Regular double p-algebras

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 RDP₁, LV (RDP₁), is isomorphic to the lattice of down sets of the poset {1} ⊕ (N × N). We describe all the subvarieties of RDP₁ and conclude that L_V (RDP₁) is countably infinite. An equational basis for each proper subvariety of RDP₁ 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.