The ordered set of principal congruences of a countable lattice

Abstract: For a lattice L, let Princ L denote the ordered set of principal congruences of L. In a pioneering paper, G. Gratzer characterized the ordered sets Princ L of finite lattices L; here we do the same for countable lattices. He also showed that each bounded ordered set H is isomorphic to Princ L of a bounded lattice L. We prove a related statement: if an ordered set H with least element is the union of a chain of principal order ideals, then H is isomorphic to Princ L of some lattice L.