ON DOUBLE-FRAMED SOFT SETS OF NON-ASSOCIATIVE ORDERED SEMIGROUPS
Print ISSN: 2319-1023 | Online ISSN: 2582-5461 | Total Downloads : 172
DOI:
Author :
Mohammed M. Khalaf (Higher Institute of Engineering and Technology, King Mariout, Alexandria, EGYPT)
Asghar Khan (Department of Mathematics, Abdul Wali Khan University, Mardan, KPK, PAKISTAN)
Tauseef Asif (Department of Mathematics, Abdul Wali Khan University, Mardan, KPK, PAKISTAN)
Faisal Yousafzai (Military College of Engineering, National University of Sciences and Technology, Islamabad, PAKISTAN)
Abstract
In this note, the authors introduce the notion of double-framed soft sets $($briefly, DFS-sets$)$ in an ordered $\mathcal{AG}$% -groupoid. An ordered $\mathcal{AG}$-groupoid can be referred to as a non-associative ordered semigroup, as the main difference between an ordered semigroup and an ordered $\mathcal{AG}$-groupoid is the switching of an associative law. We define and give the examples of DFS $l$-ideals, DFS $r$%-ideals and DFS bi-ideals in an ordered $\mathcal{AG}$-groupoid and also investigate the relationship between them. We give an alternate definition for a strongly regular element of a unitary ordered $\mathcal{AG}$-groupoid and show that how a strongly regular ordered $\mathcal{AG}$-groupoid becomes an ordered $\mathcal{AG}^{\text{**}}$-groupoid and a completely inverse ordered $\mathcal{AG}$-groupoid. As an application of our results we get characterizations of a strongly regular ordered $\mathcal{AG}$-groupoid in terms of DFS one-sided $($two-sided$)$ ideals and DFS bi-ideals. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works.
Keywords and Phrases
DFS-sets, ordered $\mathcal{AG}$-groupoid, pseudo-inverse, left invertive law and DFS ideals.
A.M.S. subject classification
20M10 and 20N99.
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