TITLE: What is the combinatorial data classifying non-normal affine toric varieties?
QUESTION [6 upvotes]: Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I take to be algebraically closed). 
It is classical that classifying normal affine toric varieties is the same thing as classifying rational polyhedral cones. The latter is a classical part of discrete and combinatorial geometry.   (Apologies for overusing the word "classical".)


 Question. Is there some combinatorial data which classifies non-normal affine toric varieties?


Background and what I am not looking for. I am not an algebraic geometer. I work a lot with non-commutative semigroups. So often people asked me what is the classification of finitely generated cancellative commutative monoids, expecting an answer like the group case. The category of affine monoids (=finitely generated submonoids of free abelian groups) is dual to the category of affine toric varieties, hence my question. The  normal affine toric varieties correspond to saturated or normal affine monoids (also called integral polyhedral cones by some). For this reason I don't consider affine monoids as an answer to the question.

REPLY [2 votes]: There is paper of Patrick Popescu-Pampu and Dmitry Stepanov called Local tropicalization, where they propose a different notion of a fan that pretends to encode nonnormal toric varieties, namely a fan of semigroups, defined as a lattice fan $\Delta$ endowed with an affine subsemigroup (affine semigroups are pointed, commutative cancellative, finite type and torsion free) $\Gamma_{\alpha}$ of $\Gamma$ for each cone $\sigma\in\Delta$ with:
i) For each cone $\sigma\in \Delta$, $\sigma(\Gamma_{\alpha})=\alpha$ and $M(\Gamma_{\alpha})=M$.
ii) $\beta$ is a face $\alpha\in \Delta$, then $\Gamma_{\beta}=\Gamma_{\alpha}+M(\beta,\Gamma_{\alpha})$.
I think this can help