For any finite poset $P$, we have the poset of isotone maps $\mathrm{Hom}(P,\mathbb{N})$, also called $P^{\mathrm{op}}$-partitions. To any poset ideal $\mathcal{J}$ in $\mathrm{Hom}(P,\mathbb{N})$, 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 $\mathbb{k}[x_p,p\in P]$ 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 maximal minors and whose initial ideals are letterplace ideals of principal poset ideals.