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Concerning intrinsic topologies on Boolean algebras and certain bicompactly generated lattices

Published online by Cambridge University Press:  18 May 2009

C. R. Atherton
C. R. Atherton
Affiliation:
Dalhousie University, Halifax, Canada
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This paper may be regarded as a continuation of the investigations begun in [2]; certain intrinsic lattice topologies are studied, especially the order and ideal topologies in Boolean algebras, bicompactly generated lattices, and other more general structures. The results of [1], [2], and [3] are shown to be closely related. It is proved that the ideal topology on any Boolean algebra has a closed subbase consisting of all sublattices, whereas the order topology on an atomic Boolean algebra has a closed subbase consisting of all sub-complete lattices. It is also shown that the order topology on an atomic Boolean algebra is autouniformizable (in the sense defined by Rema [3]) and, if the ground set is infinite, strictly coarser than the ideal topology. The conditions Cl and C3 on a lattice, introduced by Kent [1], are shown to be slightly stronger than the condition “ bicompactly generated ”, and in complete lattices, where these conditions are satisfied, the order topology is shown to be coarser than the ideal topology.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 11 , Issue 2 , July 1970 , pp. 156 - 161
Copyright
Copyright © Glasgow Mathematical Journal Trust 1970

References

REFERENCES

1.Kent, D. C., On the order topology in a lattice, Illinois J. Math. 10 (1966), 90–96.CrossRefGoogle Scholar
2.Kent, D. C. and Atherton, C. R., The order topology in a bicompactly generated lattice J. Australian Math. Soc. 8 (1968), 345–349.CrossRefGoogle Scholar
3.Rema, P. S., Auto-topologies in Boolean algebras, J. Indian Math. Soc. (N.S.) 30 (1967), 221–243.Google Scholar
4.Szasz, G., Introduction to Lattice Theory (New York, 1963).Google Scholar
5.Ward, A. J., On relations between certain intrinsic topologies in partially ordered sets, Proc. Cambridge Philos. Soc. 51 (1955), 254–261.CrossRefGoogle Scholar