  We study 2-cocycle twists, or equivalently Zhang twists, of semigroup
algebras over a field k. If the underlying semigroup is affine, that is
abelian, cancellative and finitely generated, then Spec k[S] is an affine toric
variety over k, and we refer to the twists of k[S] as quantum affine toric
varieties. We show that every quantum affine toric varieties has a "dense
quantum torus", in the sense that it has a localization isomorphic to a quantum
torus. We study quantum affine toric varieties and show that many geometric
regularity properties survive the deformation process.
