When is a concept algebra boolean?

Abstract

Concept algebras are concept lattices enriched by a weak negation and a weak opposition. The introduction of these two operations was motivated by the search of a negation on formal concepts. These weak operations form a weak dicomplementation. A weakly dicomplemented lattice is a bounded lattice equipped with a weak dicomplementation. (Weakly) dicomplemented lattices abstract (at least for finite distributive lattices) concept algebras. Distributive double p-algebras and Boolean algebras are some special subclasses of the class of weakly dicomplemented lattices. We investigate in the present work the connection between weak dicomplementations and complementation notions like semicomplementation, pseudocomplementation, complementation or orthocomplementation.