On the isomorphism problem of concept algebras

  • Authors:

  • Léonard Kwuida;Hajime Machida

  • Affiliations:

  • School of Engineering, Center of Applied Mathematics and Physics, Zurich University of Applied Sciences, Winterthur, Switzerland 8401;Department of Mathematics, Hitotsubashi University, Tokyo, Japan 186-8601

  • Venue:
  • Annals of Mathematics and Artificial Intelligence
  • Year:
  • 2010

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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 (Wille 2000; Kwuida 2004). 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 Kwuida (2004), is whether complete weakly dicomplemented lattices are isomorphic to concept algebras. In this contribution we give a negative answer to this question (Theorem 4). We also provide a new proof of a well known result due to M.H. Stone (Trans Am Math Soc 40:37---111, 1936), saying that each Boolean algebra is a field of sets (Corollary 4). Before these, we prove that the boundedness condition on the initial definition of weakly dicomplemented lattices (Definition 1) is superfluous (Theorem 1, see also Kwuida (2009)).