I have recently been studying affine semigroup (= Semigroups that is commutative, finitely generated, embeddable in a lattice) in the context of toric varieties (Cox, Little & Schenck). I was wondering where else in mathematics they 1) arise naturally, 2) are the center of study, maybe in other contexts than toric varieties and 3) If there are other topics/objects in mathematics that could be viewed from the perspective of the study of affine semigroups.
I have read some stuff about general semigroups and where they appear/are useful, however I hardly find anything about affine semigroups. (By that I mean the definition of an affine semigroup that is given by David Cox in the Book "toric varieties").
I would really appreciate some helpful insights, thank you :)