PolyTree-Soft Sets and PolyForest-Soft Sets: A Directed Acyclic Framework for Soft Set Modeling

Various mathematical frameworks have been developed to handle uncertainty, including the concepts of Fuzzy Sets and Neutrosophic Sets. Among these, Soft Sets provide a powerful and flexible approach to decision-making by mapping parameters to subsets of a universal set, effectively addressing uncertainty and vagueness. As an extension of Soft Sets, Treesoft Sets and ForestSoft Sets have been introduced to incorporate hierarchical structures into soft set theory. And a polyTree is a directed acyclic graph whose underlying undirected graph is a single tree, ensuring exactly one undirected path between any two vertices. A polyForest is a directed acyclic graph whose underlying undirected graph is a forest, possibly multiple independent tree components without any cycles. In this paper, we propose new concepts, the PolyTree-Soft Set and PolyForest-Soft Set, which generalize existing soft set models by integrating directed acyclic structures. We formally define these sets, analyze their properties, and explore their mathematical characteristics in depth