The ordered set of principal congruences of a countable lattice

TL;DR

This paper extends Gratzer's characterization of principal congruence sets from finite to countable lattices, showing that certain ordered sets can be realized as principal congruences of some lattice.

Contribution

It generalizes Gratzer's results to countable lattices, establishing conditions under which an ordered set is isomorphic to the principal congruences of a lattice.

Findings

Characterization of principal congruence sets for countable lattices

Proof that certain unions of chains of order ideals correspond to principal congruences

Extension of bounded ordered set realizations to countable lattices

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.