Perfect extensions of de Morgan algebras

TL;DR

This paper characterizes when a de Morgan algebra is a perfect extension of its Boolean subalgebra, using duality theory and Boolean product representations to provide a complete solution.

Contribution

It offers a full characterization of perfect extensions of de Morgan algebras by their Boolean skeletons, advancing understanding of algebraic extensions.

Findings

Provides necessary and sufficient conditions for perfect extensions

Utilizes duality theory and Boolean products in the analysis

Completes the classification of such extensions in de Morgan algebras

Abstract

An algebra $\mathbb A$ is called a perfect extension of its subalgebra $\mathbb B$ if every congruence of $\mathbb B$ has a unique extension to $\mathbb A$. This terminology was used by Blyth and Varlet [1994]. In the case of lattices, this concept was described by Gr\"atzer and Wehrung [1999] by saying that $\mathbb A$ is a congruence-preserving extension of $\mathbb B$. Not many investigations of this concept have been carried out so far. The present authors in another recent study faced the question of when a de Morgan algebra $\mathbb M$ is perfect extension of its Boolean subalgebra $B(\mathbb M)$, the so-called skeleton of $\mathbb M$. In this note a full solution to this interesting problem is given. The theory of natural dualities in the sense of Davey and Werner [1983] and Clark and Davey [1998], as well as Boolean product representations, are used as the main tools to obtain the solution.