On some subvarieties of double Boolean algebras

We examine algebraic properties of subvarieties generated by finite double Boolean algebras, a class of structures introduced by Rudolf Wille as abstract models of protoconcept algebras in the same way as Boolean algebras capture the equational reasoning in powerset algebras. Protoconcepts are generalisations of concepts, i.e., associated pairs of closed sets of a Galois correspondence, in such a way that the two halves of a protoconcept are not necessarily closed, but represent a pair of generating sets for the two closed sets of the same concept. A semiconcept is a protoconcept where one half is Galois closed. Double Boolean algebras reflect the abstract equational laws of ‘meet’ (extensional conjunction), ‘join’ (intensional conjunction), two ways of complementation of protoconcepts—‘negation’ (extensional complement) and ‘opposition’ (intensional complement)—and two constants ‘all’ and ‘nothing’. All of these operations result in semiconcepts, and semiconcept algebras in turn lead to an abstract representation, which is more specific, namely the so-called pure double Boolean algebras. It was shown by Kwuida and Temgoua Alomo that the latter are congruence distributive, whereas the whole variety of double Boolean algebras is not. Motivated by this result we investigate whether there are other, perhaps weaker idempotent Maľcev conditions obeyed by the variety of double Boolean algebras or by important subvarieties. The sobering reality is that even very basic locally finite subvarieties fail to admit an idempotent weak near unanimity term, placing such algebras outside the Taylor class. This highlights intrinsic structural limitations of these algebras and thus challenges of Boolean concept logic when viewed through the lens of universal algebra.