Notes on (m,n) bi-Γ-ideals in Γ-semigroups

Ansari, Moin A.; Khan, Mohd Rais

doi:10.1007/s12215-011-0024-8

Cited by 4 publications

(2 citation statements)

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“…Many definitions and results in semigroups were extended to that in Γ-semigroups. Ideal theory in Γsemigroups have been widely studied, for example, Chinram [4] studied quasi-Γ-ideals (or quasi-ideals) of Γ-semigroups in 2006 and he joined with Jirokul [3] to studied bi-Γ-ideals (or bi-ideals) of Γ-semigroups in 2007, Hila [8] focused on regular Γ-semigroups, semiprime Γ-semigroups and quasi-reflexive Γ-semigroups, and focused to study minimal quasi-ideals of Γsemigroups, Ansari and Khan [2] focused on (m, n) bi-Γ-ideal in Γ-semigroups, etc.…”

Section: Introductionmentioning

confidence: 99%

Almost Interior Gamma-ideals and Fuzzy Almost Interior Gamma-ideals in Gamma-semigroups

Jantanan¹,

Simuen²,

Yonthanthum³

et al. 2021

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Ideal theory plays an important role in studying in many algebraic structures, for example, rings, semigroups, semirings, etc. The algebraic structure Γ-semigroup is a generalization of the classical semigroup. Many results in semigroups were extended to results in Γ-semigroups. Many results in ideal theory of Γ-semigroups were widely investigated. In this paper, we first focus to study some novel ideals of Γ-semigroups. In Section 2, we define almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups by using the concept ideas of interior Γ-ideals and almost Γ-ideals of Γ-semigroups. Every almost interior Γ-ideal of a Γ-semigroup S is clearly a weakly almost interior Γ-ideal of S but the converse is not true in general. The notions of both almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups are generalizations of the notion of interior Γ-ideal of a Γ-semigroup S. We investigate basic properties of both almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups. The notion of fuzzy sets was introduced by Zadeh in 1965. Fuzzy set is an extension of the classical notion of sets. Fuzzy sets are somewhat like sets whose elements have degrees of membership. In the remainder of this paper, we focus on studying some novelties of fuzzy ideals in Γ-semigroups. In Section 3, we introduce fuzzy almost interior Γ-ideals and fuzzy weakly almost interior Γ-ideals of Γ-semigroups. We investigate their properties. Finally, we give some relationship between almost interior Γ-ideals [weakly almost interior Γ-ideals] and fuzzy almost interior Γ-ideals [fuzzy weakly almost interior Γ-ideals] of Γ-semigroups.

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Section: Introductionmentioning

confidence: 99%

Almost Interior Gamma-ideals and Fuzzy Almost Interior Gamma-ideals in Gamma-semigroups

Jantanan¹,

Simuen²,

Yonthanthum³

et al. 2021

View full text Add to dashboard Cite

show abstract

“…A bi-Γ-ideal in Γ-semigroups was sometimes called a bi-ideal (see [14]). Some generalizations of this ideal were studied in [2] and [16]. Recently, Wattanatripop and Changphas first studied the concept of almost ideals in Γ-semigroups.…”

Section: Introductionmentioning

confidence: 99%