The topics discussed in this book are Int-soft semigroup, Int-soft left (right) ideal, Int-soft (generalized) bi-ideal, Int-soft quasi-ideal, Int-soft interior ideal, Int-soft left (right) duo semigroup, starshaped (∈, ∈∨ qk)-fuzzy set, quasi-starshaped (∈, ∈∨ qk)-fuzzy set, semidetached mapping, semidetached semigroup, (∈, ∈ ∨qk)-fuzzy subsemi-group, (qk, ∈ ∨qk)-fuzzy subsemigroup, (∈, ∈ ∨ qk)-fuzzy subsemigroup, (qk, ∈ ∨ qk)-fuzzy subsemigroup, (∈ ∨ qk, ∈ ∨ qk)-fuzzy subsemigroup, (∈, ∈∨ qkδ)-fuzzy subsemigroup, ∈∨ qkδ -level subsemigroup/bi-ideal, (∈, ∈∨ qkδ )-fuzzy (generalized) bi-ideal, δ-lower (δ-upper) approximation of fuzzy set, δ-lower (δ-upper) rough fuzzy subsemigroup, δ-rough fuzzy subsemigroup, Neutrosophic N -structure, neutrosophic N -subsemigroup, ε-neutrosophic N -subsemigroup, and neutrosophic N -product.

In this paper, we introduce the notion of neutrosophic ℵ-bi-ideal for a semigroup. We infer different semigroups using neutrosophic ℵ -bi-ideal structures. Moreover, for regular semigroups, neutrosophic ℵ-product and intersection of neutrosophic ℵ-ideals are identical.

The aim of this paper is to introduce the notion of neutrosophic ℵ −ideals in semigroups and investigate their properties. Conditions for neutrosophic ℵ −structure to be a neutrosophic ℵ −ideal are provided. We also discuss the concept of characteristic neutrosophic ℵ −structure of semigroups and its related properties.

The notion of neutrosophic N-structure is introduced, and applied it to semigroup. The notions of neutrosophic N-subsemigroup, neutrosophic N-product and ε-neutrosophic N-subsemigroup are introduced, and several properties are investigated.

In this article, we will look at the definition of rough neutrosophic set in semigroups and investigate their properties. We also introduce the concept of rough Neutrosophic (right,left,two-sided,bi-) ideals in a semigroup and study some properties of the ideals of semigroup.

The intention of this paper is to introduce and study some properties of the ideals of semirings using the concept of Q-neutrosophic set.

We define the concepts of neutrosophic ℵ -interior ideal and neutrosophic ℵ −characteristic interior ideal structures of a semigroup. We infer different types of semigroups using neutrosophic ℵ-interior ideal structures. We also show that the intersection of neutrosophic ℵ-interior ideals and the union of neutrosophic ℵ-interior ideals is also a neutrosophic ℵ-interior ideal.