Increasing positive monoids of ordered fields are FF-monoids

TL;DR

This paper proves that increasing positive monoids in ordered fields are FF-monoids and hereditarily atomic, extending atomic property results from rational numbers to Archimedean fields.

Contribution

It establishes that all increasing positive monoids in ordered fields are FF-monoids and hereditarily atomic, generalizing atomic features to broader ordered field contexts.

Findings

Increasing positive monoids are FF-monoids.

Such monoids are hereditarily atomic.

Generalizations may fail outside Archimedean fields.

Abstract

Given an ambient ordered field $K$, a positive monoid is a countably generated additive submonoid of the nonnegative cone of $K$. In this paper, we first generalize several atomic features exhibited by Puiseux monoids of the field of rational numbers to the more general setting of positive monoids of Archimedean fields, accordingly arguing that such generalizations may fail if the ambient field is not Archimedean. In particular, we show that a positive monoid $P$ of an Archimedean field is a BF-monoid provided that $P \! \setminus \! \{0\}$ does not have $0$ as a limit point. Then, we prove our main result: every increasing positive monoid of an ordered field is an FF-monoid. Finally, we deduce that every increasing positive monoid is hereditarily atomic.