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,
it offers a lucid introduction to (different aspects of) the arithmetic theory of monoids (especially from the point of view of the classical theory of factorization).
From the reading of these papers, it will become clear that even a classification of rational Puiseux monoids is more or less hopeless (consistently with a statement made in a previous answer). However, some kind of classification is possible if the focus is restricted to certain families of rational Puiseux monoids. In particular, you may want to have a look at Sect. 3 of
- A. Geroldinger, F. Gotti, and S. Tringali, On strongly primary monoids, with a focus on Puiseux monoids, J. Algebra 567:1 (2021), 310-345,
where the focus is on the "strongly primary" case (an additively-written, commutative monoid $H$ is strongly primary if, for every $a \in H$, there is an integer $n \ge 1$ such that the $n$-fold sumset of the maximal ideal $H \setminus H^\times$ is contained in the coset $a+H$, where $H^\times$ is the group of units). For instance, one can show (loc. cit., Theorem 3.4) that a rational Puiseux monoid $H$ with non-empty conductor $$ (H : \widehat H) := \{x \in \mathsf q(H) \colon x + \widehat H \subseteq H\} $$ is strongly primary iff it satisfies the ACC on principal ideals, iff it is a bounded-factorization mononid, iff $0$ is not a limit point of the non-zero elements of $H$. (Here, $\mathsf q(H)$ is the quotient group of $H$ and $\widehat H$ is the complete integral closure of $H$, i.e., the set of all $x \in \mathsf q(H)$ for which there is an element $a \in H$ such that $a + nx \in H$ for all $n \in \mathbb N$.)