Definition
A toric monoid is a monoid \(P \) which is
- sharp/unit free (i.e. \(P^* = \{0\}\))
- finitely generated
- integral/cancelative (\(P \to P^{gp}\) is injective, or \(ab = cb \implies a = c\) for all \(a,b,c\in P\))
- saturated (for \(d\in \mathbb N\) and \(p \in P^{gp}\), \(d\cdot p \in P \implies p \in P\). This makes sense only when \(P \) is also integral.)
Alternatively, \(P \) is a toric monoid if and only if \(P = P^{gp} \cap \sigma\) for some rational strictly convex polyhedral cone \(\sigma \subseteq P^{gp} \otimes \mathbb R\).
See also monoid adjectives.