Varieties of Regular Pseudocomplemented de Morgan Algebras
In this paper, we investigate the varieties Mn and Kn of regular pseudocomplemented de Morgan and Kleene algebras of range n, respectively. Priestley duality as it applies to pseudocomplemented de Morgan algebras is used. We characterise the dual spaces of the simple (equivalently, subdirectly irreducible) algebras in Mn and explicitly describe the dual spaces of the simple algebras in M1 and K1. We show that the variety M1 is locally finite, but this property does not extend to Mn or even Kn for n ≥ 2. We also show that the lattice of subvarieties of K1 is an ω + 1 chain and the cardinality of the lattice of subvarieties of either K2 or M1 is 2ω. A description of the lattice of subvarieties of M1 is given.
KeywordsVariety of algebras Lattice of subvarieties Regular pseudocomplented de Morgan algebra (of range n) Discriminator variety Simple algebra Subdirectly irreducible algebra Priestley duality
Mathematics Subject Classification (2010)Primary: 06D30, 06D15, 03G25 Secondary: 08B15, 06D50, 03G10
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This work was partially supported by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the projects UID/MAT/00297/2013 and UID/MAT/00297/2019 (Centro de Matemática e Aplicações).
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