Perfect extensions of 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 Grätzer 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.