Non-associative ordered semigroups based on soft sets

Abstract

In this note, the authors introduce the notion of soft sets (briefly, S-sets) in an ordered $\mathcal{A}\mathcal{G}$-groupoid. We define and give some examples of SI-l-ideals, SI-r-ideals, and SI -bi-ideals in an ordered $\mathcal{A}\mathcal{G}$-groupoid and also investigate the relationship between them. We give an alternate definition for a strongly regular element of a unitary ordered $\mathcal{A}\mathcal{G}$-groupoid and show that how a strongly regular ordered $\mathcal{A}\mathcal{G}$-groupoid becomes an ordered $\mathcal{A}{\mathcal{G}}^{**}$-groupoid and a completely inverse ordered $\mathcal{A}\mathcal{G}$-groupoid. As an application, we get characterizations of a strongly regular ordered $\mathcal{A}\mathcal{G}$-groupoid in terms of SI-one-sided (two-sided) ideals and SI-bi-ideals via semilattices. Finally, we give the concept of an ordered ${\mathcal{A}}^{*}{\mathcal{G}}^{**}$-groupoid and give an example to show that this class is the generalization of a unitary ordered $\mathcal{A}\mathcal{G}$-groupoid.

Keywords:

• S-sets
• ordered $\mathcal{A}\mathcal{G}$-groupoid
• pseudo-inverse
• left invertive law
• SI-ideals

2000 Mathematics Subject Classification:

• 20M10
• 20N99

Disclosure statement

No potential conflict of interest was reported by the authors.