×

zbMATH — the first resource for mathematics

Distributive laws for concept lattices. (English) Zbl 0795.06006
The central theme of this fascinating paper is an investigation of the connection between various types of distributive lattices and notions involving formal concept analysis as developed by R. Wille and his coworkers at Darmstadt. The paper is much too long and detailed to allow a detailed description of the results in this review. Suffice it to say that necessary and sufficient conditions are found for a concept lattice to be one of the following types: distributive, completely distributive, a frame, isomorphic to a topology, algebraic and completely distributive. Many other types of distributive lattices are characterized.

MSC:
06B23 Complete lattices, completions
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06D05 Structure and representation theory of distributive lattices
06D10 Complete distributivity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alexandroff, P.,Diskrete Räume. Mat. Sbornik (N.S.)2 (1937), 501-518. · Zbl 0018.09105
[2] Ball, R. N.,Distributive Cauchy lattices. Algebra Universalis18 (1984), 134-174. · Zbl 0539.06008 · doi:10.1007/BF01198525
[3] Banaschewski, B. andHoffmann, R.-E. (eds.),Continuous lattices, Proc. Bremen 1979. Lecture Notes in Math.871, Springer-Verlag, Berlin1981.
[4] Bandelt, H.-J.,Regularity and complete distributivity. Semigroup Forum19 (1980), 123-126. · Zbl 0429.20056 · doi:10.1007/BF02572509
[5] Bandelt, H.-J., ?-distributive lattices. Arch. Math.39 (1982), 436-442. · Zbl 0519.06005 · doi:10.1007/BF01899545
[6] Bandelt, H.-J.,Coproducts of bounded (?, ?)-distributive lattices. Algebra Universalis17 (1983), 92-100. · Zbl 0524.06018 · doi:10.1007/BF01194517
[7] Bandelt, H.-J. andErné, M.,Representations and embeddings of ?-distributivelattices. Houston J. Math.10 (1984), 315-324. · Zbl 0551.06014
[8] Birkhoff, G.,Lattice theory. Amer. Math. Soc. Coll. Publ.25, 3rd ed., Providence, R.I. 1973. · Zbl 0063.00402
[9] Bruns, G.,Verbandstheoretische Kennzeichnung vollständiger Mengenringe. Arch. Math. (Basel)10 (1954), 109-112. · Zbl 0086.25303
[10] Büchi, J. R.,Representations of complete lattices by sets. Portugal. Math.11 (1952), 151-167.
[11] Crawley, P.,Regular embeddings which preserve lattice structure. Proc. Amer. Math. Soc.13 (1962), 748-752. · Zbl 0112.01803 · doi:10.1090/S0002-9939-1962-0140451-2
[12] Crawley, P. andDilworth, R. P.,Algebraic theory of lattices. Prentice-Hall Inc., Englewood Cliffs, N.J. 1973.
[13] Erné, M.,Verallgemeinerungen der Verbandstheorie. Preprint No.109 and Habilitationsschrift, University of Hannover 1980.
[14] Erné, M.,Scott convergence and Scott topology in partially ordered sets II. In [3], 61-96.
[15] Erné, M.,Homomorphisms of ?-generated and ?-distributive posets. Preprint No.125, University of Hannover 1981.
[16] Erné, M.,Distributivgesetze und die Dedekindsche Schnittvervollständigung. Abh. Braunschweig. Wiss. Ges.33 (1982), 117-145.
[17] Erné, M.,The ABC of order and topology. In: H. Herrlich and H.-E. Porst (eds.),Category theory at work. Heldermann, Berlin 1991, 57-83. · Zbl 0735.18005
[18] Erné, M.,The Dedekind-MacNeille completion as a reflector. Preprint1183, Techn. Hochschule Darmstadt 1988, and Order5 (1991), 159-173. · Zbl 0738.06004
[19] Erné, M.,Bigeneration in complete lattices and principal separation in partially ordered sets. Preprint1182, Techn. Hochschule Darmstadt 1988, and Order8 (1991), 197-221. · Zbl 0738.06005
[20] Erné, M. andWilke, G.,Standard completions for quasiordered sets. Semigroup Forum27 (1983), 351-376. · Zbl 0517.06009 · doi:10.1007/BF02572747
[21] Ganter, B., Wille, R. andWolff, H. E. (eds.),Beiträge zur Begriffsanalyse B. I. Wissenschaftsverlag, Mannheim 1987.
[22] Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. andScott, D. S.,A compendium of continuous lattices. Springer-Verlag, Berlin 1980. · Zbl 0452.06001
[23] Hickman, R. C. andMonro, G. P.,Distributive partially ordered sets. Fund. Math.CXX (1984), 151-166. · Zbl 0553.06005
[24] Hoffmann, R.-E.,Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications. In [3], 159-208.
[25] Hofmann, K. H. andMislove, M.,Local compactness and continuous lattices. In [3], 209-248.
[26] Katrinak, T.,Pseudokomplementäre Halbverbände. Mat. ?asopis18 (1968), 121-143.
[27] K?í?, I. andPultr, A.,A spatiality criterion and an example of a quasitopology which is not a topology. Houston J. Math.15 (1989), 215-234. · Zbl 0695.54002
[28] Lawson, J. D.,The duality of continuous posets. Houston J. Math.5 (1979), 357-386. · Zbl 0428.06003
[29] MacNeille, H. M.,Partially ordered sets. Trans. Amer. Math. Soc.42 (1937), 416-460. · Zbl 0017.33904 · doi:10.1090/S0002-9947-1937-1501929-X
[30] Monjardet, B. andWille, R.,On finite lattices generated by their doubly irreducible elements. Discr. Math.73 (1989), 163-164. · Zbl 0663.06008 · doi:10.1016/0012-365X(88)90144-6
[31] Nachbin, L.,Topology and order. Van Nostrand, Princeton 1965. · Zbl 0131.37903
[32] Priestley, H. A.,Ordered topological spaces and the representation of distributive lattices. Proc. Lond. Math. Soc. (3)24 (1972), 507-530. · Zbl 0323.06011 · doi:10.1112/plms/s3-24.3.507
[33] Raney, G. N.,Completely distributive complete lattices. Proc. Amer. Math. Soc.3 (1952), 677-680. · Zbl 0049.30304 · doi:10.1090/S0002-9939-1952-0052392-3
[34] Raney, G. N.,A subdirect union representation for completely distributive complete lattices. Proc. Amer. Math. Soc.4 (1953), 518-522. · Zbl 0053.35201 · doi:10.1090/S0002-9939-1953-0058568-4
[35] Schein, B. M.,Regular elements of the semigroup of all binary relations. Semigroup Forum13 (1976), 95-102. · Zbl 0355.20058 · doi:10.1007/BF02194925
[36] Wille, R.,Restructuring lattice theory: an approach based on hierarchies of concepts. In: I. Rival (ed.),Ordered sets. Reidel, Dordrecht-Boston 1982, 445-470. · Zbl 0491.06008
[37] Wille, R.,Subdirect decomposition of concept lattices. Algebra Universalis17 (1983), 275-287. · Zbl 0539.06006 · doi:10.1007/BF01194537
[38] Wille, R.,Tensorial decomposition of concept lattices. Order2 (1985), 81-95. · Zbl 0583.06007 · doi:10.1007/BF00337926
[39] Zaretzkii, K. A.,The semigroup of binary relations. Mat. Sbornik61 (1963), 291-305 [Russian].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.