Soft Union Quasi-Interior Ideals of Semigroups

Abstract

The generalization of algebraic structure ideals has proven to be both intriguing and beneficial for mathematicians. In this context, the notion of the quasi-interior (Ԛꟾ) ideal was introduced as a generalization of both quasi-ideals and interior ideals in semigroups. This paper applies this concept to soft set theory and semigroups, proposing a new type of soft union (S-uni) ideal called the "soft union (S-uni) quasi-interior (Ԛꟾ) ideal." The primary goal of this study is to explore the relationships between S-uni Ԛꟾ ideals and other specific types of S-uni ideals in semigroups. It has been shown that every S-uni interior ideal of a semigroup is an S-uni Ԛꟾ ideal, and every S-uni ideal is also an S-uni Ԛꟾ ideal. Furthermore, the S-uni bi-ideal of a group is an S-uni Ԛꟾ ideal, the S-uni quasi-ideal of a regular group is an S-uni Ԛꟾ ideal, and the idempotent S-uni Ԛꟾ ideal is both an S-uni bi-quasi-ideal and an S-uni bi-interior ideal. Counterexamples are provided to show that the reverse of these statements is not always true. We demonstrate that for the converses to hold, the semigroup must either be a group or regular, or the S-uni Ԛꟾ ideal must be idempotent. Our main theorem shows that if a subsemigroup of a semigroup is a Ԛꟾ ideal, its soft characteristic function is an S-uni Ԛꟾ ideal, and vice versa. This theorem establishes a link between semigroup theory and soft set theory. Through this result, we illustrate how the concept relates to existing algebraic structures in classical semigroup theory. Additionally, we provide conceptual characterizations and an analysis of the concept in terms of soft set operations, such as soft anti image and soft inverse image, and support our claims with specific examples. Lastly, we establish and present the connection between a regular semigroup and the structure of S-uni Ԛꟾ ideals.