The concept of tri-quasi ideal was presented as a generalization of quasi-ideal, interior ideal, and bi-ideal. In this paper, we transfer this concept to soft set theory and semigroups,and introduce a novel type of soft union (S-uni) ideal form called "soft union (S-uni) tri-bi-ideal”. The main aim of this study is to obtain the relations between S-uni tri-bi-ideals and other certain types of S-uni ideals of a semigroup. Our results show that every S-uni tri-bi-ideal of a band is an S-uni subsemigroup. Moreover, an S-uni tri-bi-ideal is a generalization of an S-uni ideal, interior ideal, bi-ideal, quasi-ideal, weak-interior ideal, bi-interior ideal and bi-quasi ideal, however in order to satisfy the converses, the semigroup should have specific conditions. We also demonstrate that the S-uni quasi-interior ideal of a left or right simple semigroup is an S-uni tri-bi-ideal, nevertheless the converse holds for the zero semigroup. Furthermore, the S-uni bi-quasi-interior ideal of a commutative semigroup is an S-uni tri-bi-ideal, however, for the converse to hold the semigroup must be a band. We have shown that an S-uni tri-ideal coincides with an S-uni tri-bi-ideal of a band, and every S-uni tri-bi-ideal of a group is an S-uni tri-ideal. We also obtain a relation between tri-bi-ideal and its soft characteristic function, enabling us to get the relation between semigroup and soft set theory. Furthermore, we present conceptual characterizations and analysis of the new concept in terms of the soft set operations, the soft (anti/inverse) image, supporting our assertions with illuminating examples.
Authors: Aleyna İlgin, Aslıhan Sezgin, Thiti Gaketem
DOI: https://doi.org/10.28924/2291-8639-23-2025-329
Publish Year: 2025
