Generalized Fuzzy Quasi-Ideals of an Intraregular Abel-Grassmann's Groupoid

Davvaz, Bijan; Khan, M. Rashid; Anis, Saima; Haq, Shamsul

doi:10.1155/2012/627075

Cited by 2 publications

(1 citation statement)

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“…They proved that for a locally associative Γ -LA-semigroup S with a left identity, S/ρ is a maximal weakly separative homomorphic image of S, where ρ is a relation on S defined by: aρb if and only if aγ b n = b n+1 and bγ a n = a n+1 for some positive integer n and for all γ ∈ Γ , where a, b ∈ S. In 2014, Abdullah et al [3] introduced the concept of interval-valued (∈, ∈ ∨q)-fuzzy ideals, interval-valued (∈, ∈ ∨q)-fuzzy bi-ideals and interval-valued (∈, ∈ ∨q)-fuzzy quasi-ideals of an LA-semigroup. Nowadays many scholars have studied different aspects of fuzzy subsets on LA-semigroups and Γ -LA-semigroups see [9,11,12,15,20,30,31,36,38,41,45,50]. In 2015, Abbasi and Basar [1] studied quasi-ideals and bi-ideals of locally associative Γ -LA-semigroups, (m, n) simple Γ -LA-semigroups, minimal bi-Γ -ideal, semiprime Γ -ideal and quasi-regular Γ -LA-semigroups.…”

Section: Introductionmentioning

confidence: 99%

On generalizations of quasi-prime ideals of an ordered left almost semigroups

Yiarayong

2021

Afr. Mat.

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The purposes of this paper are to introduce generalizations of quasi-prime ideals to the context of $$\phi $$ ϕ -quasi-prime ideals. Let $$\phi : {\mathcal {I}}(S) \rightarrow {\mathcal {I}}(S) \cup \left\{ \emptyset \right\} $$ ϕ : I ( S ) → I ( S ) ∪ ∅ be a function where $$ {\mathcal {I}}(S)$$ I ( S ) is the set of all left ideals of an ordered $${{\mathcal {L}}}{{\mathcal {A}}}$$ L A -semigroup S. A proper left ideal A of an ordered $${{\mathcal {L}}}{{\mathcal {A}}}$$ L A -semigroup S is called a $$\phi $$ ϕ -quasi-prime ideal, if for each $$a, b\in S$$ a , b ∈ S with $$ab \in A - \phi (A)$$ a b ∈ A - ϕ ( A ) , then $$a \in A$$ a ∈ A or $$b\in A$$ b ∈ A . Some characterizations of quasi-prime and $$\phi $$ ϕ -quasi-prime ideals are obtained. Moreover, we investigate relationships between weakly quasi-prime, almost quasi-prime, $$\omega $$ ω -quasi-prime, m-quasi-prime and $$\phi $$ ϕ -quasi-prime ideals of ordered $${{\mathcal {L}}}{{\mathcal {A}}}$$ L A -semigroups. Finally, we obtain necessary and sufficient conditions of $$\phi $$ ϕ -quasi-prime ideal in order to be a quasi-prime ideal.

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