  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.
