On finitary power monoids of linearly orderable monoids

TL;DR

The paper investigates which ideal-theoretic and atomic properties of a linearly orderable commutative monoid $M$ lift to its finitary power monoid $\mathcal{P}_{fin}(M)$. It establishes ascent results for weaker ACCP variants (quasi-ACCP and almost ACCP) and provides a precise characterization of when $\mathcal{P}_{fin}(M)$ is atomic in terms of $M$ being atomic and every nonempty finite subset of $M$ having a maximal common divisor (MCD). It also constructs explicit counterexamples showing that atomicity and near atomicity do not generally ascend, and analyzes Furstenberg-type properties and the TIDF attribute, proving they ascend in positive Archimedean cases but not in full generality. The findings deepen understanding of transfer of factorization and ideal-theoretic structure from linearly ordered monoids to their finitary power monoids, highlighting both robust ascent phenomena and sharp counterexamples.

Abstract

A commutative monoid $M$ is called a linearly orderable monoid if there exists a total order on $M$ that is compatible with the monoid operation. The finitary power monoid of a commutative monoid $M$ is the monoid consisting of all nonempty finite subsets of $M$ under the so-called sumset. In this paper, we investigate whether certain atomic and divisibility properties ascend from linearly orderable monoids to their corresponding finitary power monoids.

Theorems & Definitions (41)

• Theorem 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• Lemma 2.4
• proof
• Lemma 3.1
• proof
• Corollary 3.2
• Theorem 3.3