Discrete ordered sets whose covering graphs are median
Hans-J. Bandelt · Proceedings of the American Mathematical Society · 1984
As is well known the covering graph (= Hasse diagram) of any median semilattice is a median graph, and every median graph is the covering graph of some median semilattice. The purpose of this note is to prove that an ordered set is a median semilattice whenever (i) no interval contains an infinite chain, (ii) each pair of elements is bounded below, and (iii) the covering graph is median.