TITLE: Integer decomposition property with a partial order
QUESTION [9 upvotes]: 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. Also then the zeta polynomial of $(\mathcal{P}\cap\mathbb{Z}^n,\leq)$ is the Ehrhart polynomial of $\mathcal{P}$.

REPLY [2 votes]: Have you checked the family of marked order polytopes? These include the classical Gelfand-Tsetlin polytopes, and I think I can construct such a partial order in case of GT-polytopes.
Let $T \in kP_\lambda$, where $T$ is a GT-pattern, which is in bijection with some SSYT in $SSYT(k \lambda,m)$. We want to find a partial order, such that
$T=T_1 + \dotsb + T_k$ in a unique manner.
Well, the $T_i$ are elements in $SSYT(\lambda,m)$,
and $+$ is interpreted as concatenate-and-sort rows. So, if we take
the partial order on $SSYT(\lambda,m)$ where $T \leq T'$ iff
for each $i \leq j$, column $i$ from $T$ together with column $j$ from $T'$
form a valid SSYT when placed next to each other.
We can then find the unique decomposition $T=T_1 + \dotsb + T_k$
by letting $T_i$ be every $k$th column from $T$, starting with the $i$th one.
I think one can extend this construction to all marked order polytopes.