On the quotient of affine semigroups by a positive integer

This work delves into the {\it quotient of an affine semigroup by a positive integer}, exploring its intricate properties and broader implications. We unveil an {\it associated tree} that serves as a valuable tool for further analysis. Moreover, we successfully generalize several key irreducibility results, extending their applicability to the more general class of $\mathcal{C}$-semigroup quotients. To shed light on these concepts, we introduce the novel notion of an {\it arithmetic variety of affine semigroups}, accompanied by illuminating examples that showcase its power.