minimal system of generators Hil($M$) = {$b_1, b_2, \dots$} of $M$, the Hilbert bases.

Algorithms to find the Hilbert bases include the Normaliz algorithm [79] and the Hemmecke algorithm [80], which are available in the Normaliz package and the 4ti2 package, respectively.

**Theorem 7.**—(Gordan's Lemma.) Let $P \subseteq \mathbb{R}^d$ be a polyhedral cone. Then, $P \cap \mathbb{Z}^d$ is an affine monoid. And when $P$ is pointed, $P \cap \mathbb{Z}^d$ is a positive affine monoid.

**Definition 8.**—(Monoid homomorphisms.) A homomorphism between two affine monoids $M$ and $N$ is a function $f: M \to N$ such that (i) $f(x+y) = f(x)+f(y)$ for all $x, y$ in $M$, and (ii) $f(0) = 0$. A bijective monoid homomorphism is called a monoid isomorphism.

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*Correction:* Incorrect source information appeared in Ref. [84] and has been fixed. A new source (Ref. [85]) and its citation (Appendix E 2) were missing and have been inserted, and subsequent references renumbered.