A Study of Non-Associative Ordered Semigroups in Terms of Semilattices via Smallest (Double-Framed Soft) Ideals

• Main Authors: Faisal Yousafzai, Tauseef Asif, Asghar Khan, Bijan Davvaz
• Format: Article
• Language: English
• Published: Etamaths Publishing 2018-07-01
• Series: International Journal of Analysis and Applications
• Online Access: http://etamaths.com/index.php/ijaa/article/view/1679

Soft set theory, introduced by Molodtsov has been considered as a successful mathematical tool for modeling uncertainties. A double-framed soft set is a generalization of a soft set, consisting of union soft sets and intersectional soft sets. An ordered AG-groupoid can be referred to as a non-associative ordered semigroup, as the main difference between an ordered semigroup and an ordered AG-groupoid is the switching of an associative law. In this paper, we define the smallest left (right) ideals in an ordered AG-groupoid and use them to characterize a strongly regular class of a unitary ordered AG-groupoid along with its semilattices and double-framed soft (briefly DFS) l-ideals (r-ideals). We also give the concept of an ordered A* G**-groupoid and investigate its structural properties by using the generated ideals and DFS l-ideals (r-ideals). These concepts will verify the existing characterizations and will help in achieving more generalized results in future works.