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Ideal topology on a distrbutive lattice

Published online by Cambridge University Press:  09 April 2009

P. V. Ramana Murty
P. V. Ramana Murty
Affiliation:
Department of MathematicsAndhra University, Waltair - 530003, India
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In [1] Atherton has asked whether the ideal topology (see [1]) is Hausdorff on every distributive lattice. He also gave some sufficient conditions (Kent's conditions cl and c3 [1]) under which the ideal topology on a complete distributive lattice is Hausdorff. In this paper we determine some more classes of lattices in which the ideal topology is Hausdorff and give examples to show that the classes determined are independent even in a complete distributive lattice. We also show by an example that Atherton's conditions are not necessary even in the case of a complete distributive lattice.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 4 , December 1974 , pp. 503 - 508
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Atherton, C. R., ‘Concerning intrinsic topologies on Boolean algebras and certain bicompactly generated lattices’, Glasgow Mathematical Journal 11 (1970), 156–161.CrossRefGoogle Scholar
[2]Birkhoff, G., ‘Lattice theory’, Amer. Math. Colloquium publications (25) (1948).Google Scholar
[3]Ward, A. J., ‘On relations between certain intrinsic topologies in partially ordered sets’, Proc. Cambridge Philos. Soc. 51 (1955), 254–261.CrossRefGoogle Scholar