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Topologies of Lattice Products

Published online by Cambridge University Press:  20 November 2018

Richard A. Alo  and
Orrin Frink
Richard A. Alo
Affiliation:
Carnegie Institute of Technology, Pittsburgh, Pa., U.S.A. and The Pennsylvania State University, University Park, Pa., U.S.A.
Orrin Frink
Affiliation:
Carnegie Institute of Technology, Pittsburgh, Pa., U.S.A. and The Pennsylvania State University, University Park, Pa., U.S.A.
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A number of different ways of defining topologies in a lattice or partially ordered set in terms of the order relation are known. Three of these methods have proved to be useful and convenient for lattices of special types, namely the ideal topology, the interval topology, and the new interval topology of Garrett Birkhoff. In another paper (2) we have shown that these three topologies are equivalent for chains (totally ordered sets), where they reduce to the usual intrinsic topology of the chain.

Since many important lattices are either direct products of chains or sublattices of such products, it is natural to ask what relationships exist between the various order topologies of a direct product of lattices and those of the lattices themselves.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 1004 - 1014
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Alexander, J. W., On the concept of a topological space, Proc. Nat. Acad. Sci., 25 (1939), 52–54.Google Scholar
2. Alo, R. A. and Frink, Orrin, Topologies of chains (unpublished).Google Scholar
3. Baer, R. M., A characterization theorem for lattices with Hausdorff interval topology, J. Math. Soc. Japan, 7 (1955), 177–181.Google Scholar
4. Birkhoff, Garrett, Lattice theory, rev. ed. (New York, 1948).Google Scholar
5. Birkhoff, Garrett, A new interval topology for dually directed sets, Revista Mat. y Fis. Teorica Tucuman, 14(1962), 325–331.Google Scholar
6. Birkhoff, Garrett and Frink, Orrin, Representations of lattices by sets, Trans. Amer. Math. Soc., 64 (1948), 299–316.Google Scholar
7. Bourbaki, N., Eléments de mathématique, III, Topologie générale, chap. IX (Paris, 1948).Google Scholar
8. Frink, Orrin, Topology in lattices, Trans. Amer. Math. Soc., 51 (1942), 568–582.Google Scholar
9. Frink, Orrin, Ideals in partially ordered sets, Amer. Math. Monthly, 61 (1954), 223–233.Google Scholar
10. Gale, David, Compact sets of functions and function rings, Proc. Amer. Math. Soc., 1 (1950), 303–308.Google Scholar
11. Holland, Charles, The interval topology of a certain l-group, Czechoslovak Math. J., 15 (1965), 311–314.Google Scholar
12. Kelley, J. L., General topology, (New York, 1955).Google Scholar
13. Naito, T., Lattices with P-ideal topologies, Tôhoku Math. J., 12 (1960), 235–262.Google Scholar
14. Northam, E. S., The interval topology of a lattice, Proc Amer. Math. Soc., 4 (1953), 825–827.Google Scholar
15. Rennie, B. C., Lattices, Proc. London Math. Soc., 52 (1951), 386–400.Google Scholar
16. Ward, A. J., On relations between certain intrinsic topologies on a partially ordered set, Proc. Cambridge Phil. Soc., 51 (1955), 254–261.Google Scholar
17. Wolk, E. S., On the interval topology of an l-group, Proc. Amer. Math. Soc., 12 (1961), 304–307.Google Scholar