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

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.


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 Γsemirings and valuation Γ-ideals of a Γ-semiring.

Preliminaries and Basic Definitions
In this section we will define important terminologies based on those concepts which are being used in this paper.
Definition .Let (, +) and (Γ, +) be commutative semigroups.Then we call  a Γ-semiring, if there exists a mapping  × Γ ×  →  is written (, , ) as  such that it satisfies the following axioms for all , ,  ∈  and ,  ∈ Γ: Example (see [21]).Let  be a semiring and  , () denote the additive abelian semigroup of all  ×  matrices with identity element whose entries are from .Then  , () is a Γ-semiring with Γ =  , () ternary operation is defined by  = (  ) as the usual matrix multiplication, where   denotes the transpose of the matrix , for all ,  and  ∈  , ().
Definition .A Γ-semiring  is called an ordered Γ-semiring if it admits a compatible relation ≤; i.e., ≤ is a partial ordering on  satisfying the following conditions.
If  ≤  and  ≤  then Definition .An ordered Γ-semiring  is said to be commutative ordered Γ-semiring if  = , for all ,  ∈  and  ∈ Γ.
Definition .If  is both a left and a right ideal of ordered Γ-semirings , then  is called an ideal of .
Definition .A right ideal  of an ordered Γ-semiring  is called a right  ideal if  ∈  and  ∈  such that  +  ∈ , and then  ∈ .
A left  ideal of  is defined in a similar way.
Definition .An ordered Γ-semiring  is called totally ordered if any two elements of  are comparable.
Definition .An ordered quasi-Γ-ideals  is called ordered  quasi-Γ-ideal of  if  is an ordered sub-Γ-semiring of , and if  ∈  and  ∈  such that  +  ∈ , then  ∈ .
Clearly every ordered  quasi-Γ-ideal is an ordered quasi-Γ-ideal but converse is not true in general as shown in the following example.
Example .Let  be the set of nonnegative integers and Γ =  be additive abelian semigroups.Tennary operation is defined as (, , ) → , usual multiplication of integers.Then  is an ordered Γ-semirings.A subset  = 5 \ {5} of  is an ordered quasi-Γ-ideal of  but it is not an ordered  quasi-Γ-ideal of .
The following example shows that the converse of Theorem 16 is not true in general.
Definition .An ordered Γ-semiring  is called band if every element of  is an idempotent.
It is clear that every strongly prime ordered quasi-Γ-ideal in  is a prime ordered quasi-Γ-ideal and every prime ordered quasi-Γ-ideal in  is a semiprime ordered quasi-Γ-ideal.
Definition .An ordered quasi-Γ-ideal  of an ordered Γsemiring  is said to be maximal ordered quasi-Γ-ideal if  ̸ =  and for every ordered quasi-Γ-ideal  of  with  ⊆  ⊆ , then either  =  or  = .
(i) If  is a prime ordered quasi-Γ-ideal, then  is a strongly irreducible ordered quasi-Γ-ideal.
(ii) If  is a strongly irreducible ordered quasi-Γ-ideal, then  is an irreducible ordered quasi-Γ-ideal.
Proof.Let  be an ordered quasi-Γ-ideal of an ordered Γsemiring .
Therefore,  is an irreducible ordered quasi-Γ-ideal of .
Corollary 29.Let  be an ordered Γ-semiring.If  is a prime ordered quasi-Γ-ideal of , then  is an irreducible ordered quasi-Γ-ideal of .
Theorem 33. e intersection of family of prime (or semiprime) ordered quasi-Γ-ideal of  is a semiprime ordered quasi-Γ-ideal.
Proof.Let  be a strongly irreducible and semiprime ordered quasi-Γ-ideal of .For any ordered quasi-Γ-ideal  1 and  2 of , let ( But  is a strongly irreducible ordered quasi-Γ-ideal.Therefore,  1 ⊆  or  2 ⊆ .Thus  is a strongly prime ordered quasi-Γ-ideal of .
Theorem 35.If  is an ordered  quasi-Γ-ideal of  and  ∈  such that  ∉ , then there exists an irreducible ordered  quasi-Γ-ideal  1 of  such that  ⊆  1 and  ∉  1 .
Proof.Let  be the family of ordered  quasi-Γ-ideals of  which contain  but do not contain element .Then  is nonempty as  ∈ .This family of ordered  quasi-Γ-ideals of S forms a partially ordered set under the set inclusion.Hence by Zorn's lemma there exists a maximal ordered  quasi-Γideal say  1 in .Therefore  ⊆  1 and  ∉  1 .Now to show that  1 is an irreducible ordered  quasi-Γ-ideal of , let  and  be any two ordered  quasi-Γ-ideals of  such that  ∩  =  1 .Suppose that  and  both contain  1 properly.But  1 is a maximal ordered  quasi-Γ-ideal in .Hence we get  ∈  and  ∈ .Therefore  ∈  ∩  =  1 which is absurd.Thus either  =  1 or  =  1 .Therefore,  1 is an irreducible ordered  quasi-Γ-ideal of .
Theorem 36.e following statements are equivalent in : (1) e set of ordered  quasi-Γ-ideals of  is totally ordered set under inclusion of sets.
(1) ⇒ (2) Suppose that the set of ordered  quasi-Γideals of  is a totally ordered set under inclusion of sets.Let  be any ordered  quasi-Γ-ideal of .Then  is a strongly irreducible ordered  quasi-Γ-ideal of  for that let  1 and  2 be any two ordered  quasi-Γ-ideal of  such that  1 ∩ 2 ⊆ .
But, by the hypothesis, we have either Thus  is a strongly irreducible ordered  quasi-Γideal of .
Theorem 37. A prime ordered  quasi-Γ-ideal  of  is a prime one sided ordered  ideal of .
Theorem 38.An ordered  quasi-Γ-ideal  of  is prime if and only if, for a right ordered  ideal  and a le ordered  ideal  of , Γ ⊆  implies  ⊆  or  ⊆ .
Proof.Suppose that an ordered  quasi-Γ-ideal of  is a prime ordered  quasi-Γ-ideal of .Let  be a right ordered  ideal and  be a left ordered  ideal of  such that Γ ⊆ . and  are ordered  quasi-Γ-ideal of .Hence  ⊆  or  ⊆ .Conversely, we have to show that an ordered  quasi-Γ-ideal  of  is a prime ordered  quasi-Γ-ideal of .Let  and  be any two ordered  quasi-Γ-ideals of  such that Γ ⊆ .For any  ∈  and  ∈ , ()  ⊆  and ()  ⊆ , where ()  and ()  denote the right ordered  ideal and left ordered  ideal generated by  and , respectively.Thus ()  Γ()  ⊆ Γ ⊆ .Hence by the assumption ()  ⊆  or ()  ⊆  which further gives that  ∈  or  ∈ .In this way we get that  ⊆  or  ⊆ .Hence  is a prime ordered  quasi-Γ-ideal of .
Definition .An ordered sub-Γ-semiring  of an ordered Γsemiring  is called an ordered bi-Γ-ideal of  if ΓΓ ⊆  and if, for any ∈ ,  ∈ ,  ≤  in , then  ∈ .
Definition .An ordered bi-Γ-ideals  is called ordered  bi-Γ-ideal of  if  is an ordered sub-Γ-semiring of , and if  ∈  and  ∈  such that  +  ∈ , then  ∈ .
Clearly every ordered  bi-Γ-ideal is an ordered bi-Γ-ideal but converse is not true in general as shown in the following example.
Example .Let  be the set of nonnegative integers and Γ =  be additive abelian semigroups.Tennary operation is defined as (, , ) → , usual multiplication of integers.Then  is an ordered Γ-semirings.A subset  = 7 \ {7} of  is an ordered bi-Γ-ideal of  but it is not an ordered  bi-Γ-ideal of .
Theorem 43.Every ordered  −  bi-Γ-ideal of an ordered Γsemiring  is an ordered  bi-Γ-ideal of .
The following example shows that the converse of Theorem 43 is not true in general.
Definition .An ordered Γ-semiring  is called band if every element of  is a -idempotent.
Theorem 46.Let  be an ordered sub-Γ-semiring of an ordered Γ-semiring  in which semigroup (, +) is a band.en  is an ordered bi-Γ-ideal of  if and only if  is an ordered  bi-Γ-ideals of .
Definition .An ordered bi-Γ-ideal  of  is called a semiprime ordered bi-Γ-ideal if for any ordered bi-Γ-ideals It is clear that every strongly prime ordered bi-Γ-ideal in  is a prime ordered bi-Γ-ideal and every prime ordered bi-Γideal in  is a semiprime ordered bi-Γ-ideal.
Definition .An ordered bi-Γ-ideal  of an ordered Γsemiring  is said to be maximal ordered bi-Γ-ideal if  ̸ =  and for every ordered bi-Γ-ideal  of  with  ⊆  ⊆ , then either  =  or  = .
Theorem 55.Let  be an ordered bi-Γ-ideal of an ordered Γsemiring .
(i) If  is a prime ordered bi-Γ-ideal, then  is a strongly irreducible ordered bi-Γ-ideal.
(ii) If  is a strongly irreducible ordered bi-Γ-ideal, then  is an irreducible ordered bi-Γ-ideal.
Proof.Let  be an ordered bi-Γ-ideal of an ordered Γsemiring .
Therefore,  is an irreducible ordered bi-Γ-ideal of .
Corollary 56.Let  be an ordered Γ-semiring.If  is a prime ordered bi-Γ-ideal of , then  is an irreducible ordered bi-Γideal of .
Proof.Suppose  is an ordered bi-Γ-ideal of ,  :  →  be a homomorphism of ordered Γ-semirings and ,  ∈  Theorem 58.Let  be an ordered Γ-semiring.If  is an ordered  −  bi-Γ-ideal of , then  is a maximal ordered bi-Γ-ideal of .
Theorem 62.If  is an ordered  bi-Γ-ideal of  and  ∈  such that  ∉ , then there exists an irreducible ordered  bi-Γ-ideal  1 of  such that  ⊆  1 and  ∉  1 .
Proof.Let  be the family of ordered  bi-Γ-ideals of  which contain  but do not contain element .Then  is nonempty as  ∈ .This family of ordered  bi-Γ-ideals of S forms a partially ordered set under the set inclusion.Hence by Zorn's lemma there exists a maximal ordered  bi-Γ-ideal say  1 in .Therefore  ⊆  1 and  ∉  1 .Now to show that  1 is an irreducible ordered  bi-Γ-ideal of .Let  and  be any two ordered  bi-Γ-ideals of  such that ∩ =  1 .Suppose that  and  both contain  1 properly.But  1 is a maximal ordered  bi-Γ-ideal in .Hence we get  ∈  and  ∈ .Therefore  ∈ ∩ =  1 which is absurd.Thus either  =  1 or  =  1 .Therefore  1 is an irreducible ordered  bi-Γ-ideal of .
Theorem 63.Following statements are equivalent in : (1) e set of ordered  bi-Γ-ideals of  is totally ordered set under inclusion of sets.
Proof.(1) ⇒ (2) Suppose that the set of ordered  bi-Γideals of  is a totally ordered set under inclusion of sets.
Let  be any ordered  bi-Γ-ideal of .Then  is a strongly irreducible ordered  bi-Γ-ideal of  for that let  1 and  2 be any two ordered  bi-Γ-ideal of  such that  1 ∩ 2 ⊆ .But by the hypothesis, we have either Thus  is a strongly irreducible ordered  bi-Γ-ideal of .
Theorem 64.A prime ordered  bi-Γ-ideal  of  is a prime one sided ordered  ideal of .
Theorem 65.An ordered  bi-Γ-ideal  of  is prime if and only if, for a right ordered  ideal  and a le ordered  ideal  of , Γ ⊆  implies  ⊆  or  ⊆ .
Proof.Suppose that an ordered  bi-Γ-ideal of  is a prime ordered  bi-Γ-ideal of .Let  be a right ordered  ideal and  be a left ordered  ideal of  such that Γ ⊆ . and  are ordered  bi-Γ-ideal of .Hence  ⊆  or  ⊆ .Conversely, we have to show that an ordered  bi-Γ-ideal  of  is a prime ordered  bi-Γ-ideal of .Let  and  be any two ordered  bi-Γ-ideals of  such that Γ ⊆ .For any  ∈  and  ∈ , ()  ⊆  and ()  ⊆ , where ()  and ()  denote the right ordered  ideal and left ordered  ideal generated by  and , respectively.Thus ()  Γ()  ⊆ Γ ⊆ .Here by the assumption we get that ()  ⊆  or ()  ⊆  which gives that  ∈  or  ∈ .Thus  ⊆  or  ⊆ .Hence  is a prime ordered  bi-Γ-ideal of .