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Distributive laws for concept lattices

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Abstract

We study several kinds of distributivity for concept lattices of contexts. In particular, we find necessary and sufficient conditions for a concept lattice to be

  1. (1)

    distributive,

  2. (2)

    a frame (locale, complete Heyting algebra),

  3. (3)

    isomorphic to a topology,

  4. (4)

    completely distributive,

  5. (5)

    superalgebraic (i.e., algebraic and completely distributive).

In cases (2), (4) and (5), our criteria are first order statements on objects and attributes of the given context. Several applications are obtained by considering the completion by cuts and the completion by lower ends of a quasiordered set as special types of concept lattices. Various degrees of distributivity for concept lattices are expressed by certain “separation axioms” for the underlying contexts. Passing to complementary contexts makes some statements and proofs more elegant. For example, it leads to a one-to-one correspondence between completely distributive lattices and so-called Cantor lattices, and it establishes an equivalence between partially ordered sets and doubly founded reduced contexts with distributive concept lattices.

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  1. Institut für Mathematik, Universität Hannover, Hannover, Germany

    Marcel Erné

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Erné, M. Distributive laws for concept lattices. Algebra Universalis 30, 538–580 (1993). https://doi.org/10.1007/BF01195382

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AMS Mathematics Subject Classification 1991

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