On the isomorphism problem of concept algebras

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)).

[1]  Marcel Erné,et al.  Negations and contrapositions of complete lattices , 1998, Discret. Math..

[2]  Rudolf Wille,et al.  Boolean Concept Logic , 2000, ICCS.

[3]  R. P. Dilworth.  Lattices with Unique Complements , 1945 .

[4]  Richard Holzer,et al.  Treating Incomplete Knowledge in Formal Concept Analysis , 2005, Formal Concept Analysis.

[5]  Rokia Missaoui,et al.  Generating Positive and Negative Exact Rules Using Formal Concept Analysis: Problems and Solutions , 2008, ICFCA.

[6]  G. Boole.  An Investigation of the Laws of Thought: On which are founded the mathematical theories of logic and probabilities , 2007 .

[7]  Bernhard Ganter,et al.  Finite Distributive Concept Algebras , 2006, Order.

[8]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[9]  Rudolf Wille.  Restructuring mathematical logic: an approach based on Peirce's pragmatism , 1996 .

[10]  M. Stone.  The theory of representations for Boolean algebras , 1936 .

[11]  Rudolf Wille.  Preconcept Algebras and Generalized Double Boolean Algebras , 2004, ICFCA.

[12]  Vincent Duquenne,et al.  Familles minimales d'implications informatives résultant d'un tableau de données binaires , 1986 .

[13]  Erkko Lehtonen,et al.  On the Homomorphism Order of Labeled Posets , 2009, Order.

[14]  L. Beran,et al.  [Formal concept analysis]. , 1996, Casopis lekaru ceskych.

[15]  Bernhard Ganter,et al.  Formal Concept Analysis: Mathematical Foundations , 1998 .

[16]  Rudolf Wille,et al.  The Basic Theorem on Preconcept Lattices , 2006, ICFCA.

[17]  William M. Tepfenhart,et al.  Conceptual Structures: Standards and Practices , 1999, Lecture Notes in Computer Science.

[18]  Uta Priss,et al.  Formal concept analysis in information science , 2006, Annu. Rev. Inf. Sci. Technol..

[19]  Hilary A. Priestley,et al.  Representation of Distributive Lattices by means of ordered Stone Spaces , 1970 .

[20]  Bernhard Ganter,et al.  Contextual Attribute Logic , 1999, ICCS.

[21]  Rudolf Wille,et al.  Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts , 2009, ICFCA.

[22]  Sébastien Ferré,et al.  Negation, Opposition, and Possibility in Logical Concept Analysis , 2006, ICFCA.

[23]  Rudolf Wille,et al.  Algebras of semiconcepts and double Boolean algebras , 2000 .

[25]  George Boole,et al.  An Investigation of the Laws of Thought: Frontmatter , 2009 .

[26]  Laurence R. Horn.  A Natural History of Negation , 1989 .

[27]  M. Stone,et al.  The Theory of Representation for Boolean Algebras , 1936 .

[28]  Rudolf Wille,et al.  Semiconcept and Protoconcept Algebras: The Basic Theorems , 2005, Formal Concept Analysis.

[29]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[30]  H. Wansing,et al.  Negation : a notion in focus , 1996 .