Soft binary piecewise intersection operation

Abstract

Soft set theory has emerged as a novel approach to modelling uncertainty, addressing a wide range of theoretical and practical problems. In this study, we define a new soft set operation called “soft binary piecewise intersection operation” and explore its fundamental algebraic properties by comparing them with the properties of the intersection operation in classical set theory. Many remarkable similarities have been observed between the intersection operation and the soft binary piecewise intersection operation. Additionally, we examine the distribution of the soft binary piecewise intersection operation over other types of soft set operations. By analysing the algebraic properties of the operation and its distribution rules, we demonstrate that the collection of soft sets over the universe, as well as the collection of soft sets with a fixed parameter set, alongside the soft binary piecewise intersection operation and other types of soft sets, forms several important algebraic structures such as hemirings, near-semirings, semirings, Boolean rings, Boolean algebras, De Morgan algebras, Kleene algebras, and Stone algebras.