Rough Semigroups in Connection with Single Valued Neutrosophic (∈, ∈)-Ideal

The scheme of rough sets is an effective procedure that handle ambiguous, inexact or uncertain information configuration. Rough set theory for algebraic structures like semigroups is a formal approximation space consisting of a universal set and an equivalence relation. This article achieves a new utilization of rough sets in the theory of semigroups via single valued neutrosophic (SVN) subsemigroups/ideals. The conceptions of an SVN (∈, ∈)-subsemigroup and an SVN (∈, ∈)-ideal in semigroups are introduced, and its properties are investigated. Special congruence relations induced by an SVN (∈, ∈)-ideal are introduced in semigroups. Using these notions, the lower and upper approximations, so called the Rq-lower approximation and the Rq-upper approximation for q ∈ {T, I, F} based on an SVN (∈, ∈)-ideal in semigroups are presented, and related characteristics are discussed. The notions of lower and upper subsemigroups/ideals, so called the Rq-lower subsemigroup/ideal and the Rq-upper subsemigroup/ideal for q ∈ {T, I, F}, are defined, and then the relationships between subsemigroups/ideals and Rq-lower (upper) subsemigroups/ideals are considered.