On the atomicity of power monoids of Puiseux monoids

Abstract

A submonoid of the additive group $\mathbb{Q}$ is called a Puiseux monoid if it consists of nonnegative rationals. Given a monoid $M$, the set consisting of all nonempty finite subsets of $M$ is also a monoid under the Minkowski sum, and it is called the (finitary) power monoid of $M$. In this paper we study atomicity and factorization properties in power monoids of Puiseux monoids. We specially focus on the ascent of the property of being atomic and both the bounded and the finite factorization properties (the ascending chain on principal ideals and the length-finite factorization properties are also considered here). We prove that both the bounded and the finite factorization properties ascend from any Puiseux monoid to its power monoid. On the other hand, we construct an atomic Puiseux monoid whose power monoid is not atomic. We also prove that the existence of maximal common divisors for nonempty finite subsets is a sufficient condition for the property of being atomic to ascend from a Puiseux monoid to its power monoid.