© The Author(s) 2024.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 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.