Rough Approximations of Soft Semigroups

In the context of classical soft set theory, soft sets are primarily used to find optimal parameters and the best alternative data. A soft set can be approximated by a rough set model as either an exact or an inexact soft set. In this context, lower and upper rough approximations of soft sets serve as tools for verification. To study this concept in algebraic structures, this paper introduces suitable lower and upper rough approximations within soft ideal theory in semigroups, leading to a new rough set model. A corresponding example is provided to illustrate the proposed concepts. Fundamental properties of the model are explored through operations on soft sets. Moreover, we prove that the lower and upper rough approximations of various types of soft semigroup structures—including uni-soft semigroups, uni-soft left ideals, uni-soft right ideals, uni-soft quasi-ideals, int-soft semigroups, int-soft left ideals, int-soft right ideals, int-soft quasi-ideals, soft-covered semigroups, soft-covered left ideals, soft-covered right ideals, soft-covered quasi-ideals, soft anti-covered semigroups, soft anti-covered left ideals, soft anti-covered right ideals, and soft anti-covered quasi-ideals—preserve their respective structural properties. Overall, this study provides new results for soft sets in semigroups within the framework of rough set theory. This research is anticipated to exert a significant impact on the study of algebra and decision-making processes. Specifically, when the parametric domain is endowed with a semigroup operation.