Neutrosophic ℵ-filters in semigroups

Models of universe problems are brimming with complexities and uncertainties in almost every field of study, including engineering, mathematics, medical sciences, computer science, physics, management sciences, artificial intelligence, and operations research. To address these uncertainties, various theories have been developed, including probability, rough sets, fuzzy sets, soft ideals, and neutrosophic sets. Neutrosophic set theory is the focus of this paper. In this paper, we introduce the notions of neutrosophic ℵ-filters and neutrosophic ℵ-bi-filters in a semigroup and investigate several properties. Moreover, the relations of prime biideal subset and prime neutrosophic ℵ-bi- ideal structure; neutrosophic ℵ-bi-filter and neutrosophic ℵ-bi-ideal structure; left (resp., right) filter and neutrosophic ℵ-left (resp., right) filter; neutrosophic ℵ-left(resp., right) filter and prime neutrosophic ℵ-left (resp., right) ideals in semigroups are discussed. Finally we prove that: let X be a semigroup and XN be any neutrosophic structure. Then XN is a neutrosophic ℵ-bi-filter of X if and only if XNc is a prime neutrosophic ℵ-bi-ideal of X.