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 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.