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Regular double $p$-algebras

In: Mathematica Slovaca, vol. 69, no. 1
M. E. Adams - Hanamantagouda P. Sankappanavar - Júlia Vaz De Carvalho
Detaily:
Rok, strany: 2019, 15 - 34
Kľúčové slová:
regular double $p$-algebra, double Heyting algebra, discriminator variety, simple algebra, subdirectly irreducible algebra, equational basis, Priestley duality, lattice of subvarieties
O článku:
In this paper, we investigate the variety $\mathbf{RDP}$ of regular double $p$-algebras and its subvarieties $\mathbf{RDP}n$, $n≥ 1$, of range $n$. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety $\mathbf{RDP}1$ and show that this variety is locally finite. We also show that the lattice of subvarieties of $\mathbf{RDP}1$, $LV(\mathbf{RDP}1)$, is isomorphic to the lattice of down sets of the poset $\{1\}\oplus (\mathbb{N}× \mathbb{N})$. We describe all the subvarieties of $\mathbf{RDP}1$ and conclude that $LV(\mathbf{RDP}1)$ is countably infinite. An equational basis for each proper subvariety of $\mathbf{RDP}1$ is given. To study the subvarieties $\mathbf{RDP}n$ with $n≥ 2$, Priestley duality as it applies to regular double $p$-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its $1$-generated free algebra is infinite and that the lattice of its subvarieties has cardinality $2\aleph0$. We also use Priestley duality to prove that $\mathbf{RDP}$ and each of its subvarieties $\mathbf{RDP}n$ are generated by their finite members.
Ako citovať:
ISO 690:
Adams, M., Sankappanavar, H., Carvalho, J. 2019. Regular double $p$-algebras. In Mathematica Slovaca, vol. 69, no.1, pp. 15-34. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0200

APA:
Adams, M., Sankappanavar, H., Carvalho, J. (2019). Regular double $p$-algebras. Mathematica Slovaca, 69(1), 15-34. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0200
O vydaní:
Publikované: 24. 1. 2019