On the isomorphism problem of concept algebras
Abstract
Weakly dicomplemented lattices are bounded lattices equipped with two unary operations to encode a negation on concepts. They have been introduced to capture the equational theory of concept algebras Wi00. They generalize Boolean algebras. Concept algebras are concept lattices, thus complete lattices, with a weak negation and a weak opposition. A special case of the representation problem for weakly dicomplemented lattices, posed in Kw04, is whether complete s are isomorphic to concept algebras. In this contribution we give a negative answer to this question (Theorem T:main). We also provide a new proof of a well known result due to M.H. Stone St36, saying that each Boolean algebra is a field of sets (Corollary C:Stone). Before these, we prove that the boundedness condition on the initial definition of s (Definition D:wdl) is superfluous (Theorem T:wcl, see also Kw09).