Poset ideals of P-partitions and generalized letterplace and determinantal ideals

For any finite poset $P$ we have the poset of isotone maps $\text{Hom}(P,\mathbb{N})$, also called $P^{op}$-partitions. To any poset ideal $\mathcal{J}$ in $\text{Hom}(P,\mathbb{N})$, finite or infinite, we associate monomial ideals: the letterplace ideal $L(\mathcal{J},P)$ and the Alexander dual co-letterplace ideal $L(P,\mathcal{J})$, and study them. We derive a class of monomial ideals in $k[x_p,p\in P]$ called $P$-stable. When $P$ is a chain we establish a duality on strongly stable ideals. We study the case when $\mathcal{J}$ is a principal poset ideal. When $P$ is a chain we construct a new class of determinantal ideals which generalizes ideals of {\it maximal} minors and whose initial ideals are letterplace ideals of prinicpal poset ideals.