Perfect extensions of de Morgan algebras

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 $\mathbb A$, a terminology used by Blyth and Varlet [1994]. An another terminology saying in such case that $\mathbb A$ is a congruence-preserving extension of $\mathbb B$ was, in case of lattices, used by Gratzer and Wehrung [1999]. Not many investigations of this concept have been carried out so far. The present authors in their another recent study faced a question when a de Morgan algebra $\mathbb M$ is perfect extension of its Boolean subalgebra $B(\mathbb M)$, which is a so-called skeleton of $\mathbb M$. In this note a full solution to this interesting problem is given. 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 main tools to obtain the solution.