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

Abstract

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