On the isomorphism problem of concept algebras

TL;DR

This paper investigates the isomorphism problem for concept algebras, providing a negative answer to whether complete weakly dicomplemented lattices are always concept algebras, and offers new proofs of classical results like Stone's theorem.

Contribution

It demonstrates that not all complete weakly dicomplemented lattices are concept algebras and simplifies the conditions needed for their definition.

Findings

Complete weakly dicomplemented lattices are not necessarily concept algebras.

A new proof of Stone's theorem that Boolean algebras are fields of sets.

The boundedness condition in weakly dicomplemented lattices is unnecessary.

Abstract

Weakly dicomplemented lattices are bounded lattices equipped with two unary operations to encode a negation on {\it concepts}. They have been introduced to capture the equational theory of concept algebras \cite{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 \cite{Kw04}, is whether complete {\wdl}s are isomorphic to concept algebras. In this contribution we give a negative answer to this question (Theorem \ref{T:main}). We also provide a new proof of a well known result due to M.H. Stone \cite{St36}, saying that {\em each Boolean algebra is a field of sets} (Corollary \ref{C:Stone}). Before these, we prove that the boundedness condition on the initial definition of {\wdl}s (Definition \ref{D:wdl}) is superfluous (Theorem \ref{T:wcl}, see also \cite{Kw09}).