NeutroAlgebra of Substructures of the Semigroups built using Z_n and Z⁺

Vasantha Kandasamy ¹, Ilanthenral Kandasamy^2, * and Florentin Smarandache ³

^1.2School of Computer Science and Engineering, Vellore Institute of Technology, Vellore, Tamil Nadu, India

³ Dept. Math and Sciences, University of New Mexico, Gallup, NM, USA

Emails; vasantha.wb@vit.ac.in, ilanthenral.k@vit.ac.in  ; smarand@unm.edu

*  Correspondence: ilanthenral.k@vit.ac.in

Abstract                                                              

For the first-time authors study the NeutroAlgebraic structures of the substructures of the semigroups, { , ×},
{ , ×} and { , +} where  = {1, 2, …, ¥}. The three substructures of the semigroup studied in the context of NeutroAlgebra are subsemigroups, ideals and groups. The substructure group has meaning only if the semigroup under consideration is a Smarandache semigroup. Further in this paper, all semigroups are only commutative. It is proved the NeutroAlgebraic structure of ideals (and subsemigroups) of a semigroup can be AntiAlgebra or NeutroAlgebra in the case of infinite semigroups built on  or  =   È {0}. However, in the case of S = { , ×}; n a composite number, S is always a Smarandache semigroup. The substructures of them are completely analyzed. Further groups of Smarandache semigroups can only be a NeutroAlgebra and never an AntiAlgebra. Open problems are proposed in the final section for researchers interested in this field of study.

Keywords: NeutroAlgebra; AntiAlgebra; groupring; NeutrosubAlgebra; Partial Algebra; groups; ideals; Smarandache semigroup.