Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T07:13:43.890Z Has data issue: false hasContentIssue false
  • English
  • Français

Order-Topological lattices

Published online by Cambridge University Press:  18 May 2009

Marcel Erné
Marcel Erné
Affiliation:
Institut Für Mathematik, Universität Hannover, D 3 Hannover, Fed. Rep. of Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The observation that convergence of real sequences may be defined in terms of limits inferior and limits superior as by means of neighbourhoods in the Euclidean topology leads to the question: for which lattices does order convergence coincide with convergence in the order topology? This problem has been attacked by D. C. Kent [10], A. Gingras [7] and others. We hope to present a satisfactory solution in this paper. Although there are known several characterizations of lattices, with topological order convergence (cf. Propositions 1, 2), an evaluation of these criteria already requires some knowledge of the order topology of the given lattice. In the present paper, we establish a purely lattice-theoretical description of those lattices for which order convergence is not only topological, but moreover, the lattice operations are continuous. Henceforth, such lattices will be referred to as order-topological lattices. All convergence statements will be formulated in terms of filters rather than nets. For an introduction to convergence functions, the reader may consult D. C. Kents's paper [9].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 21 , Issue 1 , January 1980 , pp. 57 - 68
Copyright
Copyright © Glasgow Mathematical Journal Trust 1980

References

REFERENCES

1.Atherton, C. R., Concerning intrinsic topologies on Boolean algebras and certain bicompactly generated lattices, Glasgow Math. J. 11 (1970), 156–161.CrossRefGoogle Scholar
2.Balbes, R. and Dwinger, Ph., Distributive lattices (University of Missouri Press, Columbia, Missouri, 1974).Google Scholar
3.Crawley, P. and Dilworth, R. P., Algebraic theory of lattices (Prentice Hall, 1973).Google Scholar
4.Erné, M., Topologies on products of partially ordered sets III: Order convergence and order topology, to appear in Algebra Universalis.Google Scholar
5.Erné, M. and Week, S., Order convergence in lattices, to appear in Rocky Mountain J. Math.Google Scholar
6.Floyd, E. E., Boolean algebras with pathological order topologies, Pacific J. Math. 5 (1955), 687–689.CrossRefGoogle Scholar
7.Gingras, A., Order convergence and order ideals, Proceedings of the special session on convergence structures (Univ. of Nevada, Reno, 1976), 45–49.Google Scholar
8.Katĕtov, M., Remarks on Boolean algebras, Colloq. Math. 2 (1951), 229–235.CrossRefGoogle Scholar
9.Kent, D. C., Convergence functions and their related topologies, Fund. Math. 54 (1964), 125–133.CrossRefGoogle Scholar
10.Kent, D. C., On the order topology in a lattice, Illinois J. Math. 10 (1966), 90–96.CrossRefGoogle Scholar
11.Kent, D. C. and Atherton, C. R., The order topology in a bicompactly generated lattice, J.Google Scholar
12.Kowalsky, H.-J., Beiträge zur topologischen Algebra, Math. Nachr. 11 (1954), 143–185.CrossRefGoogle Scholar
13.Lawson, J. D., Intrinsic topologies in topological lattices and semilattices, Pacific J. Math. Austral. Math. Soc. 8 (1968), 345–349. 32 (1970), 459–466.Google Scholar
14.Schmidt, J., Beiträge zur Filtertheorie I, Math. Nachr. 7 (1952), 359–378.CrossRefGoogle Scholar
15.Scott, D., Continuous lattices, Toposes, algebraic geometry and logic, Lecture Notes in Mathematics No. 274 (Springer-Verlag, 1972).Google Scholar
16.Seminar on continuity in (semi-)lattices: A compendium of continuous lattices (TH Darmstadt 1978).Google Scholar
17.Strauss, D. P., Topological lattices, Proc. London Math. Soc. 18 (1968), 217–230.CrossRefGoogle Scholar
18.Ward, A. J., On relations between certain intrinsic topologies in partially ordered sets, Proc. Cambridge Philos. Soc. 51 (1955), 254–261.CrossRefGoogle Scholar