Distributive laws for concept lattices

AbstractWe 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)distributive,(2)a frame (locale, complete Heyting algebra),(3)isomorphic to a topology,(4)completely distributive,(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.

[1]  Michael W. Mislove,et al.  Local compactness and continuous lattices , 1981 .

[2]  Hilary A. Priestley,et al.  Ordered Topological Spaces and the Representation of Distributive Lattices , 1972 .

[3]  L. Nachbin.  Topology and order , 1965 .

[4]  Marcel Erné,et al.  Scott convergence and scott topology in partially ordered sets II , 1981 .

[5]  Marcel Erné,et al.  Standard completions for quasiordered sets , 1983 .

[6]  K. Hofmann,et al.  A Compendium of Continuous Lattices , 1980 .

[7]  Richard N. Ball,et al.  Distributive Cauchy lattices , 1984 .

[8]  George N. Raney,et al.  Completely distributive complete lattices , 1952 .

[9]  H. M. Macneille,et al.  Partially ordered sets , 1937 .

[10]  H. Bandelt.  Coproducts of bounded (α, β)-distributive lattices , 1983 .

[11]  Günter Bruns.  Verbandstheoretische Kennzeichnung vollständiger Mengenringe , 1959 .

[12]  ℜ-distributive lattices , 1982 .

[13]  R. P. Dilworth,et al.  Algebraic theory of lattices , 1973 .

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

[15]  Marcel Erné,et al.  The Dedekind-MacNeille completion as a reflector , 1991 .

[16]  Rudolf Wille,et al.  Tensorial decomposition of concept lattices , 1985 .

[17]  Rudolf-E. Hoffmann,et al.  Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications , 1981 .

[18]  Rudolf Wille,et al.  Subdirect decomposition of concept lattices , 1983, ICFCA.

[19]  Boris M. Schein.  Regular elements of the semigroup of all binary relations , 1976 .

[20]  George N. Raney,et al.  A subdirect-union representation for completely distributive complete lattices , 1953 .

[21]  Rudolf Wille,et al.  On finite lattices generated by their doubly irreducible elements , 1989, Discret. Math..