Perfect extensions of de Morgan algebras

Abstract

An algebra \(\mathbf{A}\) is called a perfect extension of its subalgebra \(\mathbf{B}\) if every congruence of \(\mathbf{B}\) has a unique extension to \(\mathbf{A}\). 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 \(\mathbf{A}\) is a congruence-preserving extension of \(\mathbf{B}\). 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 \(\mathbf{M}\) is perfect extension of its Boolean subalgebra \(B(\mathbf{M})\), the so-called skeleton of \(\mathbf{M}\). 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.