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

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.