Neutrosophic ${\mathcal N}$-structures and their applications in semigroups

Abstract

The notion of neutrosophic ${\mathcal N}$-structure is introduced, and applied it to semigroup. The notions of neutrosophic ${\mathcal N}$-subsemigroup, neutrosophic ${\mathcal N}$-product and $\varepsilon$-neutrosophic ${\mathcal N}$-subsemigroup are introduced, and several properties are investigated. Conditions for neutrosophic ${\mathcal N}$-structure to be neutrosophic ${\mathcal N}$-subsemigroup are {provided.} Using neutrosophic ${\mathcal N}$-product, characterization of neutrosophic ${\mathcal N}$-subsemigroup is discussed. Relations between neutrosophic ${\mathcal N}$-subsemigroup and $\varepsilon$-neutrosophic ${\mathcal N}$-subsemigroup {are discussed}. We show that the homomorphic preimage of neutrosophic ${\mathcal N}$-subsemigroup is a neutrosophic ${\mathcal N}$-subsemigroup, and the onto homomorphic image of neutrosophic ${\mathcal N}$-subsemigroup is a neutrosophic ${\mathcal N}$-subsemigroup.