Characterization and Cartesian Product of Smarandache Semigroups (S-semigroups)

Abstract

Let (S, *) be a semigroup. A semigroup S is called a Smarandache semigroup (or S-semigroup) if it contains a proper subset A ⊂ S such that (A, *) forms a group under the same binary operation defined on S. In general, not every semigroup admits a proper subset that is a group; hence, not all semigroups are S-semigroups. In this paper, several structural conditions related to Smarandache semigroups are investigated. In particular, we study the role of idempotent and completely regular elements in the structure of S-semigroups. These conditions provide a characterization of S-semigroups. Furthermore, this study investigates whether the Cartesian product of two or more S-semigroups is again an S-semigroup.