Skip to main content
SpringerLink
Log in

Algebraic Ordered Sets and Their Generalizations

  • Chapter
Algebras and Orders

Part of the book series: NATO ASI Series ((ASIC,volume 389))

Abstract

We study order-theoretical, algebraic and topological aspects of compact generation in ordered sets. Today, algebraic ordered sets (a natural generalization of algebraic lattices) have their place not only in classical mathematical disciplines like algebra and topology, but also in theoretical computer sciences. Some of the main statements are formulated in the language of category theory, because the manifold facets of algebraic ordered sets become more transparent when expressed in terms of equivalences between suitable categories. In the second part, collections of directed subsets are replaced with arbitrary selections of subsets Z. Many results on compactness remain true for the notion of Z-compactness, and the theory is now general enough to provide a broad spectrum of seemingly unrelated applications. Among other representation theorems, we present a duality theorem encompassing diverse specializations such as the Stone duality, the Lawson duality, and the duality between sober spaces and spatial frames.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 429.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 549.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Abian, S., and Brown, A.B., A theorem on partially ordered sets, with applications to fixed point theorems, Canad. J. Math. 13 (1961), 78–82.

    Article  MathSciNet  MATH  Google Scholar 

  2. Adämek, J., Herrlich, H., and Strecker, G., Abstract and Concrete Categories, Wiley-Interscience, New York 1990.

    Google Scholar 

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

    Google Scholar 

  4. Atherton, C.R., Concerning intrinsic topologies on Boolean algebras and certain bicompactly generated lattices, Glasgow Math. J. 11 (1970), 156–161.

    Article  MathSciNet  MATH  Google Scholar 

  5. Balbes, R., and Dwinger, Ph., Distributive Lattices, University of Missouri Press, Columbus, MO, 1974.

    MATH  Google Scholar 

  6. Banaschewski, B., Uber den Satz von Zorn, Math. Nachr. 10 (1953), 181–186.

    Article  MathSciNet  MATH  Google Scholar 

  7. Banaschewski, B., Hüllensysteme und Erweiterungen von Quasi-Ordnungen, Z. Math. Logik Grandlag. Math. 2 (1956), 369–377.

    MathSciNet  Google Scholar 

  8. Banaschewski, B., Essential extensions of To-spaces, General Topology and Appl. 7 (1977), 1–22.

    Article  MathSciNet  Google Scholar 

  9. Banaschewski, B., Prime elements from prime ideals, Order 2 (1985), 211–213.

    Article  MathSciNet  MATH  Google Scholar 

  10. Banaschewski, B., and Bruns, G., The fundamental duality of partially ordered sets, Order 5 (1988), 61–74.

    Article  MathSciNet  MATH  Google Scholar 

  11. Banaschewski, B., and Hoffmann, R.-E., (eds.), Continuous Lattices, Proc. Bremen 1979; Lecture Notes in Math. 871, Springer-Verlag, Berlin — Heidelberg — New York 1981.

    Google Scholar 

  12. Bandelt, H.-J., and Erné, M., The category of Z-continuous posets, J. Pure Appl. Algebra 30 (1983), 219–226.

    Article  MathSciNet  MATH  Google Scholar 

  13. Bandelt, H.-J., and Erné, M., Representations and embeddings of M-distributive lattices, Houston J. Math. 10 (1984), 315–324.

    MathSciNet  MATH  Google Scholar 

  14. Batbedat, A., Des schemas en demi-groupes commutatifs, Semigroup Forum 16 (1978), 473–481.

    Article  MathSciNet  MATH  Google Scholar 

  15. Birkhoff, G., Lattice Theory, 3rd ed., Amer. Math. Soc. Colloq. Publ. 25, Provi-dence, RI, 1973.

    Google Scholar 

  16. Birkhoff, G., and Frink, O., Representation of lattices by sets, Trans. Amer. Math. Soc. 64 (1948), 299–316.

    MathSciNet  MATH  Google Scholar 

  17. Blyth, T.S., and Janowitz, M.F., Residuation Theory, Pergamon Press, Oxford 1972. Algebraic Ordered Sets 187

    Google Scholar 

  18. Bourbaki, N., Sur le théorème de Zorn, Arch Math. (Basel) 2 (1949/50), 434–437.

    Google Scholar 

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

    Google Scholar 

  20. Crawley, P., and Dilworth, R.P., Algebraic Theory of Lattices, Prentice-Hall, En-glewood Cliffs, NJ, 1973.

    MATH  Google Scholar 

  21. Cohn, P.M., Universal Algebra, Harper and Row, New York 1965.

    MATH  Google Scholar 

  22. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., and Scott, D.S., A Compendium of Continuous Lattices, Springer-Verlag, Berlin — Heidelberg — New York 1980.

    Book  MATH  Google Scholar 

  23. Davey, B., and Priestley, H., Introduction to Lattices and Order, Cambridge Uni-versity Press, Cambridge 1990.

    MATH  Google Scholar 

  24. DE] David, E., and Erné, M., Ideal completion and Stone representation of ideal-distributive ordered sets, in: Topology and its Applications, Proc. Oxford Symposium (P. J. Collins, ed.), to appear.

    Google Scholar 

  25. Dilworth, R.P., and Gleason, A.M., A generalized Cantor theorem, Proc. Amer. Math. Soc. 13 (1962), 704–705.

    MathSciNet  MATH  Google Scholar 

  26. Erné, M., Order-topological lattices, Glasgow J. Math. 21 (1980), 57–68.

    Article  MATH  Google Scholar 

  27. Erné, M., Verallgemeinerungen der Verbandstheorie, I: Halbgeordnete Mengen und das Prinzip der Vervollständigungsinvarianz, Preprint No. 109, Inst. für Mathematik, Univ. Hannover 1980.

    Google Scholar 

  28. Erné, M., Verallgemeinerungen der Verbandstheorie, II: m-Ideale in halbgeordneten Mengen und Hüllenräumen, Habilitationsschrift, Inst. für Mathematik, Univ. Hannover 1980.

    Google Scholar 

  29. Erné, M., Homomorphisms of M-generated and M-distributive posets, Preprint No. 125, Inst. für Mathematik, Univ. Hannover 1981.

    Google Scholar 

  30. E5] Erné, M., A completion-invariant extension of the concept of continuous lattices, in [BH], 43–60.

    Google Scholar 

  31. E6] Erné, M., Scott convergence and Scott topology in partially ordered sets, in [BH], 61–96.

    Google Scholar 

  32. Erné, M., Distributivgesetze und die Dedekind’sche Schnittvervollständigung, Abh. Braunschweig. Wiss. Ges. 33 (1982), 117–145.

    MathSciNet  MATH  Google Scholar 

  33. Erné, M., Adjunctions and standard constructions for partially ordered sets, in: Contributions to General Algebra 2. Proc. Klagenfurt Conf. 1982 (G. Eigenthaler et al., eds.), Hölder-Pichler-Tempsky, Wien 1983, 77–106. 188 M. Erné

    Google Scholar 

  34. Erné, M., On the existence of decompositions in lattices, Algebra Universalis 16 (1983), 338–343.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  36. Erné, M., Chains, directed sets and continuity, Preprint No. 175, Inst. für Mathematik, Univ. Hannover 1984.

    Google Scholar 

  37. E12] Erné, M., Compactly generated and continuous closure systems, Memo Sem. Cont. Semilattices 5–7–84 (1984).

    Google Scholar 

  38. E13] Erné, M., Fixed point constructions for standard completions, Memo Sern. Cont. Semilattices 3–10–85 (1985).

    Google Scholar 

  39. Erné, M., Posets isomorphic to their extensions, Order 2 (1985), 199–210.

    Article  MathSciNet  MATH  Google Scholar 

  40. Erné, M., Order extensions as adjoint functors, Quaestiones Math. 9 (1986), 146206.

    Google Scholar 

  41. Erné, M., Compact generation in partially ordered sets, J. Austral. Math. Soc. 42 (1987), 69–83.

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Erné, M., The ABC of order and topology, in: Category Theory at Work ( H. Herrlich and H.-E. Porst, eds.), Heldermann, Berlin 1981, 57–83.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Fleischer, I., Even every join-extension solves a universal problem, J. Austral. Math. Soc. 21 (1976), 220–223.

    Article  MathSciNet  MATH  Google Scholar 

  47. Frink, O., Ideals in partially ordered sets, Amer. Math. Monthly 61 (1954), 223–234.

    Article  MathSciNet  MATH  Google Scholar 

  48. Geissinger, L., and Graves, W., The category of complete algebraic lattices, J. Combin. Theory Ser.A 13 (1972), 332–338.

    Article  MathSciNet  MATH  Google Scholar 

  49. Godement, R., Théorie des faisceaux, Hermann, Paris 1958.

    MATH  Google Scholar 

  50. Grätzer, G., On the family of certain subalgebras of a universal algebra, Indag. Math. 27 (1965), 790–802.

    Google Scholar 

  51. Grätzer, G., General Lattice Theory, Birkhäuser, Basel 1978.

    Book  Google Scholar 

  52. Gunter, C., Profinite Solutions for Recursive Domain Equations, Ph. D. Thesis, University of Wisconsin, Madison 1985.

    Google Scholar 

  53. Gunter, C., Comparing categories of domains, in: Mathematical foundations of programming semantics (A. Melton, ed.), Lecture Notes in Comput. Sci. 239, Springer-Verlag, Berlin — Heidelberg — New York 1985, 101–121.

    Google Scholar 

  54. Halpern, J.D., The independence of the axiom of choice from the Boolean prime ideal theorem, Fund. Math. 55 (1964), 57–66.

    MathSciNet  MATH  Google Scholar 

  55. Higgs, D., Lattices isomorphic to their ideal lattices, Algebra Universalis 1 (1971), 71–72.

    Article  MathSciNet  MATH  Google Scholar 

  56. Hoffmann, R.-E., Sobrification of partially ordered sets, Semigroup Forum 17 (1979), 123–138.

    Article  MathSciNet  MATH  Google Scholar 

  57. Hoffmann, R.-E., Continuous posets and adjoint sequences, Semigroup Forum 18 (1979), 173–188.

    Article  MathSciNet  MATH  Google Scholar 

  58. Hf3] Hoffmann, R.-E., Projective sober spaces, in [BH], 125–158.

    Google Scholar 

  59. Hf4] Hoffmann, R.-E., Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications, in [BH], 159–208.

    Google Scholar 

  60. Hoffmann, R.-E., Topological spaces admitting a dual, in: Categorical Topology (H. Herrlich and G. Preuß, eds.), Lecture Notes in Math. 719, Springer-Verlag, Berlin — Heidelberg — New York 1979, 157–166.

    Google Scholar 

  61. Hoffmann, R.-E., and Hofmann, K.H. (eds.), Continuous Lattices and Their Appli-cations, Proc. Bremen 1982, Lecture Notes in Pure and Appl. Math. 101, Marcel Decker, New York 1985.

    Google Scholar 

  62. HM1] Hofmann, K.H., and Mislove, M., Local compactness and continuous lattices, in [BH], 209–248.

    Google Scholar 

  63. HM2] Hofmann, K.H., and Mislove, M., Free objects in the category of completely distributive lattices, in [HH], 129–151.

    Google Scholar 

  64. Isbell, J.R., Directed unions and chains, Proc. Amer. Math. Soc. 17 (1966), 1467–1468.

    MATH  Google Scholar 

  65. Isbell, J.R., Completion of a construction of Johnstone, Proc. Amer. Math. Soc. 85 (1982), 333–334.

    Article  MathSciNet  MATH  Google Scholar 

  66. Iwamura, T., A lemma on directed sets, Zenkoku Shijo Sugaku Danwakai 262 (1944), 107–111 [Japanese].

    Google Scholar 

  67. Johnstone, P.T., Stone Spaces, Cambridge University Press, Cambridge 1982. 190 M. Erné

    Google Scholar 

  68. Johnstone, P.T., Scott is not always sober, in [BH], 282–283.

    Google Scholar 

  69. Jul Jung, A., Cartesian Closed Categories of Domains, Ph. D. Thesis, Technische Hochschule Darmstadt 1988.

    Google Scholar 

  70. Katrina, T., Pseudokomplementäre Halbverbände, Mat. Casopis 18 (1968), 121–143.

    Google Scholar 

  71. Krasner, M., Un type d’ensembles semi-ordonnés et ses rapports avec une hypothèse de M. A. Weil, Bull. Soc. Math. France 67 (1939), 162–176.

    MathSciNet  Google Scholar 

  72. Lawson, J.D., The duality of continuous posets, Houston J. Math. 5 (1979), 357–386.

    MathSciNet  MATH  Google Scholar 

  73. Lawson, J.D., The versatile continuous order, in: Lecture Notes in Comput. Sci. 298, Springer-Verlag, Berlin — Heidelberg — New York 1987, 134–160.

    Google Scholar 

  74. Levy, A., Axioms of multiple choice, Fund. Math. 50 (1962), 475–483.

    MathSciNet  MATH  Google Scholar 

  75. Maeda, F., Kontinuierliche Geometrien, Springer-Verlag, Berlin — Göttingen — Hei-delberg 1958.

    Book  Google Scholar 

  76. MacLane, S., Categories for the Working Mathematician, Springer-Verlag, Berlin — Heidelberg — New York 1971.

    Google Scholar 

  77. Markowsky, G., Chain-complete posets and directed sets with applications, Algebra Universalis 6 (1976), 53–68.

    Article  MathSciNet  MATH  Google Scholar 

  78. Mayer-Kalkschmidt, J., and Steiner, E., Some theorems in set theory and appli-cations in the ideal theory of partially ordered sets, Duke Math. J. 31 (1964), 287–290.

    Article  MathSciNet  MATH  Google Scholar 

  79. McKenzie, R.N., McNulty, G.F., and Taylor, W.F., Algebras, Lattices and Varieties I, Wadsworth and Brooks, Monterey, CA. 1987.

    Google Scholar 

  80. Meseguer, J., Order completion monads, Algebra Universalis 16 (1983), 63–82.

    Article  MathSciNet  MATH  Google Scholar 

  81. Moore, G.H., Zermelo’s Axiom of Choice, Springer-Verlag, Berlin — Heidelberg —New York 1982.

    Book  Google Scholar 

  82. Mostowski, A., Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungs-prinzip, Fund. Math. 32 (1939), 201–259.

    Google Scholar 

  83. Ne] Nelson, E., Z-continuous algebras, in [BH], 315–334.

    Google Scholar 

  84. Novak, D., Generalization of continuous posets, Trans. Amer. Math. Soc. 272 (1982), 645–667.

    Article  MathSciNet  MATH  Google Scholar 

  85. Papert, S., Which distributive lattices are lattices of closed sets? Proc. Cambridge Philos. Soc. 55 (1959), 172–176. Algebraic Ordered Sets 191

    Google Scholar 

  86. Plotkin, G., The category of complete partial orders: a tool for making meanings, in: Proc. Summer School on Foundations of Artificial Intelligence and Computer Science, Istituto di Scienze dell’ Informazione, Univ. di Pisa 1978.

    Google Scholar 

  87. Rosickÿ, J., On a characterization of the lattice of m-ideals of an ordered set, Arch. Math. (Brno) 8 (1972), 137–142.

    MathSciNet  Google Scholar 

  88. Schmidt, J., Über die Rolle der transfiniten Schlußweisen in einer allgemeinen Idealtheorie, Math. Nachr. 7 (1952), 165–182.

    Article  MathSciNet  MATH  Google Scholar 

  89. Schmidt, J., Einige grundlegende Begriffe und Sätze aus der Theorie der Hüllenoperatoren, Ber. Math. Tagung Berlin 1953, 21–48.

    Google Scholar 

  90. Schmidt, J., Each join-completion of a partially ordered set is the solution of a universal problem, J. Austral. Math. Soc. 17 (1974), 406–413.

    Article  MathSciNet  MATH  Google Scholar 

  91. Scott, D.S., Continuous lattices, in: Toposes, Algebraic Geometry and Logic (F.W. Lawvere, ed.), Lecture Notes in Math. 274, Springer-Verlag, Berlin — Heidelberg — New York 1972.

    Google Scholar 

  92. Scott, D.S., Domains for denotational semantics, in: ICALP 82 (M. Nielsen and E. M. Schmidt, eds.), Lecture Notes in Comput. Sci. 140, Springer-Verlag, Berlin Heidelberg New York 1982.

    Google Scholar 

  93. Smyth, M.B., The largest cartesian closed category of domains, Theoret. Comput. Sci. 27 (1983), 109–119.

    Article  MathSciNet  MATH  Google Scholar 

  94. Smyth, M.B., and Plotkin, G., The category theoretic solution of recursive domain equations, SIAM J. Comput. 11 (1982), 761–783.

    Article  MathSciNet  MATH  Google Scholar 

  95. Steen, L.A., and Seebach, J.A., Counterexamples in Topology, Holt, Rinehart & Winston, New York 1970.

    Google Scholar 

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

    Google Scholar 

  97. Stone, M.H., Topological representation of distributive lattices and Brouwerian logics, G’asopis Pest. Mat. 67 (1937), 1–25.

    Google Scholar 

  98. Stoy, J.E., Denotational Semantics: The Scott - Strachey Approach to Programming Language Theory, MIT Press, Cambridge, MA. 1977.

    Google Scholar 

  99. Szpilrajn, E., Sur l’extension de l’ordre partiel, Fund. Math. 16 (1930), 386–389.

    MATH  Google Scholar 

  100. Venugopalan, P., 2-continuous posets, Houston J. Math. 12 (1986), 275–294.

    MathSciNet  MATH  Google Scholar 

  101. Weihrauch, K., Computability,EATCS Monogr. Theoret. Comp. Sci. 9 Springer-Verlag, Berlin — Heidelberg — New York 1987. 192 M. Ern é

    Google Scholar 

  102. Witt, E., Beweisstudien zum Satz von M. Zorn, Math. Nachr. 4 (1951), 434–438.

    Article  MathSciNet  MATH  Google Scholar 

  103. Wright, J.B., Wagner, E.G., and Thatcher, J.W., A uniform approach to inductive posets and inductive closure, Theoret. Comput. Sci. 7 (1978), 57–77.

    Article  MathSciNet  MATH  Google Scholar 

  104. Zermelo, E., Beweis, daß jede Menge wohlgeordnet werden kann, Math. Ann. 59 (1904), 514–516.

    Article  MathSciNet  MATH  Google Scholar 

  105. Zermelo, E., Neuer Beweis für die Möglichkeit einer Wohlordnung, Math. Ann. 65 (1908), 107–128.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Hannover, Welfengarten 1, DW-3000, Hannover, Germany

    Marcel Erné

Authors
  1. Marcel Erné
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succ. A, H3C 3J7, Montréal, Québec, Canada

    Ivo G. Rosenberg (Scientific Director of the ASI) & Gert Sabidussi (Scientific Director of the ASI) & 

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Erné, M. (1993). Algebraic Ordered Sets and Their Generalizations. In: Rosenberg, I.G., Sabidussi, G. (eds) Algebras and Orders. NATO ASI Series, vol 389. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0697-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-0697-1_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4243-9

  • Online ISBN: 978-94-017-0697-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation