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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.

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Acknowledgements

The authors express their thanks to the editor and the referee for useful suggestions concerning the final improvement of the text.

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Correspondence to Miroslav Haviar.

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Dedicated to the memory of Dr. Milan Demko (1963–2021)

Miroslav Haviar acknowledges his appointment as a Visiting Professor at the University of Johannesburg in the years 2020–2023. Miroslav Ploščica acknowledges support from Slovak grant VEGA 1/0097/18.

Communicated by Presented by B.A. Davey.

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Haviar, M., Ploščica, M. Perfect extensions of de Morgan algebras. Algebra Univers. 82, 58 (2021). https://doi.org/10.1007/s00012-021-00750-5

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Keywords

  • De Morgan algebra
  • Priestley space
  • MS-algebra
  • Boolean skeleton
  • Perfect extension

Mathematics Subject Classification

  • 06B10
  • 06D30