Subvarieties of Pseudocomplemented Kleene Algebras

In this paper we study the subdirectly irreducible algebras in the variety \({\cal PCDM}\) of pseudocomplemented De Morgan algebras by means of their De Morgan \(p\)-spaces. We introduce the notion of \(body\) of an algebra \({\bf L} \in {\cal PCDM}\) and determine \(Body({\bf L})\) when \({\bf L}\)...

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Published in:arXiv.org 2020-06
Main Authors: Castaño, Diego, Castaño, Valeria, José Patricio Díaz Varela, Marcela Muñoz Santis
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Castaño, Valeria
José Patricio Díaz Varela
Marcela Muñoz Santis
description In this paper we study the subdirectly irreducible algebras in the variety \({\cal PCDM}\) of pseudocomplemented De Morgan algebras by means of their De Morgan \(p\)-spaces. We introduce the notion of \(body\) of an algebra \({\bf L} \in {\cal PCDM}\) and determine \(Body({\bf L})\) when \({\bf L}\) is subdirectly irreducible. As a consequence of this, in the case of pseudocomplemented Kleene algebras, three special subvarieties arise naturally, for which we give explicit identities that characterize them. We also introduce a subvariety \({\cal BPK}\) of \({\cal PCDM}\), namely the variety of \(bundle\) \(pseudocomplemented\) \(Kleene\) \(algebras\), determine the whole subvariety lattice and find explicit equational bases for each of the subvarieties. In addition, we study the subvariety \({\cal BPK}_0\) of \({\cal BPK}\) generated by the simple members of \({\cal BPK}\), determine the structure of the free algebra over a finite set and their finite weakly projective algebras.
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title Subvarieties of Pseudocomplemented Kleene Algebras
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