Exploring Algebraic Structures in Power Monoids Derived from Numerical Monoids

Abstract

 Let  be a numerical monoid. We consider two algebraic constructions associated with : the power monoid , consisting of all finite nonempty subsets of , and the restricted power monoid , which is formed by those finite nonempty subsets that include the zero element. Both structures are endowed with the binary operation induced by setwise addition.

Recent studies have examined various arithmetic aspects of such power monoids. In this work, we extend the existing body of research by focusing on intrinsic algebraic features, with particular emphasis on structural properties such as the prime ideal spectrum. In addition, we establish that, within the restricted power monoid , the vast majority of elements cannot be decomposed into nontrivial sumsets, and hence are irreducible.