  In this paper, we explore affine semigroup versions of the convex geometry
theorems of Helly, Tverberg, and Caratheodory. Additionally, we develop a new
theory of colored affine semigroups, where the semigroup generators each
receive a color and the elements of the semigroup take into account the colors
used (the classical theory of affine semigroups coincides with the case in
which all generators have the same color). We prove an analog of Tverberg's
theorem and colorful Helly's theorem for semigroups, as well as a version of
colorful Caratheodory's theorem for cones. We also demonstrate that colored
numerical semigroups are particularly rich by introducing a colored version of
the Frobenius number.
