On semilattice decomposition of an Abel-Grassmann’s groupoid

Khan, M. Rashid; Anis, Saima

doi:10.1007/s10114-012-0014-6

Cited by 6 publications

(6 citation statements)

References 12 publications

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“…They proved that an ideal A of an AG-band is prime iff if ideal (S) is totally ordered; it is prime iff it is strongly irreducible. In 2012, Khan and Anis [17] proved that S/γ is a maximal separative semilattice homomorphic image of an LA-semigroup S. In 2013, Shah and Rehman [17] studied several properties of locally associative Γ -LA-semigroups. 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.…”

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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Section: Introductionmentioning

confidence: 99%

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

Yiarayong

2021

Afr. Mat.

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“…As a generalization of a commutative semigroup, the notion of an Abel Grassmann's groupoid was introduced by Kazim and Naseeruddin [12] in 1972 and this structure is known as the left almost semigroup (LA-semigroup). An AG-groupoid is a non-associative algebraic structure and many features of the AG-groupoid can be studied in [13]. In [14][15][16][17][18][19][20][21], some properties and connections of AG-groupoid, with some classes of algebraic structures, have been investigated.…”

Section: Introductionmentioning

confidence: 99%

The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops

Zhang

2019

Mathematics

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In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every AG-NET-Loop is the disjoint union of its maximal subgroups. At the same time, the new notion of Abel Grassmann’s (l, l)-Loop (AG-(l, l)-Loop), which is the Abel-Grassmann’s groupoid with the local left identity and local left inverse, were introduced. The strong AG-(l, l)-Loops were systematically analyzed, and the following decomposition theorem was proved: every strong AG-(l, l)-Loop is the disjoint union of its maximal sub-AG-groups.

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“…In [2], the same structure is called a left invertive groupoid. In [3][4][5][6][7][8][9], some properties and different classes of an AG-groupoid are investigated.…”

Section: Introductionmentioning

confidence: 99%

Generalized Abel-Grassmann’s Neutrosophic Extended Triplet Loop

Zhang

2019

Mathematics

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A group is an algebraic system that characterizes symmetry. As a generalization of the concept of a group, semigroups and various non-associative groupoids can be considered as algebraic abstractions of generalized symmetry. In this paper, the notion of generalized Abel-Grassmann’s neutrosophic extended triplet loop (GAG-NET-Loop) is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) an algebraic system is an AG-NET-Loop if and only if it is a strong inverse AG-groupoid; (2) an algebraic system is a GAG-NET-Loop if and only if it is a quasi strong inverse AG-groupoid; (3) an algebraic system is a weak commutative GAG-NET-Loop if and only if it is a quasi Clifford AG-groupoid; and (4) a finite interlaced AG-(l,l)-Loop is a strong AG-(l,l)-Loop.

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On semilattice decomposition of an Abel-Grassmann’s groupoid

Cited by 6 publications

References 12 publications

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

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

The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops

Generalized Abel-Grassmann’s Neutrosophic Extended Triplet Loop

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