On the ascent of almost and quasi-atomicity to monoid semidomains

TL;DR

This paper investigates whether weaker forms of atomicity, namely almost atomicity and quasi-atomicity, extend from base semidomains to their polynomial and monoid domain extensions, providing new results and counterexamples.

Contribution

It establishes conditions under which almost atomicity and quasi-atomicity ascend to polynomial extensions and constructs explicit counterexamples for their non-ascent to monoid domains.

Findings

Almost atomicity and quasi-atomicity ascend to polynomial extensions under certain conditions.

Counterexamples show these properties do not always ascend to monoid domains.

The results improve understanding of atomicity properties in algebraic structures.

Abstract

A commutative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral (semi)domain is atomic if its multiplicative monoid is atomic. Notions weaker than atomicity have been introduced and studied during the past decade, including almost atomicity and quasi-atomicity, which were coined and first investigated by Boynton and Coykendall in their study of graphs of divisibility of integral domains. The ascent of atomicity to polynomial extensions was settled by Roitman back in 1993 while the ascent of atomicity to monoid domains was settled by Coykendall and the second author in 2019 (in both cases the answer was negative). The main purpose of this paper is to study the ascent of almost atomicity and quasi-atomicity to polynomial extensions and monoid domains. Under certain reasonable conditions, we establish the ascent of both properties to polynomial extensions (over semidomains). Then we construct an explicit example illustrating that, with no extra conditions, quasi-atomicity does not ascend to polynomial extensions. Finally, we show that, in general, neither almost atomicity nor quasi-atomicity ascend to monoid domains, improving upon a construction first provided by Coykendall and the second author for the non-ascent of atomicity.