  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.
