Ockham Algebras with Balanced Double Pseudocomplementation

Abstract

In this paper, we introduce a variety bdO of Ockham algebras with balanced double pseudocomplementation, consisting of those algebras ${(L; \wedge, \vee, f,\,^{*},^{+}, 0, 1)}$ of type ${\langle2,\,2,\,1,\,1,\,1,\,0,\,0\rangle}$ where ${(L; \wedge, \vee, f, 0, 1)}$ is an Ockham algebra, ${(L; \wedge, \vee, f,\,^{*},^{+}, 0, 1)}$ is a double p-algebra, and the operations ${x \mapsto f(x), x \mapsto x^{*}}$ and ${x \mapsto x^{+}}$ are linked by the identities [f(x)]* = [f(x)]⁺ = f ²(x), f(x*) = x ^** and f(x ⁺) = x ⁺⁺. We give a description of the congruences on the algebras, and show that there are precisely nine non-isomorphic subdirectly irreducible members in the class of the algebras via the Priestley duality. We also describe all axioms in the variety bdO, and provide a characterization of all subvarieties of bdO determined by 12 none-equivalent axioms, identifying therein the biggest subvariety in which every principal congruence is complemented.