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Sezgin, Aslihan
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A New Type of Extended Soft Set Operation: Complementary Extended Intersection Operation Sezgin, Aslihan; Sarialioglu, Murat; Aygün, Emin
Jurnal Matematika Vol 14 No 1 (2024)
Publisher : Publisher : Mathematics Department, Faculty of Mathematics and Natural Sciences, Udayana University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24843/JMAT.2024.v14.i01.p173

Abstract

Soft set theory is seen as an effective mathematical tool in solving problems involving uncertainty, and has been applied in many theoretical and practical areas since its introduction. The basic concept of the theory is soft set operations. In this context, in this paper, a new kind of soft set operations called complementary extended soft set operation is defined in order to contribute to the theory. The properties of the operation are examined in detail together with its distributions over other soft set operations in order to obtain the relationship between complementary extended intersection operation and the others. We demonstrate that the collection of soft sets over with a fixed parameter set, along with the complementary extended intersection operation and other certain types of soft sets, form many well-known and important algebraic structures in classical algebra, including semiring, hemiring, Boolean ring, Boolean Algebra, De Morgan Algebra, Kleene Algebra, and Stone Algebra.
Soft union bi-quasi-interior ideals of semigroups BAŞ, Zeynep Hare; SEZGİN, Aslıhan; STOJANOVİĆ, Nenad
Jurnal Matematika Integratif Vol 21, No 1: April 2025
Publisher : Department of Matematics, Universitas Padjadjaran

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24198/jmi.v21.n1.62779.55-74

Abstract

The concept of the soft union (S-uni) bi-quasi-interior ₿ꝖĪ) ideal of semigroups is proposed in this study, along with its equivalent definition. We derive the relationships between S-uni ideals and S-uni ₿ꝖĪ ideal. The S-uni ₿ꝖĪ ideal is shown to be S-uni bi-ideal, left ideal, right ideal, interior ideal, quasi-ideal, bi-interior ideal, left/right bi-quasi ideal, and left/right quasi-interior ideal. It is shown that certain additional requirements, such as regularity or right/left simplicity, are necessary for the converses, and counterexamples are given to demonstrate that the converses are not true. Additionally, it is demonstrated that the soft anti characteristic function of a subsemigroup of a semigroup is an S-uni ₿ꝖĪ ideal if the subsemigroup itself is a ₿ꝖĪ ideal, and vice versa. Consequently, a significant connection between soft set theory and classical semigroup theory is established. Additionally, it is demonstrated that while the finite soft OR-products and union of S-uni ₿ꝖĪ ideals are also S-uni ₿ꝖĪ ideals, the intersection and finite soft AND-products are not. A broad conceptual characterization and analysis of S-uni ₿ꝖĪ ideals are presented in this paper. 
Co-Authors Aygün, Emin BAŞ, Zeynep Hare Sarialioglu, Murat STOJANOVİĆ, Nenad