Endomorphism classes of ordered sets, graphs and lattices (Q1272942)

Mathematical structures of a certain type with an underlying set \(X\) are considered; they form the class \(C\). If \(S\in C\), then \(\text{End} S\) is the set of endomorphisms of \(S\). The following question is considered: Which structures \(S'\in C\) on the set \(X\) have the property that \(\text{End} S\subseteq\text{End} S'\)? If for some \(S\in C\) all structures from \(C\) on \(X\) have this property, then \(S\) is called rigid. The question is answered for reflexive partial orders, reflexive undirected graphs, irreflexive bipartite undirected graphs and distributive lattices.