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

Abstract We describe how one can explicitly obtain all atoms of an arbitrary root-closed monoid, whose quotient group is isomorphic to $$\mathbb {Z}^2$$ Z2 . For this purpose, we solve task for three special types such monoids in Theorems 5 and 6, then transfer these results the general case. It turns out that be obtained from (regular) continued fraction expansion slopes bounding rays cone, which spanned by monoid.