## Abstract

In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties 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 RDP1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP_{1}, LV (RDP_{1}), is isomorphic to the lattice of down sets of the poset {1} ⊕ (N × N). We describe all the subvarieties of RDP_{1} and conclude that L_{V} (RDP_{1}) is countably infinite. An equational basis for each proper subvariety of RDP_{1} is given. To study the subvarieties 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^{ℵ:0}. We also use Priestley duality to prove that RDP and each of its subvarieties RDP_{n} are generated by their finite members.

Original language | English |
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Pages (from-to) | 15-34 |

Number of pages | 20 |

Journal | MATHEMATICA SLOVACA |

Volume | 69 |

Issue number | 1 |

DOIs | |

Publication status | Published - 25 Feb 2019 |

## Keywords

- discriminator variety
- double Heyting algebra
- equational basis
- lattice of subvarieties
- Priestley duality
- regular double p-algebra
- simple algebra
- subdirectly irreducible algebra