Arithmetic of additively reduced monoid semidomains

TL;DR

This paper investigates the factorization properties of additively reduced monoid semidomains, establishing conditions for various factorization types and exploring their elasticity, with numerous examples illustrating the arithmetic structure.

Contribution

It provides necessary and sufficient conditions for additively reduced monoid semidomains to be bounded, finite, or unique factorization domains, advancing understanding of their arithmetic properties.

Findings

Characterization of when semidomains are bounded, finite, or UFDs

Identification of classes with full and infinite elasticity

Examples illustrating the factorization behavior

Abstract

A subset $S$ of an integral domain $R$ is called a semidomain if the pairs $(S,+)$ and $(S, \cdot)$ are semigroups with identities; additionally, we say that $S$ is additively reduced provided that $S$ contains no additive inverses. Given an additively reduced semidomain $S$ and a torsion-free monoid $M$, we denote by $S[M]$ the semidomain consisting of polynomial expressions with coefficients in $S$ and exponents in $M$; we refer to these objects as additively reduced monoid semidomains. We study the factorization properties of additively reduced monoid semidomains. Specifically, we determine necessary and sufficient conditions for an additively reduced monoid semidomain to be a bounded factorization semidomain, a finite factorization semidomain, and a unique factorization semidomain. We also provide large classes of semidomains with full and infinity elasticity. Throughout the paper we provide examples aiming to shed some light upon the arithmetic of additively reduced semidomains.