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Topologies on Boolean algebras defined by ideals and dual ideals
Published online by Cambridge University Press: 18 May 2009
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In the paper [5], Rema used the well-known fact that in a Boolean algebra 
the binary operation d: B × B → B defined by 
is a “metric“ operation to show that, if D is any dual ideal of ^, then the sets Up = {(x, y): d(x, y) <p}, where p ∈ D, form a base for a uniformity of }, the resulting topological space <B; T[D]> being called an auto-topologized Boolean algebra. Recently, Kent and Atherton [1, 4] exhibited a family of topologies on an arbitrary lattice ℒ defined in terms of ideals and dual ideals. More specifically, if I and D are respectively an ideal and a dual ideal of ℒ, then the T[I:D] topology on ℒ is the topology defined by taking the sets of the form a*⋂b+, where 
, as sub-base for the open sets. It is these topologies that are studied in this paper.
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- Copyright © Glasgow Mathematical Journal Trust 1973
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.Birkhoff, G., Lattice theory, 3rd edition, Amer. Math. Soc. Colloquium Publications 25 (Providence, R.I., 1967).Google Scholar
4.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
5.Rema, P. S., Auto-topologies in Boolean algebras, J. Indian Math. Soc. 30 (4) (1966), 221–243.Google Scholar
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