Homomorphisms and principal congruences of bounded lattices. III. The Independence Theorem

TL;DR

This paper presents an alternative proof of a theorem showing that for any ordered set with at least two elements and any group, there exists a bounded lattice whose principal congruences and automorphism group are isomorphic to them.

Contribution

It offers a new proof of Czédli's theorem using a historical result, expanding understanding of lattice automorphisms and principal congruences.

Findings

Existence of bounded lattices with prescribed principal congruences

Construction of lattices with automorphism groups isomorphic to any given group

Alternative proof method based on a 1960s result

Abstract

A new result of G. Cz\'edli states that for an ordered set $P$ with at least two elements and a group $G$, there exists a bounded lattice $L$ such that the ordered set of principal congruences of $L$ is isomorphic to $P$ and the automorphism group of $L$ is isomorphic to $G$. I provide an alternative proof utilizing a result of mine with J. Sichler from the late 1960-s.