# Ordered Quasi(BI)-Γ-Ideals in Ordered Γ-Semirings

Authors: Ali N. A. Koam; Azeem Haider; Moin A. Ansari
Journal: Journal of Mathematics
 (2019)

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## Abstract

In this paper, we have defined ordered quasi-Γ-ideals and ordered bi-Γ-ideals in ordered Γ-semirings by defining the relation “≤” in ordered Γ semiring S as a≤b if a+x=b for any a,b,x∈S. By using this relation we have shown that ordered quasi-Γ-ideals and ordered bi-Γ-ideals in ordered Γ-semirings are generalization of quasi-ideals and bi-ideals in ordered semirings. Properties of many types of ordered Γ-ideals including (semi)prime, (strongly) irreducible, and maximal ordered quasi-Γ-ideals and ordered bi-Γ-ideals in ordered Γ-semirings S have been studied.


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## Body

## 1. Introduction

The notion of semiring was introduced by Vandiver [1] in 1934 whereas Γ-semiring was introduced and studied by Rao [2] in 1995 as a generalization of the notion of Γ-rings as well as of semirings. It is well known that semiring is a generalization of a ring and Γ-semiring is a generalization of semiring but interestingly ideals of semirings do not coincide with ideals of rings. Applications of semirings are not limited to theoretical computer science, graph theory, optimization theory, automata, coding theory, and formal languages but in other branches of sciences and technologies as well.Quasi-ideal was first introduced by O. Steinfeld for semigroups [3] and then for rings by the same author O. Steinfeld [4] where it is seen that quasi-ideal is a generalization of left and right ideals. Kiyoshi Iseki [5] introduced quasi-ideals for semirings without zero and showed results based on them. Donges studied quasi-ideals in semirings in [6] whereas Shabir et al. [7] characterized semirings by using quasi-ideals. Minimal quasi-ideals for Γ-semiring was studied by Iseki [5].The concept of bi-ideal for semigroups was given by Good and Hughes [8] in 1952. Generalized bi-ideal was then introduced for rings in 1970 by Szasz [9, 10] and then for semigroups by Lajos [11]. Many types of ideals on the algebraic structures were characterized by several authors such as the following. In 2000, Dutta and Sardar studied the characterization of semiprime ideals and irreducible ideals 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 2008, Pianskool 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 inΓ-semirings [16], whereas, in 2009, quasi-ideals and minimal quasi-ideals in Γ-semiring were studied by Jagatap and Pawar [17]. In 2014, Jagatap [18] studied prime k-Bi-ideals in Γ-semirings. In 2015, Lavanya et al. [19] studied prime biternary Γ-ideals in ternary Γ-semirings. In 2016, Jagatap [20] defined the concept of bi-ideals in a Γ-semiring and studied properties of bi-ideals in a regular Γ-semiring. In 2018, Rao introduced ideals in ordered Γ-semirings [21]. It is natural to define and study properties of ordered quasi-Γ-ideals and ordered bi-Γ-ideals in ordered Γ-semirings. Therefore, we have extended the concept of ordered ideals in ordered Γ-semirings to that of ordered quasi-Γ-ideals and ordered bi-Γ-ideals in ordered Γ-semiring S by considering different order than the order defined in [21].


## 2. Preliminaries and Basic Definitions

In this section we will define important terminologies based on those concepts which are being used in this paper.Definition 1.
Let(S,+) and (Γ,+) be commutative semigroups. Then we call S a Γ-semiring, if there exists a mapping S×Γ×S→S is written (x,α,y) as xαy such that it satisfies the following axioms for all x,y,z∈S and α,β∈Γ:(i)
xα(y+z)=xαy+xαz(ii)
(x+y)αz=xαz+yαz(iii)
x(α+β)y=xαy+xβy(iv)
xα(yβz)=(xαy)βz.Example 2 (see [21]).
LetS be a semiring and Mp,q(S) denote the additive abelian semigroup of all p×q matrices with identity element whose entries are from S. Then Mp,q(S) is a Γ-semiring with Γ=Mp,q(S) ternary operation is defined by xαz=x(αt)z as the usual matrix multiplication, where αt denotes the transpose of the matrix α, for all x,z and α∈Mp,q(S).Definition 3.
AΓ-semiring S is called an ordered Γ-semiring if it admits a compatible relation ≤; i.e., ≤ is a partial ordering on S satisfying the following conditions.
Ifa≤b and c≤d then(i)
a+c≤b+d(ii)
aγc≤bγd(iii)
cγa≤dγb, ∀a,b,c,d∈S and γ∈Γ.Definition 4.
An orderedΓ-semiring S is said to be commutative ordered Γ-semiring if xγy=yγx, for all x,y∈S and γ∈Γ.Definition 5 (see [21]).
A nonempty subsetA of an ordered Γ-semiring S is called a sub-Γ-semiring of S if (A,+) is a subsemigroup of (S,+) and aγb∈A for all a,b∈A and γ∈Γ.Definition 6 (see [21]).
A nonempty subsetA of an ordered Γ-semiring S is called a left (right) ideal if A is closed under addition, SΓA⊆A(AΓS⊆A) and if, for any ∈S, b∈A, a≤b then a∈A.Definition 7.
IfA is both a left and a right ideal of ordered Γ-semirings S, then A is called an ideal of S.Definition 8.
A right idealA of an ordered Γ-semiring S is called a right k ideal if a∈A and x∈S such that a+x∈A, and then x∈A.A leftk ideal of S is defined in a similar way.Definition 9 (see [21]).
A nonempty subsetK of an ordered Γ-semiring S is called a k ideal if K is an ideal and x∈S,x+y∈K, y∈K; then x∈K.Definition 10.
An orderedΓ-semiring S is called totally ordered if any two elements of S are comparable.Definition 11.
LetK and L be ordered Γ-semirings. A mapping f:K→L is called a homomorphism if(i)
f(a+b)=f(a)+f(b),(ii)
f(aαb)=f(a)αf(b), for all a,b∈K,α∈Γ.


## 3. Ordered Quasi-Γ-Ideals in Ordered-Γ-Semirings

In this section we define ordered quasi-Γ-ideal, ordered k quasi-Γ-ideal, ordered m-k quasi-Γ-ideal, ordered prime quasi-Γ-ideal, ordered semiprime quasi-Γ-ideal, ordered maximal quasi-Γ-ideal, ordered irreducible, and ordered strongly irreducible quasi-Γ-ideal of an ordered Γ-semiring S. We define the relation “≤” in b as a≤b if a+x=b for any a,b,x∈S. For a nonempty subset X of S and a,b∈S, we say that a≤b in X if a+x=b for any x∈X.Definition 12.
An ordered sub-Γ-semiring Q of an ordered Γ-semiring S is called an ordered quasi-Γ-ideal of S if SΓQ∩QΓS⊆Q and if, for any ∈S, q∈Q, a≤q then a∈Q in Q.Definition 13.
An ordered quasi-Γ-ideals Q is called ordered k quasi-Γ-ideal of S if Q is an ordered sub-Γ-semiring of S, and if q∈Q and x∈S such that x+q∈Q, then x∈Q.
Clearly every orderedk quasi-Γ-ideal is an ordered quasi-Γ-ideal but converse is not true in general as shown in the following example.Example 14.
LetS be the set of nonnegative integers and Γ=N be additive abelian semigroups. Tennary operation is defined as (x,γ,y)→xγy, usual multiplication of integers. Then S is an ordered Γ-semirings. A subset Q=5S∖5 of S is an ordered quasi-Γ-ideal of S but it is not an ordered k quasi-Γ-ideal of S.Definition 15.
An ordered quasi-Γ-ideal Q of an ordered Γ-semiring S is called ordered m-k quasi-Γ-ideal of S if q∈Q,x∈S, qγx∈Q for γ∈Γ; then x∈Q.Theorem 16.
Every orderedm-k quasi-Γ-ideal of an ordered Γ-semiring S is an ordered k quasi-Γ-ideal of S.Proof.
LetQ be an ordered m-k quasi-Γ-ideal of an ordered Γ-semiring S. Consider x+q∈Q, q∈Q, x∈S, and γ∈Γ; then (x+q)γx∈Q. Then x∈Q, since Q is an ordered m-k quasi-Γ-ideal. Hence Q is an ordered k quasi-Γ-ideal of S.The following example shows that the converse of Theorem16 is not true in general.Example 17.
LetS be the set of all natural numbers. Then S with usual ordering is an ordered semiring. If Γ=S, then S is an ordered Γ-semiring. If Q=2,4,6,… then Q forms an ordered k quasi-Γ-ideal but not ordered m-k quasi-Γ-ideals of ordered Γ-semiring S.Definition 18.
An orderedΓ-semiring S is called band if every element of S is an idempotent.Theorem 19.
LetQ be an ordered sub-Γ-semiring of an ordered Γ-semiring S in which semigroup (S,+) is a band. Then Q is an ordered quasi-Γ-ideal of S if and only if Q is an ordered k quasi-Γ-ideals of S.Proof.
LetQ be an ordered quasi-Γ-ideal of an ordered Γ-semiring S and x+q∈Q, q∈Q, and then x+(x+q)=(x+x)+q=x+q⇒x≤x+q⇒x∈Q as Q is an ordered quasi-Γ-ideal of S. Hence, Q is a an ordered k quasi-Γ-ideal of S.
Conversely suppose thatQ is an ordered k quasi-Γ-ideal of an ordered Γ-semiring S. Let x∈S,q∈Q and x≤q⇒x+q′=q⇒x+q′∈Q⇒x∈Q, since Q is an ordered k quasi-Γ-ideal of an ordered Γ-semiring S. Hence Q is an ordered quasi-Γ-ideal of S.Definition 20.
An ordered quasi-Γ-ideal Q of S is called an irreducible ordered quasi-Γ-ideal if Q1∩Q2=Q implies Q1=Q or Q2=Q, for any ordered quasi-Γ-ideal Q1 and Q2 of S.Definition 21.
An ordered quasi-Γ-ideal Q of S is called strongly irreducible ordered quasi-Γ-ideal if, for any ordered quasi-Γ-ideal Q1 and Q2 of S, Q1∩Q2⊆Q implies Q1⊆Q or Q2⊆Q. It is obvious that every strongly irreducible ordered quasi-Γ-ideal is an irreducible ordered quasi-Γ-ideal.Definition 22.
An ordered quasi-Γ-ideal Q of S is called a prime ordered quasi-Γ-ideal if Q1ΓQ2⊆Q implies Q1⊆Q or Q2⊆Q, for any ordered quasi-Γ-ideals Q1,Q2 of S.Definition 23.
An orderedk quasi-Γ-ideal Q of S is called a prime ordered k quasi-Γ-ideal if Q1ΓQ2⊆Q implies Q1⊆Q or Q2⊆Q, for any ordered k quasi-Γ-ideals Q1,Q2 of S.Definition 24.
An ordered quasi-Γ-ideal Q of S is called a strongly prime ordered quasi-Γ-ideal if (Q1ΓQ2)∩(Q2ΓQ1)⊆Q implies that Q1⊆Q or Q2⊆Q for any ordered quasi-Γ-ideals Q1, Q2 of S.Definition 25.
An ordered quasi-Γ-ideal Q of S is called a semiprime ordered quasi-Γ-ideal if for any ordered quasi-Γ-ideals Q1, of SQ12=Q1ΓQ1⊆Q implies Q1⊆Q.It is clear that every strongly prime ordered quasi-Γ-ideal in S is a prime ordered quasi-Γ-ideal and every prime ordered quasi-Γ-ideal in S is a semiprime ordered quasi-Γ-ideal.Definition 26.
An ordered quasi-Γ-ideal Q of an ordered Γ-semiring S is said to be maximal ordered quasi-Γ-ideal if Q≠S and for every ordered quasi-Γ-ideal C of S with Q⊆C⊆S, then either Q=C or Q=S.Theorem 27.
In an orderedΓ-semiring S, every maximal ordered quasi-Γ-ideal of S is irreducible ordered quasi-Γ-ideal of S.Proof.
LetQm be a maximal ordered quasi-Γ-ideal of an ordered Γ-semiring S. Suppose Qm is not irreducible and Qm=U∩V⇒Qm≠U and Qm≠V⇒Qm⊂U⊂S and Qm⊂V⊂S a contradiction. Hence Qm is irreducible ordered quasi-Γ-deal of an ordered Γ-semiring S.Theorem 28.
LetQ be an ordered quasi-Γ-ideal of an ordered Γ-semiring S.(i)
IfQ is a prime ordered quasi-Γ-ideal, then Q is a strongly irreducible ordered quasi-Γ-ideal.(ii)
IfQ is a strongly irreducible ordered quasi-Γ-ideal, then Q is an irreducible ordered quasi-Γ-ideal.Proof.
LetQ be an ordered quasi-Γ-ideal of an ordered Γ-semiring S.
(i) SupposeQ is a prime ordered quasi-Γ-ideal, J and K are ordered quasi-Γ-ideal of an ordered Γ-semiring S such that J∩K⊆B. Then JΓK⊆Q⇒J⊆Q or K⊆Q. Hence, Q is a strongly irreducible ordered quasi-Γ-ideal.
(ii) SupposeQ is a strongly irreducible ordered quasi-Γ-ideal; J and K are ordered quasi-Γ-ideal of an ordered Γ-semiring S such that J∩K=Q. Then certainly J∩K⊆Q⇒J⊆Q or K⊆Q. Hence J=Q or K=Q.
Therefore,Q is an irreducible ordered quasi-Γ-ideal of S.Corollary 29.
LetS be an ordered Γ-semiring. If Q is a prime ordered quasi-Γ-ideal of S, then Q is an irreducible ordered quasi-Γ-ideal of S.Theorem 30.
Letf:K→L be a homomorphism of ordered Γ-semirings. If J is an ordered quasi-Γ-ideal of L, then f-1(J) is an ordered quasi-Γ-ideal of an ordered Γ-semiring K.Proof.
SupposeJ is an ordered quasi-Γ-ideal of L, f:K→L is a homomorphism of ordered Γ-semirings, and x,y∈f-1(J)⇒f(x), f(y)∈J⇒f(x)+f(y)=f(x+y)∈J⇒x+y∈f-1(J). If x,y∈f-1(J)⇒f(x), f(y)∈J⇒f(x)γf(y)∈J⇒f(xγy)∈J⇒xγy∈f-1(J). We need to show here that f-1(J) is an ordered quasi-Γ-ideal of K; i.e., KΓf-1(J)∩f-1(J)ΓK⊆f-1(J). For that we suppose j∈KΓf-1(J)∩f-1(J)ΓK⇒j=sαx and j=yβs′ for s,s′∈K,x,y∈f-1(J) and α,β∈Γ⇒f(x),f(y)∈J, which is a quasi-ideal. Now f(j)=f(sαx)=f(s)αf(x) and f(j)=f(y)βf(s′)⇒f(j)∈LΓJ∩JΓL, as f(s),f(s′)∈L⇒f(j)∈J⇒j∈f-1(J).
Letx∈K,y∈f-1(J) such that x≤y⇒x+y=y⇒f(x+y)=f(y)⇒f(x)+f(y)=f(y)∈J⇒f(x)≤f(y)⇒f(x)∈J⇒x∈f-1(J). Hence f-1(J) is an ordered quasi-Γ-ideal of an ordered Γ-semiring K.Theorem 31.
LetS be an ordered Γ-semiring. If Q is an ordered m-k quasi-Γ-ideal of S, then Q is a maximal ordered quasi-Γ-ideal of S.Proof.
LetQ be an ordered m-k quasi-Γ-ideal of an ordered Γ-semiring S. Suppose J is an ordered quasi-Γ-ideal of S such that Q⊆J,x∈J,y∈Q and γ∈Γ. We have yγx∈Q⇒x∈Q, since Q is an ordered m-k quasi-Γ-ideal of S. Therefore Q=J. Hence ordered m-k quasi-Γ-ideal Q of S is maximal ordered quasi-Γ-ideal of S.Theorem 32.
LetS be an ordered Γ-semiring in which (S,+) is cancellative semigroup and H=x∈S∣x+x=x. If H≠∅, then H is an ordered m-k quasi-Γ-ideal of an ordered Γ-semiring S.Proof.
Letx∈H,y∈S and β∈Γ. Then x=x+x⇒xβy=(x+x)βy=xβy+xβy⇒xβy∈H. Similarly yβx∈H. Suppose x,y∈H. Then x+x=x,y+y=y⇒(x+y)+(x+y)=(x+x)+(y+y)=x+y⇒x+y∈H. Suppose x≤y,y∈H. Then x+y=y⇒x+x+y=x+y⇒x+x=x. Let sαh∈SΓH for s∈S,α∈Γ and h∈H. Next, sαh+sαh=sα(h+h)=sαh⇒sαh∈H⇒SΓH⊆H. In a similar way, we can show that HΓS⊆H. Therefore, SΓH∩HΓS⊆H. Hence H is a quasi-ideal.
Therefore,H is an ordered quasi-Γ-ideal of an ordered Γ-semiring. Suppose xγy∈H,x∈H, and γ∈Γ. Then xγy+xγy=xγy⇒xγ(y+y)=xγy⇒y+y=y⇒y∈H. Hence H is an ordered m-k quasi-Γ-ideal of S.Theorem 33.
The intersection of family of prime (or semiprime) ordered quasi-Γ-ideal of S is a semiprime ordered quasi-Γ-ideal.Proof.
LetPi∣i∈Λ be the family of prime ordered quasi-Γ-ideal of S and P=⋂i=1kPi. Since SΓP=SΓ(⋂i=1kPi)⊆⋂i=1k(SΓPi) and PΓS=(⋂i=1kPi)ΓS⊆⋂i=1kPiΓS⇒SΓP⋂PΓS⊆((⋂i=1kSΓPi)∩(⋂i=1kPiΓS)=⋂i=1kSΓPi∩PiΓS⊆⋂i=1kPi=P. Hence P is a quasi-Γ-ideal.
Now for anyy∈P such that x≤y⇒x∈Pi∀i⇒x∈P. Hence P is an ordered quasi-Γ-ideal of S. For any ordered quasi-Γ-ideal Q of S, QΓQ⊆P implies QΓQ⊆Pi for all i∈Λ. As Pi are semiprime ordered quasi-Γ-ideals, Q⊆Pi for all i∈Λ. Hence Q⊆P.Theorem 34.
Every strongly irreducible, semiprime ordered quasi-Γ-ideal of S is a strongly prime ordered quasi-Γ-ideal.Proof.
LetQ be a strongly irreducible and semiprime ordered quasi-Γ-ideal of S. For any ordered quasi-Γ-ideal Q1 and Q2 of S, let (Q1ΓQ2)∩(Q2ΓQ1)⊆Q. Q1∩Q2 is a quasi-Γ-ideal of S, since (Q1∩Q2)2=(Q1∩Q2)Γ(Q1∩Q2)⊆Q1ΓQ2. Similarly we get (Q1∩Q2)2=(Q1∩Q2)Γ(Q1∩Q2)⊆Q2ΓQ1. Therefore (Q1∩Q2)2⊆Q1ΓQ2∩Q2ΓQ1⊆Q. As Q is a semiprime ordered quasi-Γ-ideal, Q1∩Q2⊆Q. But Q is a strongly irreducible ordered quasi-Γ-ideal. Therefore, Q1⊆Q or Q2⊆Q. Thus Q is a strongly prime ordered quasi-Γ-ideal of S.Theorem 35.
IfQ is an ordered k quasi-Γ-ideal of S and q∈S such that q∉Q, then there exists an irreducible ordered k quasi-Γ-ideal Q1 of S such that Q⊆Q1 and q∉Q1.Proof.
LetF be the family of ordered k quasi-Γ-ideals of S which contain Q but do not contain element q. Then F is nonempty as Q∈F. This family of ordered k quasi-Γ-ideals of S forms a partially ordered set under the set inclusion. Hence by Zorn’s lemma there exists a maximal ordered k quasi-Γ-ideal say Q1 in F. Therefore Q⊆Q1 and q∉Q1. Now to show that Q1 is an irreducible ordered k quasi-Γ-ideal of S, let K and L be any two ordered k quasi-Γ-ideals of S such that K∩L=Q1. Suppose that K and L both contain Q1 properly. But Q1 is a maximal ordered k quasi-Γ-ideal in F. Hence we get q∈K and q∈L. Therefore q∈K∩L=Q1 which is absurd. Thus either K=Q1 or L=Q1. Therefore, Q1 is an irreducible ordered k quasi-Γ-ideal of S.Theorem 36.
The following statements are equivalent inS:(1)
The set of orderedk quasi-Γ-ideals of S is totally ordered set under inclusion of sets.(2)
Each orderedk quasi-Γ-ideal of S is strongly irreducible.(3)
Each orderedk quasi-Γ-ideal of S is irreducible.Proof.
( 1 ) ⇒ ( 2 ) Suppose that the set of ordered k quasi-Γ-ideals of S is a totally ordered set under inclusion of sets. Let B be any ordered k quasi-Γ-ideal of S. Then B is a strongly irreducible ordered k quasi-Γ-ideal of S for that let Q1 and Q2 be any two ordered k quasi-Γ-ideal of S such that Q1∩Q2⊆Q. But, by the hypothesis, we have either Q1⊆Q2 or Q2⊆Q1. Therefore, Q1∩Q2=Q1 or Q1∩Q2=Q2. Hence Q1⊆Q or Q2⊆Q. Thus Q is a strongly irreducible ordered k quasi-Γ-ideal of S.
( 2 ) ⇒ ( 3 ) Suppose that each ordered k quasi-Γ-ideal of S is strongly irreducible. Let Q be any ordered k quasi-Γ-ideal of S such that Q=Q1∩Q2 for any ordered k quasi-Γ-ideal Q1 and Q2 of S. But by hypothesis Q1⊆Q or Q2⊆Q. As Q⊆Q1 and Q⊆Q2, we get Q1=Q or Q2=Q. Hence Q is an irreducible ordered k quasi-Γ-ideal of S.
( 3 ) ⇒ ( 1 ) Suppose that each ordered k quasi-Γ-ideal of S is an irreducible ordered k quasi-Γ-ideal. Let Q1 and Q2 be any two ordered k quasi-Γ-ideal of S. Then Q1∩Q2 is also ordered k quasi-Γ-ideal of S. Hence Q1∩Q2=Q1∩Q2 will imply Q1∩Q2=Q1 or Q1∩Q2=Q2 by assumption. Therefore either Q1⊆Q2 or Q2⊆Q1. This shows that the set of ordered k quasi-Γ-ideal of S is totally ordered set under inclusion of sets.Theorem 37.
A prime orderedk quasi-Γ-ideal Q of S is a prime one sided ordered k ideal of S.Proof.
LetQ be a prime k quasi-Γ-ideal of S. Suppose Q is not a one sided k ideal of S. Therefore, QΓS⊈Q and SΓQ⊈Q. As Q is a prime k quasi-Γ-ideal, (QΓS)Γ(SΓQ)⊈Q. (QΓS)Γ(SΓQ)⊆(QΓS)∩(SΓQ)⊆Q which is a contradiction. Therefore, QΓS⊆Q or SΓQ⊆Q. Thus Q is a prime one sided k ideal of S.Theorem 38.
An orderedk quasi-Γ-ideal Q of S is prime if and only if, for a right ordered k ideal R and a left ordered k ideal L of S, RΓL⊆Q implies R⊆Q or L⊆Q.Proof.
Suppose that an orderedk quasi-Γ-ideal of S is a prime ordered k quasi-Γ-ideal of S. Let R be a right ordered k ideal and L be a left ordered k ideal of S such that RΓL⊆Q. R and L are ordered k quasi-Γ-ideal of S. Hence R⊆Q or L⊆Q. Conversely, we have to show that an ordered k quasi-Γ-ideal Q of S is a prime ordered k quasi-Γ-ideal of S. Let A and C be any two ordered k quasi-Γ-ideals of S such that AΓC⊆Q. For any a∈A and c∈C, (a)r⊆A and (c)l⊆C, where (a)r and (c)l denote the right ordered k ideal and left ordered k ideal generated by a and c, respectively. Thus (a)rΓ(c)l⊆AΓC⊆Q. Hence by the assumption (a)r⊆B or (c)l⊆Q which further gives that a∈Q or c∈Q. In this way we get that A⊆Q or C⊆Q. Hence Q is a prime ordered k quasi-Γ-ideal of S.


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

In this section we define ordered bi-Γ-ideal, ordered k bi-Γ-ideal, ordered m-k bi-Γ-ideal, ordered prime bi-Γ-ideal, ordered semiprime bi-Γ-ideal, ordered maximal bi-Γ-ideal, ordered irreducible, and ordered strongly irreducible bi-Γ-ideal of an ordered Γ-semiring S. Recall the relation in the previous section “≤” in b as a≤b if a+x=b for any a,b,x∈S. Further for a nonempty subset X of S and a,b∈S, a≤b in X if a+x=b for any x∈X.Definition 39.
An ordered sub-Γ-semiring B of an ordered Γ-semiring S is called an ordered bi-Γ-ideal of S if BΓSΓB⊆B and if, for any ∈S, b∈B, a≤b in B, then a∈B.Definition 40.
An ordered bi-Γ-ideals B is called ordered k bi-Γ-ideal of S if B is an ordered sub-Γ-semiring of S, and if b∈B and x∈S such that x+b∈B, then x∈B.
Clearly every orderedk bi-Γ-ideal is an ordered bi-Γ-ideal but converse is not true in general as shown in the following example.Example 41.
LetS be the set of nonnegative integers and Γ=N be additive abelian semigroups. Tennary operation is defined as (x,γ,y)→xγy, usual multiplication of integers. Then S is an ordered Γ-semirings. A subset B=7S∖7 of S is an ordered bi-Γ-ideal of S but it is not an ordered k bi-Γ-ideal of S.Definition 42.
An ordered bi-Γ-ideal B of an ordered Γ-semiring S is called ordered m-k bi-Γ-ideal of S if b∈B,x∈S, bγx∈B for γ∈Γ, and then x∈B.Theorem 43.
Every orderedm-k bi-Γ-ideal of an ordered Γ-semiring S is an ordered k bi-Γ-ideal of S.Proof.
LetB be an ordered m-k bi-Γ-ideal of an ordered Γ-semiring S. Consider x+b∈B, b∈B, x∈S and γ∈Γ; then (x+b)γx∈B. Then x∈B, since B is an ordered m-k bi-Γ-ideal. Hence B is an ordered k bi-Γ-ideal of S.The following example shows that the converse of Theorem43 is not true in general.Example 44.
LetS be the set of all natural numbers. Then S with usual ordering is an ordered semiring. If Γ=S, then S is an ordered Γ-semiring. If B=2,4,6,… then B forms an ordered k bi-Γ-ideal but not ordered m-k bi-Γ-ideals of ordered Γ-semiring S.Definition 45.
An orderedΓ-semiring S is called band if every element of S is a γ-idempotent.Theorem 46.
LetB be an ordered sub-Γ-semiring of an ordered Γ-semiring S in which semigroup (S,+) is a band. Then B is an ordered bi-Γ-ideal of S if and only if B is an ordered k bi-Γ-ideals of S.Proof.
LetB be an ordered bi-Γ-ideal of an ordered Γ-semiring S and x+b∈B, x∈B. Then x+(x+b)=(x+x)+b=x+b⇒x≤x+b⇒x∈B as B is an ordered bi-Γ-ideal of S. Hence B is a an ordered k bi-Γ-ideal of S.
Conversely suppose thatB is an ordered k bi-Γ-ideal of an ordered Γ-semiring S. Let x∈S,b∈B and x≤b in B⇒x+b′=b for any b′∈B⇒x+b′∈B⇒x∈B, since B is an ordered k bi-Γ-ideal of an ordered Γ-semiring S. Hence B is an ordered bi-Γ-ideal of S.Definition 47.
An ordered bi-Γ-ideal B of S is called an irreducible ordered bi-Γ-ideal if B1∩B2=B implies B1=B or B2=B, for any ordered bi-Γ-ideal B1 and B2 of S.Definition 48.
An ordered bi-Γ-ideal B of S is called strongly irreducible ordered bi-Γ-ideal if, for any ordered bi-Γ-ideal B1 and B2 of S, B1∩B2⊆B implies B1⊆B or B2⊆B. It is obviously that every strongly irreducible ordered bi-Γ-ideal is an irreducible ordered bi-Γ-ideal.Definition 49.
An ordered bi-Γ-ideal B of S is called a prime ordered bi-Γ-ideal if B1ΓB2⊆B implies B1⊆B or B2⊆B, for any ordered bi-Γ-ideals B1,B2 of S.Definition 50.
An orderedk bi-Γ-ideal B of S is called a prime ordered k bi-Γ-ideal if B1ΓB2⊆B implies B1⊆B or B2⊆B, for any ordered k bi-Γ-ideals B1,B2 of S.Definition 51.
An ordered bi-Γ-ideal B of S is called a strongly prime ordered bi-Γ-ideal if (B1ΓB2)∩(B2ΓB1)⊆B implies that B1⊆B or B2⊆B for any ordered bi-Γ-ideals B1, B2 of S.Definition 52.
An ordered bi-Γ-ideal B of S is called a semiprime ordered bi-Γ-ideal if for any ordered bi-Γ-ideals B1 of SB12=B1ΓB1⊆B implies B1⊆B.It is clear that every strongly prime ordered bi-Γ-ideal in S is a prime ordered bi-Γ-ideal and every prime ordered bi-Γ-ideal in S is a semiprime ordered bi-Γ-ideal.Definition 53.
An ordered bi-Γ-ideal B of an ordered Γ-semiring S is said to be maximal ordered bi-Γ-ideal if B≠S and for every ordered bi-Γ-ideal C of S with B⊆C⊆S, then either B=C or B=S.Theorem 54.
In an orderedΓ-semiring S, every maximal ordered bi-Γ-ideal of S is irreducible ordered bi-Γ-ideal of S.Proof.
LetBm be a maximal ordered bi-Γ-ideal of an ordered Γ-semiring S. Suppose Bm is not irreducible and Bm=U∩V⇒Bm≠U and Bm≠V⇒Bm⊂U⊂S and Bm⊂V⊂S a contradiction. Hence Bm is irreducible ordered bi-Γ-deal of an ordered Γ-semiring S.Theorem 55.
LetB be an ordered bi-Γ-ideal of an ordered Γ-semiring S.(i)
IfB is a prime ordered bi-Γ-ideal, then B is a strongly irreducible ordered bi-Γ-ideal.(ii)
IfB is a strongly irreducible ordered bi-Γ-ideal, then B is an irreducible ordered bi-Γ-ideal.Proof.
LetB be an ordered bi-Γ-ideal of an ordered Γ-semiring S.
(i) SupposeB is a prime ordered bi-Γ-ideal; J and K are ordered bi-Γ-ideal of an ordered Γ-semiring S such that J∩K⊆B. Then JΓK⊆B⇒J⊆B or K⊆B. Hence B is a strongly irreducible ordered bi-Γ-ideal.
(ii) SupposeB is a strongly irreducible ordered bi-Γ-ideal; J and K are ordered bi-Γ-ideal of an ordered Γ-semiring S such that J∩K=B. Then certainly J∩K⊆B⇒J⊆B or K⊆B. Hence J=B or K=B.
Therefore,B is an irreducible ordered bi-Γ-ideal of S.Corollary 56.
LetS be an ordered Γ-semiring. If B is a prime ordered bi-Γ-ideal of S, then B is an irreducible ordered bi-Γ-ideal of S.Theorem 57.
Letf:K→L be a homomorphism of ordered Γ-semirings. If J is an ordered bi-Γ-ideal of L then f-1(J) is an ordered bi-Γ-ideal of an ordered Γ-semiring K.Proof.
SupposeJ is an ordered bi-Γ-ideal of L, f:K→L be a homomorphism of ordered Γ-semirings and x,y∈f-1(J)⇒f(x),f(y)∈J⇒f(x)+f(y)=f(x+y)∈J⇒x+y∈f-1(J). If x,y∈f-1(J)⇒f(x),f(y)∈J⇒f(x)γf(y)∈J⇒f(xγy)∈J⇒xγy∈f-1(J). Now for x,y∈f-1(J),α,β∈Γ,s∈K⇒f(x),f(y)∈J and f(s)∈L. Since J is an ordered bi-Γ-ideal, so we have f(x)αf(s)βf(y)∈J⇒f(xαsβy)∈J⇒xαsβy∈f-1(J)⇒f-1(J) that is a bi-Γ-ideal of S.
Letx∈K,y∈f-1(J) such that x≤y in f-1(J)⇒x+y′=y for any y′∈f-1(J)⇒f(x+y′)=f(y)⇒f(x)+f(y′)=f(y)∈J⇒f(x)≤f(y) in J⇒f(x)∈J⇒x∈f-1(J). Hence f-1(J) is an ordered bi-Γ-ideal of an ordered Γ-semiring K.Theorem 58.
LetS be an ordered Γ-semiring. If B is an ordered m-k bi-Γ-ideal of S, then B is a maximal ordered bi-Γ-ideal of S.Proof.
LetB be an ordered m-k bi-Γ-ideal of an ordered Γ-semiring S. Suppose J is an ordered bi-Γ-ideal of S such that B⊆J,x∈J,y∈B and γ∈Γ. We have yγx∈B⇒x∈B, since B is an ordered m-k bi-Γ-ideal of S. Therefore, B=J. Hence ordered m-k bi-Γ-ideal B of S is maximal ordered bi-Γ-ideal of S.Theorem 59.
LetS be an ordered Γ-semiring in which (S,+) is cancellative semigroup and H=x∈S∣x+x=x. If H≠∅, then H is an ordered k and m-k bi-Γ-ideal of an ordered Γ-semiring S.Proof.
Letx∈H,y∈S and β∈Γ. Then x=x+x⇒xβy=(x+x)βy=xβy+xβy. Therefore, xβy∈H. Similarly yβx∈H. Suppose x,y∈H. Then x+x=x,y+y=y⇒(x+y)+(x+y)=(x+x)+(y+y)=x+y⇒x+y∈H. Suppose x≤y,y∈H. Then x+y=y⇒x+x+y=x+y⇒x+x=x. Further x,y∈H,α,β∈Γ and s∈S, and then xαsβy=(x+x)αsβy=xαsβy+aαsβy⇒xαsβy∈H⇒HΓSΓH⊆H.
Therefore,H is an ordered bi-Γ-ideal of an ordered Γ-semiring. Suppose y,x+y∈H. Then x+x=x,(x+y)+(x+y)=x+y⇒(x+x)+(y+y)=x+y⇒x+(y+y)=x+y⇒y+y=y⇒y∈H. Hence H is an ordered k bi-Γ-ideal of an ordered Γ-semiring S. Suppose xγy∈H,x∈H and γ∈Γ. Then xγy+xγy=xγy⇒xγ(y+y)=xγy⇒y+y=y⇒y∈H. Hence H is an ordered m-k bi-Γ-ideal of S.Theorem 60.
The intersection of family of prime (or semiprime) ordered bi-Γ-ideal of S is a semiprime ordered bi-Γ-ideal.Proof.
LetPi∣i∈Λ be the family of prime ordered bi-Γ-ideal of S and P=∩i=1kPi. For x,y∈P⇒x,y∈Pi∀i⇒xαsβy∈Pi∀α,β∈Γ,s∈S⇒xαsβy∈P. Now for any y∈P such that x≤y⇒x∈Pi∀i⇒x∈P. Hence P is an ordered bi-Γ-ideal of S. For any ordered bi-Γ-ideal B of S, BΓB⊆P implies BΓB⊆Pi for all i∈Λ. As Pi are semiprime ordered bi-Γ-ideals, B⊆Pi for all i∈Λ. Hence B⊆P.Theorem 61.
Every strongly irreducible, semiprime ordered bi-Γ-ideal of S is a strongly prime ordered bi-Γ-ideal.Proof.
LetB be a strongly irreducible and semiprime ordered bi-Γ-ideal of S. For any ordered bi-Γ-ideal B1 and B2 of S, let (B1ΓB2)∩(B2ΓB1)⊆B. B1∩B2 is a bi-Γ-ideal of S. Since (B1∩B2)2=(B1∩B2)Γ(B1∩B2)⊆B1ΓB2. Similarly we get (B1∩B2)2=(B1∩B2)Γ(B1∩B2)⊆B2ΓB1. Therefore, (B1∩B2)2⊆B1ΓB2∩B2ΓB1⊆B. As B is a semiprime ordered bi-Γ-ideal, B1∩B2⊆B. But B is a strongly irreducible ordered bi-Γ-ideal. Therefore B1⊆B or B2⊆B. Thus B is a strongly prime ordered bi-Γ-ideal of S.Theorem 62.
IfB is an ordered k bi-Γ-ideal of S and b∈S such that b∉B, then there exists an irreducible ordered k bi-Γ-ideal B1 of S such that B⊆B1 and b∉B1.Proof.
LetF be the family of ordered k bi-Γ-ideals of S which contain B but do not contain element b. Then F is nonempty as B∈F. This family of ordered k bi-Γ-ideals of S forms a partially ordered set under the set inclusion. Hence by Zorn’s lemma there exists a maximal ordered k bi-Γ-ideal say B1 in F. Therefore B⊆B1 and b∉B1. Now to show that B1 is an irreducible ordered k bi-Γ-ideal of S. Let K and L be any two ordered k bi-Γ-ideals of S such that K∩L=B1. Suppose that K and L both contain B1 properly. But B1 is a maximal ordered k bi-Γ-ideal in F. Hence we get b∈K and b∈L. Therefore b∈K∩L=B1 which is absurd. Thus either K=B1 or L=B1. Therefore B1 is an irreducible ordered k bi-Γ-ideal of S.Theorem 63.
Following statements are equivalent inS: (1)
The set of orderedk bi-Γ-ideals of S is totally ordered set under inclusion of sets.(2)
Each orderedk bi-Γ-ideal of S is strongly irreducible.(3)
Each orderedk bi-Γ-ideal of S is irreducible.Proof.
( 1 ) ⇒ ( 2 ) Suppose that the set of ordered k bi-Γ-ideals of S is a totally ordered set under inclusion of sets. Let B be any ordered k bi-Γ-ideal of S. Then B is a strongly irreducible ordered k bi-Γ-ideal of S for that let B1 and B2 be any two ordered k bi-Γ-ideal of S such that B1∩B2⊆B. But by the hypothesis, we have either B1⊆B2 or B2⊆B1. Therefore, B1∩B2=B1 or B1∩B2=B2. Hence B1⊆B or B2⊆B. Thus B is a strongly irreducible ordered k bi-Γ-ideal of S.
( 2 ) ⇒ ( 3 ) Suppose that each ordered k bi-Γ-ideal of S is strongly irreducible. Let B be any ordered k bi-Γ-ideal of S such that B=B1∩B2 for any ordered k bi-Γ-ideal B1 and B2 of S. But by hypothesis B1⊆B or B2⊆B. As B⊆B1 and B⊆B2, we get B1=B or B2=B. Hence B is an irreducible ordered k bi-Γ-ideal of S.
( 3 ) ⇒ ( 1 ) Suppose that each ordered k bi-Γ-ideal of S is an irreducible ordered k bi-Γ-ideal. Let B1 and B2 be any two ordered k bi-Γ-ideal of S. Then B1∩B2 is also ordered k bi-Γ-ideal of S. Hence B1∩B2=B1∩B2 will imply B1∩B2=B1 or B1∩B2=B2 by assumption. Therefore either B1⊆B2 or B2⊆B1. This shows that the set of ordered k bi-Γ-ideal of S is totally ordered set under inclusion of sets.Theorem 64.
A prime orderedk bi-Γ-ideal B of S is a prime one sided ordered k ideal of S.Proof.
LetB be a prime k bi-Γ-ideal of S. Suppose B is not a one sided k ideal of S. Therefore, BΓS⊈B and SΓB⊈B. As B is a prime k bi-Γ-ideal, (BΓS)Γ(SΓB)⊈B. (BΓS)Γ(SΓB) =BΓ(SΓS)ΓB⊆BΓSΓB⊆B, which is a contradiction. Therefore, BΓS⊆B or SΓB⊆B. Thus B is a prime one sided k ideal of S.Theorem 65.
An orderedk bi-Γ-ideal B of S is prime if and only if, for a right ordered k ideal R and a left ordered k ideal L of S, RΓL⊆B implies R⊆B or L⊆B.Proof.
Suppose that an orderedk bi-Γ-ideal of S is a prime ordered k bi-Γ-ideal of S. Let R be a right ordered k ideal and L be a left ordered k ideal of S such that RΓL⊆B. R and L are ordered k bi-Γ-ideal of S. Hence R⊆B or L⊆B. Conversely, we have to show that an ordered k bi-Γ-ideal B of S is a prime ordered k bi-Γ-ideal of S. Let A and C be any two ordered k bi-Γ-ideals of S such that AΓC⊆B. For any a∈A and c∈C, (a)r⊆A and (c)l⊆C, where (a)r and (c)l denote the right ordered k ideal and left ordered k ideal generated by a and c, respectively. Thus (a)rΓ(c)l⊆AΓC⊆B. Here by the assumption we get that (a)r⊆B or (c)l⊆B which gives that a∈B or c∈B. Thus A⊆B or C⊆B. Hence B is a prime ordered k bi-Γ-ideal of S.


## 5. Conclusion

In this work we have given many types of ordered(k), (m-k)-Γ-ideals mainly ordered k-(quasi, bi)-Γ-ideal, ordered m-k (quasi, bi)-Γ-ideal, ordered (semi)prime, (k)-(semi)prime (quasi, bi)-Γ-ideals, ordered maximal(k)-(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 f:K→L be a homomorphism of ordered Γ-semirings. We have defined homomorphism and prove that if J is an ordered quasi-Γ-ideal of L, then f-1(J) is an ordered quasi-Γ-ideal of an ordered Γ-semiring K. The same is true in case of ordered bi-Γ-ideals. For further research work the extended identical study can be taken under consider for other algebraic structures.



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