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

Abbasi, Mohammad Yahya; Basar, Abul

doi:10.1155/2013/565848

Cited by 8 publications

(16 citation statements)

References 9 publications

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“…For the purpose of the following analysis, we need a new notion. An ordered Γ-semigroup with apartness (S, = S , = S , w S ) under a co-order S is called regular if for each x ∈ S and for every pair a, b ∈ Γ there exists an element x ∈ S such that x S xax bx according to a definition given in the paper [1]. In this case, a co-ordered left L and a right R co-ideals in a co-ordered regular Γ-semigroup with apartness S have some specific features.…”

Section: Notesmentioning

confidence: 99%

The Concept of Dual of $\Gamma$-Ideals in a Co-ordered $\Gamma$-Semigroup with Apartness

2021

Electronic Journal of Mathematics

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In 1981, Sen introduced the notion of Γ-semigroups as a generalization of semigroups. The idea of Γ-semigroups with apartness in the Bishop's constructive framework was introduced and analyzed in 2019 by the present author. The concept of co-filters as well as some other related concepts (and their interrelationships) in a co-ordered Γ-semigroup with apartness is the focus of the present author's interest. In this paper, as a continuation of the previous research, the author discusses the design of the concept of (left, right, both-sided, prime) co-ideals as duals of corresponding ideals in the before-mentioned algebraic systems.

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Section: Notesmentioning

confidence: 99%

The Concept of Dual of $\Gamma$-Ideals in a Co-ordered $\Gamma$-Semigroup with Apartness

2021

Electronic Journal of Mathematics

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“…for all a, b, x ∈ S and α, β ∈ Γ. For other basic definitions and properties, we refer [1], [8], [2], [3], [11], [13], [14], [16]. In what follows we denote the ordered Γ-semigroup (S, Γ, •, ≤) by S unless otherwise specified.…”

Section: Introductionmentioning

confidence: 99%

A characterization of ordered Γ-semigroups by ordered (m,n)-Γ-ideals

Abbasi

Basar

Ali

2021

bspm

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In this paper, we study (m,n)-regular ordered Γ-semigroups through ordered (m,n)-Γ-ideals. It is shown that if (S,Γ,·,≤) is an ordered Γ-semigroup; m,n are non-negative integers and A(m,n) is the set of all ordered (m,n)-Γ-ideals of S. Then, S is (m,n)-regular⇐⇒ ∀A ∈ A(m,n), A = (AmΓSΓAn]. It is also proved that if (S,Γ,·,≤) is an ordered Γ-semigroup and m,n are nonnegative integers and R(m,0) and L(0,n) is the set of all (m,0)-Γideals and (0,n)-Γ-ideals of S, respectively. Then, S is (m,n)-regular ordered Γ semigroup ⇐⇒∀R ∈R(m,0)∀L ∈L(0,n),R∩L = (RmΓL∩RΓLn].

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“…We go on further to prove that two characterizations of regularity, one for ordered Γ-semigroups and the other for ordered semigroups are logically equivalent. The characterization of the regularity of ordered Γ-semigroups is Theorem 8(iii) of [2] and also Theorem 3 of [3], which states that an ordered Γ-semigroup ( ≤ ) S, Γ, S is regular if and only if one-sided ideals of ( ≤ ) S, Γ, S are idempotent, and for every right ideal R and every left ideal L of ( ≤ ) S, Γ, S , ( ] R L Γ is a quasi ideal of ( ≤ ) S, Γ, S . On the other hand, the characterization of the regularity of ordered semigroups is Theorem 3.1(iii) of [4], which states that an ordered semigroup ( ⋅ ≤ ) S, , S is regular if and only if, one-sided ideals of ( ⋅ ≤ ) S, , S are idempotent, and for every right ideal R and every left ideal L of…”

Section: Introductionmentioning

confidence: 99%