On interval-valued fuzzy soft set theory applied to semigroups

In this paper, we focus on combining the theories of interval-valued fuzzy soft sets over semigroups, and establishing a new framework for interval-valued fuzzy soft semigroups. The aim of this manuscript is to apply interval-valued fuzzy soft set for dealing with several kinds of theories in semigroups. First, we present the concepts of interval-valued fuzzy soft sets, interval-valued fuzzy soft semigroups, interval-valued fuzzy soft ideals, interval-valued fuzzy soft quasi-ideals, interval-valued fuzzy soft interior ideals and interval-valued fuzzy soft bi-ideals. Meanwhile, some illustrative examples are given to show the rationality of the definitions introduced in this paper. Also, we prove that a non-empty subset of a semigroup S is a subsemigroup (left ideal, right ideal, ideal) of S if and only if the interval-valued fuzzy soft set over S is the interval-valued fuzzy soft subsemigroup (interval-valued fuzzy soft left ideal, interval-valued fuzzy soft right ideal, interval-valued fuzzy soft ideal) over S. Second, several new kinds of generalized fuzzy soft sets over semigroups are proposed, and related properties and mutual relationships are also investigated. Moreover, we study relation between quasi-ideals and interval-valued fuzzy soft quasi-ideals over semigroups. Finally, we obtain necessary and sufficient conditions of an interval-valued fuzzy soft ideal in order to be an interval-valued fuzzy soft interior ideal.