In settings where parametric variability is present, soft set theory has evolved into a robust and versatile mathematical framework for modeling and analyzing uncertainty. Central to this framework are the operations and product constructions on soft sets, which together provide a powerful algebraic infrastructure for addressing complex parameter-dependent problems. Formally, it is shown that under the union operation, the collection of all soft sets defined over a fixed parameter set forms a bounded semilattice, thereby supplying essential algebraic structure and coherence. Through careful analysis, it is further demonstrated that the algebraic system consisting of all soft sets over a fixed parameter set endowed with a group structure, equipped with the union operation and the proposed product, satisfies the axiomatic framework of a hemiring. This structural characterization yields two major theoretical implications: it strengthens the algebraic foundations of soft set theory and lays the groundwork for constructing a soft group theory analogous to its classical counterpart. A new product on soft sets, termed the soft symmetric difference–union product, is then introduced for the case in which the parameter set carries a group structure. This operation is examined in detail from both axiomatic and structural perspectives, with particular attention to its compatibility with soft equality and soft subsethood. The approach presented here makes a substantial contribution to the ongoing algebraic refinement and theoretical advancement of soft set theory, as the formal development of soft algebraic systems fundamentally relies on rigorously defined operations and product constructions.
Authors: Aslıhan Sezgin, Zeynep Ay, İbrahim Durak
DOI: https://doi.org/10.31181/jdaic10012122025s
Publish Year: 2025
