On the quotient of affine semigroups by a positive integer

Abstract 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 $$\mathcal {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> -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.