The ordered set of principal congruences of a countable lattice. (Q292837)

For a lattice \(L\), denote by \(\mathrm{Princ}(L)\) the ordered set of principal congruences. It was proved by G. Grätzer that each bounded poset \(P\) is isomorphic to \(\mathrm{Princ}(L)\) for some bounded lattice \(L\). Motivated by this, the author proves that if \(H\) is an ordered set with a least element which is the union of a chain of principal ideals, then \(H\) is isomorphic to \(\mathrm{Princ}(L)\) for some lattice \(L\).