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

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

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Abstract

For the first-time authors study the NeutroAlgebraic structures of the substructures of the semigroups, {Zn , ×}, {Z+ , ×} and {Z+ , +} where Z+ = {1, 2, …, inf}. 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 Z+ or Z* = Z+ U {0}. However, in the case of S = {Zn , ×}; 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 group ring NeutrosubAlgebra Partial Algebra groups ideals Smarandache semigroup

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