Let $\mathcal{P}$ be a convex lattice polytope in $\mathbb{R}^n$. We say that $\mathcal{P}$ has the  *integer decomposition property* (or "is IDP") if for all $k\in \mathbb{N}$ and $\alpha \in k\mathcal{P}\cap\mathbb{Z}^n$, there are $\alpha_1,\ldots,\alpha_k \in \mathcal{P}\cap\mathbb{Z}^n$ such that $\alpha=\alpha_1+\cdots+\alpha_k$. IDP polytopes are a well-studied class with connections to commutative algebra, algebraic geometry, etc.

Here's a new/nonstandard notion. Let's say $\mathcal{P}$ is IDP$\leq$ if there exists a partial order $\leq$ on $\mathcal{P}\cap\mathbb{Z}^n$ such that for all $k\in \mathbb{N}$ and $\alpha \in k\mathcal{P}\cap\mathbb{Z}^n$, there is a **unique** multichain $\alpha_1 \leq \cdots \leq \alpha_k \in \mathcal{P}\cap\mathbb{Z}^n$ with $\alpha=\alpha_1+\cdots+\alpha_k$.

For example, the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ of a poset $P$ are IDP$\leq$ where $\leq$ is the natural distributive lattice order (on order filters/antichains).

**Question**: Are there other interesting families of IDP$\leq$ polytopes?

Motivation: this property gives a canonical/algorithmic choice of decomposition for all latice points of dilates.