On the quotient of affine semigroups by a positive integer

This work delves into the quotient of an affine semigroup by a positive integer, exploring its intricate properties and broader implications. We unveil an 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 \(\mathcal {C}\) $\mathcal{C}$ -semigroup quotients. To shed light on these concepts, we introduce the novel notion of an arithmetic variety of affine semigroups, accompanied by illuminating examples that showcase its power.