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

Laila Karimatul Fadhilah (Department of Mathematics, Universitas Diponegoro, Semarang, 50275, Indonesia)
Suryoto (Department of Mathematics, Universitas Diponegoro, Semarang, 50275, Indonesia)
Nikken Prima Puspita (Department of Mathematics, Universitas Diponegoro, Semarang, 50275, Indonesia)
Titi Udjiani (Department of Mathematics, Universitas Diponegoro, Semarang, 50275, Indonesia)

Integra: Journal of Integrated Mathematics and Computer Science
Vol. 3 No. 1 (2026): March

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.