Congruences on ockham algebras with pseudocomplementation

Abstract

The variety pOconsists of those algebras (L;⋁,⋀,f,^*,0,1) where (L;⋁,⋀,f,0,1) is an Ockham algebra, (L;⋁,⋀,f,^*,0,1) is a p-algebra, and the unary operations fand ^*. commute. For an algebra in pK _ωwe show that the compact congruences form a dual Stone lattice and use this to determine necessary and sufficient conditions for a principal congruence to be complemented. We also describe the lattice of subvarieties of pK _1,1identifying therein the biggest subvariety in which every principal congruence is complemented, and the biggest subvariety in which the intersection of two principal congruences is principal.