Uni-Soft Ideals in Ordered Semigroups, Delineated by Soft Union Products (European Journal of Pure and Applied Mathematics, 2026)
In this paper, we introduce the concepts of soft left and soft right ideals, soft quasi-ideals, and soft bi-ideals within the context of ordered semigroups. We demonstrate that in ordered semigroups, both soft right and soft left ideals exhibit properties of soft quasi-ideals. Similarly, soft quasi-ideals possess characteristics of soft bi-ideals. Furthermore, our analysis establishes that the definitions of soft quasi-ideals and soft bi-ideals align, indicating their equivalence within this specific class of semigroups. Additionally, we prove that in an ordered semigroup, soft quasi-ideals can be understood simply as the unions of soft right and soft left ideals. This elucidates the relationship between these concepts, shedding light on their fundamental role in the structure of ordered semigroups.
Keywords: ordered semigroup; soft set; soft union product; soft left (right) ideal; soft quasi-ideal; soft bi-ideal