Order Preserving Maps of Posets

For any two finite posets $P$ and $Q$, let $\text{Hom}(P,Q)$ be the hom-poset consisting of all order preserving maps from $P$ to $Q$, and $J(Q)$ the collection of all order ideals of $Q$. In this paper, we study some basic properties of the hom-poset $\text{Hom}(P,Q)$ and prove that $\text{Hom}(P,J(Q))$ is a distributive lattice and characterized by $$\text{Hom}(P,J(Q))\cong J(P^{\ast }\times Q),$$ where $P^{\ast }$ is the dual of $P$. Consequently, we obtain that $\text{Hom}(P,J(Q))$ and $\text{Hom}(Q,J(P))$ are dual isomorphic, i.e., $$\text{Hom}(P,J(Q))\cong {\text{Hom}}^{\ast }(Q,J(P))\mathrm{.}$$ As applications, we calculate the number of order preserving maps from any poset to the boolean algebra, and the characteristic polynomial of $\text{Hom}(P,J(Q))$.