Homomorphism-homogeneous partially ordered sets (Q2476634)

A homomorphism between posets \((A,\leq)\) and \((B,\leq)\) is a map preserving the ordering \(\leq\). A poset \(P\) is called homomorphism-homogeneous if every homomorphism \(A\to B\) of sub-posets \(A,B\subset P\) can be extended to a homomorphism \(P\to P\). After giving several examples, the author shows that there are five types of homomorphism-homogeneous posets: trees, dual trees, posets whose connected components are chains, posets splitting into a tree and a dual tree, and a class of locally bounded posets.