Increasing positive monoids of ordered fields are FF-monoids

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.