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Complete Congruences on Topologies and Down-set Lattices

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Abstract

From the work of Simmons about nuclei in frames it follows that a topological space X is scattered if and only if each congruence Θ on the frame of open sets is induced by a unique subspace A so that \(\Theta = \{ (U,V) | U\mathop{\cap} A = V\mathop{\cap} A\}\), and that the same holds without the uniqueness requirement iff X is weakly scattered (corrupt). We prove a seemingly similar but substantially different result about quasidiscrete topologies (in which arbitrary intersections of open sets are open): each complete congruence on such a topology is induced by a subspace if and only if the corresponding poset is (order) scattered, i.e. contains no dense chain. More questions concerning relations between frame, complete, spatial, induced and open congruences are discussed as well.

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References

  1. Alexandroff, P.: Diskrete Räume. Mat. Sb. (N.S.) 2, 501–518 (1937)

    MATH  Google Scholar 

  2. Aull, C.E., Thron, W.J.: Separation axioms between T0 and T1. Indag. Math. 24, 26–37 (1963)

    MathSciNet  Google Scholar 

  3. Banaschewski, B.: Hüllensysteme und Erweiterungen von Quasiordnungen. Z. Math. Log. Grundl. Math. 2, 369–377 (1956)

    MathSciNet  Google Scholar 

  4. Banaschewski, B.: Remarks on dual adjointness. Nordwestdeutsches Kategorienseminar, Bremen 1976. Math. Arbeitspapiere Universität Bremen. 7, 3–10 (1976)

    MathSciNet  Google Scholar 

  5. Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeille completion. Arch. Math. (Basel) 43, 369–377 (1967)

    MathSciNet  Google Scholar 

  6. Banaschewski, B., Harting, R.: Lattice aspects of radical ideals and choice principles. Proc. London Math. Soc. 50, 385–404 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  7. Banaschewski, B., Niefield, S.: Projective and supercoherent frames. J. Pure Appl. Algebra 70, 45–51 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Banaschewski, B., Pultr, A.: Adjointness aspects of the down-set functor. Appl. Categ. Structures 9, 419–436

  9. Bandelt, H.-J.: Regularity and complete distributivity. Semigroup Forum 19, 123–126 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bandelt, H.-J., Erné, M.: The category of \(\mathcal{Z}\)-continuous posets. J. Pure Appl. Algebra 30, 219–226 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  11. Birkhoff, G.: Lattice Theory, 3rd edn. AMS, Providence (1973)

    Google Scholar 

  12. Büchi, J.R.: Representations of complete lattices by sets. Portugal. Math. 11, 151–167 (1952)

    Google Scholar 

  13. Bruns, G.: Verbandstheoretische Kennzeichnung vollständiger Mengenringe. Arch. Math. (Basel) 10, 109–112 (1959)

    MATH  MathSciNet  Google Scholar 

  14. Crawley, P., Dilworth, R.P.: Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs, NJ (1973)

    MATH  Google Scholar 

  15. Davey, B., Priestley, H.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  16. Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symbolic Logic 70(3), 713–740 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Erné, M.: Adjunctions and standard constructions for partially ordered sets. In: Eigenthaler, G. et al. (eds.) Contributions to General Algebra, vol. 2, pp. 77–106. Hölder–Pichler–Tempsky, Wien (1983)

    Google Scholar 

  18. Erné, M.: Lattice representations for categories of closure spaces. In: Bentley, L.H. et al. (eds.) Categorical Topology, Proceedings Toledo, Ohio, 1983, pp. 197–222. Heldermann, Berlin (1984)

    Google Scholar 

  19. Erné, M.: Ordnungs- und Verbandstheorie. Fernuniversität Hagen, Germany (1987)

    Google Scholar 

  20. Erné, M.: The ABC of order and topology. In: Herrlich, H., Porst, H.-E. (eds.) Category Theory at Work, pp. 57–83. Heldermann Verlag, Berlin (1991)

    Google Scholar 

  21. Erné, M.: The Dedekind-MacNeille completion as a reflector. Order 8, 159–173 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  22. Erné, M.: Bigeneration in complete lattices and principal separation in partially ordered sets. Order 8, 197–221 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Erné, M.: Distributors and wallmanufacture. J. Pure Appl. Algebra 68, 109–125 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Erné, M.: Algebraic ordered sets and their generalizations. In: Rosenberg, I., Sabidussi, G. (eds.) Algebras and Orders. Proc. Montreal 1992. Kluwer, Amsterdam (1994)

    Google Scholar 

  25. Erné, M.: \(\mathcal{Z}\)-continuous posets and their topological manifestation. Appl. Categ. Structures 7, 31–70 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Erné, M.: Congruences on frames and superalgebraic lattices. University of Hannover (2005) (Preprint)

  27. Erné, M., Wilke, G.: Standard completions for quasiordered sets. Semigroup Forum 27, 351–376 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  28. Fraïssé, R.: Theory of Relations Studies in Logic, vol. 118. North-Holland, Amsterdam (1986)

    Google Scholar 

  29. Gehrke, M.: The order structure of Stone spaces and the T D -separation axiom. Z. Math. Logic Grundl. Math. 37, 5–15 (1991)

    MATH  MathSciNet  Google Scholar 

  30. Gehrke, M.: Generalized Kripke Semantics. 40 years of possible world semantics. Special issue of Stud. Log. (2006) (to appear)

  31. Gehrke, M., Jónsson, B.: Bounded distributive lattices expansions. Math. Scand. 94, 13–45 (2004)

    MATH  MathSciNet  Google Scholar 

  32. Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logics. Ann. Pure Appl. Logic 102(1–3), 65–102 (2005)

    Article  MathSciNet  Google Scholar 

  33. Gehrke, M., Priestley, H.A.: Canonical extensions of certain algebras with binary operations: an algebraic perspective on duality. J. Pure Appl. Algebra (to appear)

  34. Gehrke, M., Priestley, H.A.: Duality for certain algebras with binary operations via their canonical extensions. Stud. Log. (2006) (submitted)

  35. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: A Compendium of Continuous Lattices. Springer, Berlin Heidelberg New York (1980)

    MATH  Google Scholar 

  36. Goldblatt, R., Hodkinson, I., Venema, Y.: On canonical modal logics that are not elementarily generated. Logique et Anal. 181, 77–101 (2003)

    MathSciNet  Google Scholar 

  37. Hausdorff, F.: Grundzüge einer Theorie der geordneten Mengen. Math. Ann. 65, 435–505 (1908)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hoffmann, R.-E.: Bemerkungen über T D -Räume. Manuscripta Math. 12, 195–196 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  39. Hoffmann, R.-E.: Irreducible filters and sober spaces. Manuscripta Math. 22, 365–380 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  40. Hofmann, K.H., Mislove, M.: Free objects in the category of completely distributive lattices. In: Hoffmann, R.-E., Hofmann, K.H. (eds.) Continuous Lattices and Applications. Marcel Dekker, New York (1985)

    Google Scholar 

  41. Isbell, J.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)

    MATH  MathSciNet  Google Scholar 

  42. Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  43. Jónsson, B., Tarski, A.: Boolean algebras with operators, I. Amer. J. Math. 73, 891–939 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  44. Kříž, I., Pultr, A.: A spatiality criterion and a quasitopology which is not a topology. Houston J. Math. 15, 215–233 (1989)

    MathSciNet  MATH  Google Scholar 

  45. MacNeille, H.M.: Partially ordered sets. Trans. Amer. Math. Soc. 42, 416–460 (1937)

    Article  MATH  MathSciNet  Google Scholar 

  46. Menon, V.G.: Separating points in posets. Houston J. Math. 21, 283–290 (1995)

    MATH  MathSciNet  Google Scholar 

  47. Mislove, M.: Order scattered distributive continuous lattices are topologically scattered. Houston J. Math. 11, 559–573 (1985)

    MATH  MathSciNet  Google Scholar 

  48. Mislove, M.: When are order scattered and topologically scattered the same?. In: Pouzet, M., Richard, D. (eds.) Orders: Decription and Roles. North-Holland, Amsterdam (1984)

    Google Scholar 

  49. Niefield, S., Rosenthal, K.: Spatial sublocales and essential primes. Topology Appl. 26, 263–269 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  50. Priestley, H.A.: Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc. 24(3), 507–530 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  51. Pultr, A.: Frames. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 3, pp. 791–858. Elsevier, Amsterdam, The Netherlands (2003)

    Google Scholar 

  52. Pultr, A., Tozzi, A.: Separation axioms and frame representation of some topological facts. Appl. Categ. Structures 2, 107–118 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  53. Raney, G.N.: Completely distributive complete lattices. Proc. Amer. Math. Soc. 3, 667–680 (1952)

    Article  MathSciNet  Google Scholar 

  54. Raney, G.N.: A subdirect-union representation for completely distributive complete lattices. Proc. Amer. Math. Soc. 4, 518–522 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  55. Rosenthal, K.I.: Quantales and Their Applications. Longman, Harlow (1990)

    MATH  Google Scholar 

  56. Simmons, H.: The lattice theoretic part of topological separation axioms. Proc. Edinburgh Math. Soc. 21, 41–48 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  57. Simmons, H.: A framework for topology. In: Logic Colloquium 77, Studies in Logic and the Foundations of Mathematics, vol. 87, pp. 239–251. North-Holland, Amsterdam (1978)

    Google Scholar 

  58. Stone, M.H.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40, 37–111 (1936)

    Article  MATH  MathSciNet  Google Scholar 

  59. Stone, M.H.: Topological representation of distributive lattices and Brouwerian logics. Časopis Pěst. Mat. Fys. 67, 1–25 (1937)

    Google Scholar 

  60. Thron, W.J.: Lattice equivalence of topological spaces. Duke Math. J. 29, 671–679 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  61. Wallman, H.: Lattices and topological spaces. Ann. of Math. 39, 112–126 (1938)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty for Mathematics and Physics, University of Hannover, D30167, Hannover, Germany

    Marcel Erné

  2. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM88003, USA

    Mai Gehrke

  3. Department of Applied Mathematics and ITI, MFF, Charles University, CZ 11800, Prague, Czech Republic

    Aleš Pultr

Authors
  1. Marcel Erné
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  2. Mai Gehrke
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  3. Aleš Pultr
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Correspondence to Aleš Pultr.

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Dedicated to Bernhard Banaschewski.

The second author was partially supported by NSF grant no. NSF01-4- 21760, and both the second and the third author would like to express their thanks for support by the Carlsberg Foundation. The third author would like to express his thanks for support by project 1M0021620808 of the Ministry of Education of the Czech Republic.

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Erné, M., Gehrke, M. & Pultr, A. Complete Congruences on Topologies and Down-set Lattices. Appl Categor Struct 15, 163–184 (2007). https://doi.org/10.1007/s10485-006-9054-3

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