right ordered 119896 ideal 119877 and a le ordered 119896 ideal 119871of 119878 119877Γ119871 sube 119876 implies 119877 sube 119876 or 119871 sube 119876

Proof Suppose that an ordered 119896 quasi-Γ-ideal of 119878 is a primeordered 119896 quasi-Γ-ideal of 119878 Let 119877 be a right ordered 119896 idealand 119871 be a left ordered 119896 ideal of 119878 such that 119877Γ119871 sube 119876 119877 and119871 are ordered 119896 quasi-Γ-ideal of 119878 Hence 119877 sube 119876 or 119871 sube 119876Conversely we have to show that an ordered 119896 quasi-Γ-ideal119876 of 119878 is a prime ordered 119896 quasi-Γ-ideal of 119878 Let 119860 and 119862 beany two ordered 119896 quasi-Γ-ideals of 119878 such that 119860Γ119862 sube 119876 Forany 119886 isin 119860 and 119888 isin 119862 (119886)119903 sube 119860 and (119888)119897 sube 119862 where (119886)119903 and(119888)119897 denote the right ordered 119896 ideal and left ordered 119896 idealgenerated by 119886 and 119888 respectively Thus (119886)119903Γ(119888)119897 sube 119860Γ119862 sube 119876Hence by the assumption (119886)119903 sube 119861 or (119888)119897 sube 119876 which further


gives that 119886 isin 119876 or 119888 isin 119876 In this way we get that 119860 sube 119876 or119862 sube 119876 Hence 119876 is a prime ordered 119896 quasi-Γ-ideal of 119878

4 Ordered Bi-Γ-Ideals in Ordered-Γ-Semirings

In this section we define ordered bi-Γ-ideal ordered 119896 bi-Γ-ideal ordered 119898 minus 119896 bi-Γ-ideal ordered prime bi-Γ-idealordered semiprime bi-Γ-ideal ordered maximal bi-Γ-idealordered irreducible and ordered strongly irreducible bi-Γ-ideal of an ordered Γ-semiring 119878 Recall the relation in theprevious section ldquolerdquo in 119887 as119886 le 119887 if 119886+119909 = 119887 for any 119886 119887 119909 isin 119878Further for a nonempty subset119883 of 119878 and 119886 119887 isin 119878 119886 le 119887 in119883if 119886 + 119909 = 119887 for any 119909 isin 119883

Definition 39 An ordered sub-Γ-semiring 119861 of an ordered Γ-semiring 119878 is called an ordered bi-Γ-ideal of 119878 if 119861Γ119878Γ119861 sube 119861and if for any isin 119878 119887 isin 119861 119886 le 119887 in 119861 then 119886 isin 119861

Definition 40 An ordered bi-Γ-ideals 119861 is called ordered 119896bi-Γ-ideal of 119878 if 119861 is an ordered sub-Γ-semiring of 119878 and if119887 isin 119861 and 119909 isin 119878 such that 119909 + 119887 isin 119861 then 119909 isin 119861

Clearly every ordered 119896 bi-Γ-ideal is an ordered bi-Γ-idealbut converse is not true in general as shown in the followingexample

Example 41 Let 119878 be the set of nonnegative integers andΓ = 119873 be additive abelian semigroups Tennary operation isdefined as (119909 120574 119910) 997888rarr 119909120574119910 usual multiplication of integersThen 119878 is an ordered Γ-semirings A subset 119861 = 7119878 7 of 119878 isan ordered bi-Γ-ideal of 119878 but it is not an ordered 119896 bi-Γ-idealof 119878

Definition 42 An ordered bi-Γ-ideal 119861 of an ordered Γ-semiring 119878 is called ordered119898minus 119896 bi-Γ-ideal of 119878 if 119887 isin 119861 119909 isin119878 119887120574119909 isin 119861 for 120574 isin Γ and then 119909 isin 119861

Theorem 43 Every ordered 119898 minus 119896 bi-Γ-ideal of an ordered Γ-semiring 119878 is an ordered 119896 bi-Γ-ideal of 119878

Proof Let 119861 be an ordered 119898 minus 119896 bi-Γ-ideal of an ordered Γ-semiring 119878 Consider 119909 + 119887 isin 119861 119887 isin 119861 119909 isin 119878 and 120574 isin Γthen (119909 + 119887)120574119909 isin 119861Then 119909 isin 119861 since 119861 is an ordered 119898 minus 119896bi-Γ-ideal Hence 119861 is an ordered 119896 bi-Γ-ideal of 119878


Example 44 Let 119878 be the set of all natural numbers Then 119878with usual ordering is an ordered semiring If Γ = 119878 then119878 is an ordered Γ-semiring If 119861 = 2 4 6 then 119861 formsan ordered 119896 bi-Γ-ideal but not ordered 119898 minus 119896 bi-Γ-ideals ofordered Γ-semiring 119878

Definition 45 An ordered Γ-semiring 119878 is called band if everyelement of 119878 is a 120574-idempotent

Theorem 46 Let 119861 be an ordered sub-Γ-semiring of anordered Γ-semiring 119878 in which semigroup (119878 +) is a band13en119861 is an ordered bi-Γ-ideal of 119878 if and only if 119861 is an ordered 119896bi-Γ-ideals of 119878

Proof Let 119861 be an ordered bi-Γ-ideal of an ordered Γ-semiring 119878 and 119909+119887 isin 119861 119909 isin 119861Then119909+(119909+119887) = (119909+119909)+119887 =119909 + 119887 997904rArr 119909 le 119909+ 119887 997904rArr 119909 isin 119861 as 119861 is an ordered bi-Γ-ideal of119878Hence 119861 is a an ordered 119896 bi-Γ-ideal of 119878

Conversely suppose that 119861 is an ordered 119896 bi-Γ-ideal of anordered Γ-semiring 119878 Let 119909 isin 119878 119887 isin 119861 and 119909 le 119887 in 119861 997904rArr119909 + 1198871015840 = 119887 for any 1198871015840 isin 119861 997904rArr 119909 + 1198871015840 isin 119861 997904rArr 119909 isin 119861 since 119861 isan ordered 119896 bi-Γ-ideal of an ordered Γ-semiring 119878 Hence 119861is an ordered bi-Γ-ideal of 119878

Definition 47 An ordered bi-Γ-ideal 119861 of 119878 is called anirreducible ordered bi-Γ-ideal if 1198611 cap 1198612 = 119861 implies 1198611 = 119861or 1198612 = 119861 for any ordered bi-Γ-ideal 1198611 and 1198612 of 119878

Definition 48 An ordered bi-Γ-ideal 119861 of 119878 is called stronglyirreducible ordered bi-Γ-ideal if for any ordered bi-Γ-ideal1198611 and 1198612 of 119878 1198611 cap 1198612 sube 119861 implies 1198611 sube 119861 or 1198612 sube 119861 It isobviously that every strongly irreducible ordered bi-Γ-ideal isan irreducible ordered bi-Γ-ideal

Definition 49 An ordered bi-Γ-ideal 119861 of 119878 is called a primeordered bi-Γ-ideal if 1198611Γ1198612 sube 119861 implies 1198611 sube 119861 or 1198612 sube 119861 forany ordered bi-Γ-ideals 1198611 1198612 of 119878

Definition 50 An ordered 119896 bi-Γ-ideal 119861 of 119878 is called a primeordered 119896 bi-Γ-ideal if 1198611Γ1198612 sube 119861 implies 1198611 sube 119861 or 1198612 sube 119861for any ordered 119896 bi-Γ-ideals 1198611 1198612 of 119878

Definition 51 An ordered bi-Γ-ideal 119861 of 119878 is called a stronglyprime ordered bi-Γ-ideal if (1198611Γ1198612)cap(1198612Γ1198611) sube 119861 implies that1198611 sube 119861 or 1198612 sube 119861 for any ordered bi-Γ-ideals 1198611 1198612 of 119878

Definition 52 An ordered bi-Γ-ideal 119861 of 119878 is called asemiprime ordered bi-Γ-ideal if for any ordered bi-Γ-ideals1198611 of 119878 119861


It is clear that every strongly prime ordered bi-Γ-ideal in119878 is a prime ordered bi-Γ-ideal and every prime ordered bi-Γ-ideal in 119878 is a semiprime ordered bi-Γ-ideal

Definition 53 An ordered bi-Γ-ideal 119861 of an ordered Γ-semiring 119878 is said to be maximal ordered bi-Γ-ideal if 119861 = 119878and for every ordered bi-Γ-ideal 119862 of 119878 with 119861 sube 119862 sube 119878 theneither 119861 = 119862 or 119861 = 119878

Theorem 54 In an ordered Γ-semiring 119878 every maximalordered bi-Γ-ideal of 119878 is irreducible ordered bi-Γ-ideal of 119878

Proof Let 119861119898 be a maximal ordered bi-Γ-ideal of an orderedΓ-semiring 119878 Suppose 119861119898 is not irreducible and 119861119898 = 119880 cap119881 997904rArr 119861119898 = 119880 and 119861119898 = 119881 997904rArr 119861119898 sub 119880 sub 119878 and 119861119898 sub 119881 sub 119878a contradiction Hence 119861119898 is irreducible ordered bi-Γ-deal ofan ordered Γ-semiring 119878

Theorem 55 Let 119861 be an ordered bi-Γ-ideal of an ordered Γ-semiring 119878

(i) If 119861 is a prime ordered bi-Γ-ideal then 119861 is a stronglyirreducible ordered bi-Γ-ideal


(ii) If 119861 is a strongly irreducible ordered bi-Γ-ideal then 119861is an irreducible ordered bi-Γ-ideal

Proof Let 119861 be an ordered bi-Γ-ideal of an ordered Γ-semiring 119878

(i) Suppose 119861 is a prime ordered bi-Γ-ideal J and K areordered bi-Γ-ideal of an ordered Γ-semiring 119878 such that 119869 cap119870 sube 119861 Then 119869Γ119870 sube 119861 997904rArr 119869 sube 119861 or 119870 sube 119861 Hence 119861 is astrongly irreducible ordered bi-Γ-ideal

(ii) Suppose119861 is a strongly irreducible ordered bi-Γ-ideal119869 and 119870 are ordered bi-Γ-ideal of an ordered Γ-semiring 119878such that 119869 cap 119870 = 119861 Then certainly 119869 cap 119870 sube 119861 997904rArr 119869 sube 119861 or119870 sube 119861 Hence 119869 = 119861 or 119870 = 119861

Therefore 119861 is an irreducible ordered bi-Γ-ideal of 119878

Corollary 56 Let 119878 be an ordered Γ-semiring If 119861 is a primeordered bi-Γ-ideal of 119878 then 119861 is an irreducible ordered bi-Γ-ideal of 119878

Theorem 57 Let 119891 119870 997888rarr 119871 be a homomorphism of orderedΓ-semirings If 119869 is an ordered bi-Γ-ideal of 119871 then 119891minus1(119869) is anordered bi-Γ-ideal of an ordered Γ-semiring 119870

Proof Suppose 119869 is an ordered bi-Γ-ideal of 119871 119891 119870 997888rarr119871 be a homomorphism of ordered Γ-semirings and 119909 119910 isin119891minus1(119869) 997904rArr 119891(119909) 119891(119910) isin 119869 997904rArr 119891(119909) + 119891(119910) = 119891(119909 + 119910) isin119869 997904rArr 119909 + 119910 isin 119891minus1(119869) If 119909 119910 isin 119891minus1(119869) 997904rArr 119891(119909) 119891(119910) isin119869 997904rArr 119891(119909)120574119891(119910) isin 119869 997904rArr 119891(119909120574119910) isin 119869 997904rArr 119909120574119910 isin 119891minus1(119869)Now for 119909 119910 isin 119891minus1(119869) 120572 120573 isin Γ 119904 isin 119870 997904rArr 119891(119909) 119891(119910) isin 119869and 119891(119904) isin 119871 Since 119869 is an ordered bi-Γ-ideal so we have119891(119909)120572119891(119904)120573119891(119910) isin 119869 997904rArr 119891(119909120572119904120573119910) isin 119869 997904rArr 119909120572119904120573119910 isin119891minus1(119869) 997904rArr 119891minus1(119869) that is a bi-Γ-ideal of 119878

Let 119909 isin 119870 119910 isin 119891minus1(119869) such that 119909 le 119910 in 119891minus1(119869) 997904rArr119909 + 1199101015840 = 119910 for any 1199101015840 isin 119891minus1(119869) 997904rArr 119891(119909 + 1199101015840) = 119891(119910) 997904rArr119891(119909) + 119891(1199101015840) = 119891(119910) isin 119869 997904rArr 119891(119909) le 119891(119910) in 119869 997904rArr 119891(119909) isin119869 997904rArr 119909 isin 119891minus1(119869) Hence 119891minus1(119869) is an ordered bi-Γ-ideal of anordered Γ-semiring 119870

Theorem58 Let 119878 be an ordered Γ-semiring If119861 is an ordered119898minus119896 bi-Γ-ideal of 119878 then 119861 is a maximal ordered bi-Γ-ideal of119878

Proof Let 119861 be an ordered 119898 minus 119896 bi-Γ-ideal of an ordered Γ-semiring 119878 Suppose 119869 is an ordered bi-Γ-ideal of 119878 such that119861 sube 119869 119909 isin 119869 119910 isin 119861 and 120574 isin Γ We have 119910120574119909 isin 119861 997904rArr 119909 isin 119861since 119861 is an ordered 119898 minus 119896 bi-Γ-ideal of 119878 Therefore 119861 = 119869Hence ordered 119898 minus 119896 bi-Γ-ideal 119861 of 119878 is maximal orderedbi-Γ-ideal of 119878

Theorem 59 Let 119878 be an ordered Γ-semiring in which (119878 +) iscancellative semigroup and119867 = 119909 isin 119878 | 119909 + 119909 = 119909 If119867 = 0then 119867 is an ordered 119896 and 119898 minus 119896 bi-Γ-ideal of an ordered Γ-semiring 119878

Proof Let 119909 isin 119867 119910 isin 119878 and 120573 isin Γ Then 119909 = 119909 + 119909 997904rArr119909120573119910 = (119909+119909)120573119910 = 119909120573119910+119909120573119910Therefore 119909120573119910 isin 119867 Similarly119910120573119909 isin 119867 Suppose 119909 119910 isin 119867 Then 119909 + 119909 = 119909 119910 + 119910 = 119910 997904rArr(119909 + 119910) + (119909 + 119910) = (119909 + 119909) + (119910 + 119910) = 119909 + 119910 997904rArr 119909 + 119910 isin 119867Suppose 119909 le 119910 119910 isin 119867 Then 119909 + 119910 = 119910 997904rArr 119909 + 119909 + 119910 =

119909+119910 997904rArr 119909+119909 = 119909 Further 119909 119910 isin 119867 120572 120573 isin Γ and 119904 isin 119878 andthen 119909120572119904120573119910 = (119909 + 119909)120572119904120573119910 = 119909120572119904120573119910 + 119886120572119904120573119910 997904rArr 119909120572119904120573119910 isin119867 997904rArr 119867Γ119878Γ119867 sube 119867

Therefore 119867 is an ordered bi-Γ-ideal of an ordered Γ-semiring Suppose 119910 119909 + 119910 isin 119867Then 119909 + 119909 = 119909 (119909 + 119910) +(119909+119910) = 119909+119910 997904rArr (119909+119909)+ (119910+119910) = 119909+119910 997904rArr 119909+(119910+119910) =119909 + 119910 997904rArr 119910 + 119910 = 119910 997904rArr 119910 isin 119867Hence119867 is an ordered 119896 bi-Γ-ideal of an ordered Γ-semiring 119878 Suppose 119909120574119910 isin 119867 119909 isin 119867and 120574 isin Γ Then 119909120574119910 + 119909120574119910 = 119909120574119910 997904rArr 119909120574(119910 + 119910) = 119909120574119910 997904rArr119910+119910 = 119910 997904rArr 119910 isin 119867 Hence119867 is an ordered119898minus119896 bi-Γ-idealof 119878

Theorem 60 13e intersection of family of prime (orsemiprime) ordered bi-Γ-ideal of 119878 is a semiprime orderedbi-Γ-ideal

Proof Let 119875119894 | 119894 isin Λ be the family of prime ordered bi-Γ-ideal of S and 119875 = cap119896119894=1119875119894 For 119909 119910 isin 119875 997904rArr 119909 119910 isin 119875119894 forall119894 997904rArr119909120572119904120573119910 isin 119875119894 forall120572 120573 isin Γ 119904 isin 119878 997904rArr 119909120572119904120573119910 isin 119875 Now for any119910 isin 119875 such that 119909 le 119910 997904rArr 119909 isin 119875119894 forall119894 997904rArr 119909 isin 119875 Hence P isan ordered bi-Γ-ideal of 119878 For any ordered bi-Γ-ideal 119861 of 119878119861Γ119861 sube 119875 implies 119861Γ119861 sube 119875119894 for all 119894 isin Λ As 119875119894 are semiprimeordered bi-Γ-ideals 119861 sube 119875119894 for all 119894 isin Λ Hence 119861 sube 119875

Theorem61 Every strongly irreducible semiprime ordered bi-Γ-ideal of 119878 is a strongly prime ordered bi-Γ-ideal

Proof Let 119861 be a strongly irreducible and semiprime orderedbi-Γ-ideal of 119878 For any ordered bi-Γ-ideal 1198611 and 1198612 of 119878 let(1198611Γ1198612) cap (1198612Γ1198611) sube 119861 1198611 cap 1198612 is a bi-Γ-ideal of 119878 Since(1198611 cap 1198612)

2 = (1198611 cap 1198612)Γ(1198611 cap 1198612) sube 1198611Γ1198612 Similarly weget (1198611 cap 1198612)

2 = (1198611 cap 1198612)Γ(1198611 cap 1198612) sube 1198612Γ1198611Therefore(1198611cap1198612)

2 sube 1198611Γ1198612 cap1198612Γ1198611 sube 119861As 119861 is a semiprime orderedbi-Γ-ideal 1198611cap1198612 sube 119861 But 119861 is a strongly irreducible orderedbi-Γ-ideal Therefore 1198611 sube 119861 or 1198612 sube 119861 Thus 119861 is a stronglyprime ordered bi-Γ-ideal of 119878

Theorem62 If119861 is an ordered 119896 bi-Γ-ideal of 119878 and 119887 isin 119878 suchthat 119887 notin 119861 then there exists an irreducible ordered 119896 bi-Γ-ideal1198611 of 119878 such that 119861 sube 1198611 and 119887 notin 1198611

Proof Let 119865 be the family of ordered 119896 bi-Γ-ideals of 119878 whichcontain 119861 but do not contain element 119887 Then 119865 is nonemptyas 119861 isin 119865 This family of ordered 119896 bi-Γ-ideals of S forms apartially ordered set under the set inclusion Hence by Zornrsquoslemma there exists a maximal ordered 119896 bi-Γ-ideal say 1198611 in119865 Therefore 119861 sube 1198611 and 119887 notin 1198611 Now to show that 1198611 is anirreducible ordered 119896 bi-Γ-ideal of 119878 Let119870 and 119871 be any twoordered 119896 bi-Γ-ideals of 119878 such that119870cap119871 = 1198611 Suppose that119870and 119871 both contain 1198611 properly But 1198611 is a maximal ordered119896 bi-Γ-ideal in 119865 Hence we get 119887 isin 119870 and 119887 isin 119871 Therefore119887 isin 119870cap119871 = 1198611 which is absurdThus either119870 = 1198611 or 119871 = 1198611Therefore 1198611 is an irreducible ordered 119896 bi-Γ-ideal of 119878

Theorem 63 Following statements are equivalent in 119878

(1) 13e set of ordered 119896 bi-Γ-ideals of 119878 is totally ordered setunder inclusion of sets

(2) Each ordered 119896 bi-Γ-ideal of 119878 is strongly irreducible


(3) Each ordered 119896 bi-Γ-ideal of 119878 is irreducible

Proof (1) 997904rArr (2) Suppose that the set of ordered 119896 bi-Γ-ideals of 119878 is a totally ordered set under inclusion of setsLet 119861 be any ordered 119896 bi-Γ-ideal of 119878 Then 119861 is a stronglyirreducible ordered 119896 bi-Γ-ideal of 119878 for that let 1198611 and 1198612 beany two ordered 119896 bi-Γ-ideal of 119878 such that 1198611cap1198612 sube 119861 But bythe hypothesis we have either 1198611 sube 1198612 or 1198612 sube 1198611 Therefore1198611 cap 1198612 = 1198611 or 1198611 cap 1198612 = 1198612 Hence 1198611 sube 119861 or 1198612 sube 119861 Thus119861 is a strongly irreducible ordered 119896 bi-Γ-ideal of 119878(2) 997904rArr (3) Suppose that each ordered 119896 bi-Γ-ideal of 119878

is strongly irreducible Let 119861 be any ordered 119896 bi-Γ-ideal of 119878such that 119861 = 1198611 cap 1198612 for any ordered 119896 bi-Γ-ideal 1198611 and 1198612of 119878 But by hypothesis 1198611 sube 119861 or 1198612 sube 119861 As 119861 sube 1198611 and119861 sube 1198612 we get 1198611 = 119861 or 1198612 = 119861 Hence 119861 is an irreducibleordered 119896 bi-Γ-ideal of 119878(3) 997904rArr (1) Suppose that each ordered 119896 bi-Γ-ideal of 119878 is

an irreducible ordered 119896 bi-Γ-ideal Let 1198611 and 1198612 be any twoordered 119896 bi-Γ-ideal of 119878 Then 1198611 cap1198612 is also ordered 119896 bi-Γ-ideal of 119878 Hence 1198611 cap 1198612 = 1198611 cap 1198612 will imply 1198611 cap 1198612 = 1198611or 1198611 cap 1198612 = 1198612 by assumption Therefore either 1198611 sube 1198612 or1198612 sube 1198611 This shows that the set of ordered 119896 bi-Γ-ideal of 119878 istotally ordered set under inclusion of sets

Theorem 64 A prime ordered 119896 bi-Γ-ideal 119861 of 119878 is a primeone sided ordered 119896 ideal of 119878

Proof Let 119861 be a prime 119896 bi-Γ-ideal of 119878 Suppose 119861 is nota one sided 119896 ideal of 119878 Therefore 119861Γ119878 sube 119861 and 119878Γ119861 sube 119861As 119861 is a prime 119896 bi-Γ-ideal (119861Γ119878)Γ(119878Γ119861) sube 119861 (119861Γ119878)Γ(119878Γ119861)=119861Γ(119878Γ119878)Γ119861 sube 119861Γ119878Γ119861 sube 119861 which is a contradictionTherefore 119861Γ119878 sube 119861 or 119878Γ119861 sube 119861 Thus 119861 is a prime one sided119896 ideal of 119878

Theorem 65 An ordered 119896 bi-Γ-ideal 119861 of 119878 is prime if andonly if for a right ordered 119896 ideal 119877 and a le ordered 119896 ideal 119871of 119878 119877Γ119871 sube 119861 implies 119877 sube 119861 or 119871 sube 119861

Proof Suppose that an ordered 119896 bi-Γ-ideal of 119878 is a primeordered 119896 bi-Γ-ideal of 119878 Let 119877 be a right ordered 119896 ideal and119871 be a left ordered 119896 ideal of 119878 such that 119877Γ119871 sube 119861 119877 and 119871 areordered 119896 bi-Γ-ideal of 119878 Hence 119877 sube 119861 or 119871 sube 119861 Converselywe have to show that an ordered 119896 bi-Γ-ideal 119861 of 119878 is a primeordered 119896 bi-Γ-ideal of 119878 Let 119860 and 119862 be any two ordered119896 bi-Γ-ideals of 119878 such that 119860Γ119862 sube 119861 For any 119886 isin 119860 and119888 isin 119862 (119886)119903 sube 119860 and (119888)119897 sube 119862 where (119886)119903 and (119888)119897 denote theright ordered 119896 ideal and left ordered 119896 ideal generated by 119886and 119888 respectively Thus (119886)119903Γ(119888)119897 sube 119860Γ119862 sube 119861 Here by theassumption we get that (119886)119903 sube 119861 or (119888)119897 sube 119861 which gives that119886 isin 119861 or 119888 isin 119861 Thus 119860 sube 119861 or 119862 sube 119861 Hence 119861 is a primeordered 119896 bi-Γ-ideal of 119878

5 Conclusion

In this work we have given many types of ordered (119896)(119898 minus 119896) minus Γ-ideals mainly ordered 119896-(quasi bi)-Γ-idealordered 119898 minus 119896 (quasi bi)-Γ-ideal ordered (semi)prime(119896)-(semi)prime (quasi bi)-Γ-ideals ordered maximal(119896)-(quasi bi)-Γ-ideals ordered (strongly) irreducible (quasi bi)-Γ-ideals We have studied properties of these different ideals

and some of their relations in ordered Γ-semirings Let 119891 119870 997888rarr 119871 be a homomorphism of ordered Γ-semiringsWe have defined homomorphism and prove that if 119869 is anordered quasi-Γ-ideal of 119871 then 119891minus1(119869) is an ordered quasi-Γ-ideal of an ordered Γ-semiring 119870 The same is true incase of ordered bi-Γ-ideals For further research work theextended identical study can be taken under consider forother algebraic structures

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

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[2] M M Rao ldquoΓ-semirings Irdquo Southeast Asian Bulletin of Mathe-matics vol 19 no 1 pp 49ndash54 1995

[3] O Steinfeld ldquoOn ideal-quotients and prime idealsrdquo Acta Math-ematica Hungarica vol 4 pp 289ndash298 1953

[4] O Steinfeld ldquoUher die quasi-ideals von halbgruppenrdquo Publica-tiones Mathematicae vol 4 pp 262ndash275 1956

[5] K Iseki ldquoQuasi-ideals in semirings without zerordquo Proceedingsof the Japan Academy Series A Mathematical Sciences vol 34pp 79ndash81 1958

[6] C DongesOnQuasi-Ideals of Semirings vol 17 no 1 GermanyDW-3392 Clausthal-Zellerfeld Germany 1994

[7] M Shabir A Ali and S Batool ldquoA note on quasi-ideals insemiringsrdquo Southeast Asian Bulletin of Mathematics vol 27 no5 pp 923ndash928 2004

[8] R A Good and D R Hughes ldquoAssociated groups for asemigrouprdquo Bulletin of the American Mathematical Society vol58 pp 624-625 1952

[9] F A Szasz ldquoGeneralized biideals of rings Irdquo MathematischeNachrichten vol 47 pp 355ndash360 1970

[10] F A Szasz ldquoGeneralized biideals of rings IIrdquo MathematischeNachrichten vol 47 pp 361ndash364 1970

[11] S Lajos ldquoNotes on generalized bi-ideals in semigroupsrdquo Soo-chow Journal of Mathematics vol 10 pp 55ndash59 1984

[12] T K Dutta and S K Sardar ldquoSemiprime ideals and irreducibleideals of Γ-semiringsrdquoNovi Sad Journal of Mathematics vol 30no 1 pp 97ndash108 2000

[13] S K Sardar and U Dasgupta ldquoOn primitive Γ-semiringsrdquo NoviSad Journal of Mathematics vol 34 no 1 pp 1ndash12 2004

[14] J P Kaushik M A Ansari and M Khan ldquoOn bi-Γ-ideal in Γ-semiringsrdquo International Journal of Contemporary Mathemati-cal Sciences vol 3 no 25-28 pp 1255ndash1260 2008

[15] S Pianskool S Sangwirotjanapat and S Tipyota ldquoValuation Γ-semirings and valuation Γ-idealsrdquo13ai Journal of MathematicsSpecial Issue (Annual Meeting in Mathematics) vol 6 no 3 pp93ndash102 2008


[16] R Chinram ldquoA note on quasi-ideals in Γ-semiringsrdquo Interna-tionalMathematical Forum Journal for13eory andApplicationsvol 3 no 25-28 pp 1253ndash1259 2008

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

[18] R D Jagatap ldquoPrime k-Bi-ideals in Γ-semiringsrdquo PalestineJournal of Mathematics vol 3 (Spec 1) pp 489ndash494 2014

[19] M S Lavanya D M Rao and V S J Rajendra ldquoPrime bi-ternary G-ideals in ternary G-semiringsrdquo British Journal ofResearch vol 2 no 6 pp 156ndash166 2015

[20] R D Jagatap and Y S Pawar ldquoBi-ideals in Γ-semiringsrdquo Bulletinof International Mathematical Virtual Institute vol 6 no 2 pp169ndash179 2016

[21] M M Rao ldquoIdeals in ordered Γ-semiringsrdquo Discussiones Math-ematicae General Algebra and Applications vol 38 no 1 pp47ndash68 2018

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(i) 119909120572(119910 + 119911) = 119909120572119910 + 119909120572119911

(ii) (119909 + 119910)120572119911 = 119909120572119911 + 119910120572119911

(iii) 119909(120572 + 120573)119910 = 119909120572119910 + 119909120573119910

(iv) 119909120572(119910120573119911) = (119909120572119910)120573119911

Example 2 (see [21]) Let 119878 be a semiring and119872119901119902(119878) denotethe additive abelian semigroup of all 119901 times 119902 matrices withidentity element whose entries are from 119878 Then 119872119901119902(119878) isa Γ-semiring with Γ = 119872119901119902(119878) ternary operation is definedby 119909120572119911 = 119909(120572119905)119911 as the usual matrix multiplication where120572119905 denotes the transpose of the matrix 120572 for all 119909 119911 and120572 isin 119872119901119902(119878)

Definition 3 A Γ-semiring 119878 is called an ordered Γ-semiringif it admits a compatible relation le ie le is a partial orderingon 119878 satisfying the following conditions

If 119886 le 119887 and 119888 le 119889 then

(i) 119886 + 119888 le 119887 + 119889

(ii) 119886120574119888 le 119887120574119889

(iii) 119888120574119886 le 119889120574119887 forall119886 119887 119888 119889 isin 119878 and 120574 isin Γ

Definition 4 An ordered Γ-semiring 119878 is said to be commu-tative ordered Γ-semiring if 119909120574119910 = 119910120574119909 for all 119909 119910 isin 119878 and120574 isin Γ

Definition 5 (see [21]) A nonempty subset 119860 of an orderedΓ-semiring 119878 is called a sub-Γ-semiring of 119878 if (119860 +) is asubsemigroup of (119878 +) and 119886120574119887 isin 119860 for all 119886 119887 isin 119860 and 120574 isin Γ

Definition 6 (see [21]) A nonempty subset 119860 of an orderedΓ-semiring 119878 is called a left (right) ideal if 119860 is closed underaddition 119878Γ119860 sube 119860(119860Γ119878 sube 119860) and if for any isin 119878 119887 isin 119860 119886 le 119887then 119886 isin 119860

Definition 7 If 119860 is both a left and a right ideal of orderedΓ-semirings 119878 then 119860 is called an ideal of 119878

Definition 8 A right ideal 119860 of an ordered Γ-semiring 119878 iscalled a right 119896 ideal if 119886 isin 119860 and 119909 isin 119878 such that 119886 + 119909 isin 119860and then 119909 isin 119860

A left 119896 ideal of 119878 is defined in a similar way

Definition 9 (see [21]) A nonempty subset119870 of an ordered Γ-semiring 119878 is called a 119896 ideal if119870 is an ideal and 119909 isin 119878 119909+119910 isin119870 119910 isin 119870 then 119909 isin 119870

Definition 10 An ordered Γ-semiring 119878 is called totallyordered if any two elements of 119878 are comparable

Definition 11 Let119870 and119871 be ordered Γ-semirings Amapping119891 119870 997888rarr 119871 is called a homomorphism if

(i) 119891(119886 + 119887) = 119891(119886) + 119891(119887)

(ii) 119891(119886120572119887) = 119891(119886)120572119891(119887) for all 119886 119887 isin 119870 120572 isin Γ

3 Ordered Quasi-Γ-Ideals inOrdered-Γ-Semirings

In this section we define ordered quasi-Γ-ideal ordered 119896quasi-Γ-ideal ordered 119898 minus 119896 quasi-Γ-ideal ordered primequasi-Γ-ideal ordered semiprime quasi-Γ-ideal orderedmaximal quasi-Γ-ideal ordered irreducible and orderedstrongly irreducible quasi-Γ-ideal of an ordered Γ-semiring119878We define the relation ldquolerdquo in 119887 as 119886 le 119887 if 119886 + 119909 = 119887 for any119886 119887 119909 isin 119878 For a nonempty subset 119883 of 119878 and 119886 119887 isin 119878 we saythat 119886 le 119887 in 119883 if 119886 + 119909 = 119887 for any 119909 isin 119883

Definition 12 An ordered sub-Γ-semiring 119876 of an ordered Γ-semiring 119878 is called an ordered quasi-Γ-ideal of 119878 if 119878Γ119876 cap119876Γ119878 sube 119876 and if for any isin 119878 119902 isin 119876 119886 le 119902 then 119886 isin 119876 in119876

Definition 13 An ordered quasi-Γ-ideals 119876 is called ordered119896 quasi-Γ-ideal of 119878 if 119876 is an ordered sub-Γ-semiring of119878 and if 119902 isin 119876 and 119909 isin 119878 such that 119909 + 119902 isin 119876 then119909 isin 119876

Clearly every ordered 119896 quasi-Γ-ideal is an ordered quasi-Γ-ideal but converse is not true in general as shown in thefollowing example

Example 14 Let 119878 be the set of nonnegative integers andΓ = 119873 be additive abelian semigroups Tennary operation isdefined as (119909 120574 119910) 997888rarr 119909120574119910 usual multiplication of integersThen 119878 is an ordered Γ-semirings A subset 119876 = 5119878 5 of119878 is an ordered quasi-Γ-ideal of 119878 but it is not an ordered 119896quasi-Γ-ideal of 119878

Definition 15 An ordered quasi-Γ-ideal 119876 of an ordered Γ-semiring 119878 is called ordered 119898 minus 119896 quasi-Γ-ideal of 119878 if 119902 isin119876 119909 isin 119878 119902120574119909 isin 119876 for 120574 isin Γ then 119909 isin 119876

Theorem 16 Every ordered119898minus 119896 quasi-Γ-ideal of an orderedΓ-semiring 119878 is an ordered 119896 quasi-Γ-ideal of 119878

Proof Let119876 be an ordered119898minus 119896 quasi-Γ-ideal of an orderedΓ-semiring 119878 Consider 119909 + 119902 isin 119876 119902 isin 119876 119909 isin 119878 and 120574 isin Γthen (119909 + 119902)120574119909 isin 119876Then 119909 isin 119876 since 119876 is an ordered119898 minus 119896quasi-Γ-ideal Hence119876 is an ordered 119896 quasi-Γ-ideal of 119878


Example 17 Let 119878 be the set of all natural numbers Then 119878with usual ordering is an ordered semiring If Γ = 119878 then 119878is an ordered Γ-semiring If 119876 = 2 4 6 then 119876 forms anordered 119896 quasi-Γ-ideal but not ordered119898minus 119896 quasi-Γ-idealsof ordered Γ-semiring 119878

Definition 18 An ordered Γ-semiring 119878 is called band if everyelement of 119878 is an idempotent

Theorem19 Let119876 be an ordered sub-Γ-semiring of an orderedΓ-semiring 119878 in which semigroup (119878 +) is a band 13en119876 is anordered quasi-Γ-ideal of 119878 if and only if119876 is an ordered 119896 quasi-Γ-ideals of 119878


































119896119894=1 119875119894) sube


119878Γ119875⋂119875Γ119878 sube ((⋂119896119894=1(119878Γ119875119894)) cap (⋂119896119894=1(119875119894)Γ119878) = ⋂

119896119894=1(119878Γ119875119894 cap
























































































5 Conclusion



Data Availability




References






























Journal of












Journal of







Volume 2018






Volume 2018









































119896119894=1 119875119894) sube


119878Γ119875⋂119875Γ119878 sube ((⋂119896119894=1(119878Γ119875119894)) cap (⋂119896119894=1(119875119894)Γ119878) = ⋂

119896119894=1(119878Γ119875119894 cap
























































































5 Conclusion



Data Availability




References






























Journal of












Journal of







Volume 2018






Volume 2018











































































5 Conclusion



Data Availability




References






























Journal of












Journal of







Volume 2018






Volume 2018






















Journal of












Journal of







Volume 2018






Volume 2018








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