Constructing Q-Ideals for Boolean Semiring Partitioning Using Seeds

Abstract

Semiring partitioning is widely used in mathematics, computer science, and data analysis. The purpose of this paper is to add to the theory of semirings by proposing a novel method to construct Q-ideals for partitioning Boolean semirings. We introduce the set of all seeds—all s-tuples over a particular Boolean algebra—and the notion of their weight and complement. Utilizing this new method for constructing Q-ideals, we develop a nested hierarchical partitioning algorithm based on the weight of selected seeds. Additionally, we determine the maximal semiring homomorphism corresponding to this proposed method.