Atoms of root-closed submonoids of $\mathbb{Z}^2$

TL;DR

This paper provides a method to explicitly determine all atoms of root-closed submonoids of a5^2, linking the structure of these atoms to continued fraction expansions of the cone slopes.

Contribution

It introduces a novel approach to find atoms in root-closed monoids in a5^2, connecting geometric and number-theoretic techniques.

Findings

Atoms are characterized via continued fraction expansions.

Explicit descriptions are provided for three special monoid types.

Results are extended to the general case of such monoids.

Abstract

We describe how one can explicitly obtain all atoms of an arbitrary root-closed monoid, whose quotient group is isomorphic to $\mathbb{Z}^2$. For this purpose, we solve this task for three special types of such monoids in Theorems 5 and 6, and then transfer these results to the general case. It turns out that all atoms can be obtained from the (regular) continued fraction expansion of the slopes of the bounding rays of the cone, which is spanned by the monoid.