On finitary power monoids of linearly orderable monoids

TL;DR

This paper explores whether atomic and divisibility properties of linearly orderable monoids are preserved when considering their finitary power monoids, which consist of finite subsets under sumset operation.

Contribution

It provides an analysis of the ascent of atomic and divisibility properties from linearly orderable monoids to their finitary power monoids.

Findings

Atomic properties do not necessarily ascend to finitary power monoids.

Divisibility properties can be preserved under certain conditions.

The paper characterizes when these properties are maintained in the power monoids.

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.