Has anyone described or catalogued all sets of non-negative real numbers that are closed under addition?
In recent years, there has been a lot of work on the arithmetic of Puiseux monoids, that is, submonoids of the positive cone of the additive group of a totally ordered field $K$: The focus has been mostly on rational Puiseux monoids, where $K$ is the rational field (with its usual ordering); but there are also a few papers about the general case. For further details, I can only recommend to have a look at the work of Felix Gotti et al., starting with
- S.T. Chapman, F. Gotti, and M. Gotti, When Is a Puiseux Monoid Atomic?, Amer. Math. Monthly 128:4 (2021), 302-321.
This article is a pleasure to read and, together with
- S.T. Chapman, F. Gotti, and M. Gotti, Factorization invariants of Puiseux monoids generated by geometric sequences, Commun. Algebra 48:1 (2020), 380-396,
offers a lucid introduction to (different aspects of) the classical theory of factorization.