The ordered set of principal congruences of a countable lattice

摘要

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 set Princ(L) of a finite lattice L; here we do the same for a countable lattice. He also showed that every 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 a least element is the union of a chain of principal ideals (equivalently, if 0 H and H has a cofinal chain), then H is isomorphic to Princ(L) of some lattice L.