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Essential completions of distributive lattices

Published online by Cambridge University Press:  17 April 2009

Gerhard Gierz
Affiliation:
Department of Mathematics, University of California, Riverside, California 92521, U.S.A.
Albert R. Stralka
Affiliation:
Department of Mathematics, University of California, Riverside, California 92521, U.S.A.
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Abstract

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The salient feature of the essential completion process is that for most common distributive lattices it will yield a completely distributive lattice. In this note it is shown that for those distributive lattices which have at least one completely distributive essential extension the essential completion is minimal among the completions by infinitely distributive lattices. Thus in its setting the essential completion of a distributive lattice behaves in much the some way as the one-point compactification of locally compact topological space does in its setting.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Ball, R. “Distributive Cauchy lattices”, Algebra Universalis (to appear).Google Scholar
[2]Banaschewski, B. and Bruns, G., “Injective hulls in the category of distributive lattices”, J. reine und angewandte Math. 232 (1968), 102103.Google Scholar
[3]Birkhoff, G.Lattice Theory, (Amer. Math. Soc. Colloq. Publications, Providence, Rhode Island 1967).Google Scholar
[4]Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., and Scott, D.S., A Compendium of Continuous Lattices, (Springer Verlag, Berlin, Heidelberg, New York 1980).CrossRefGoogle Scholar
[5]Gierz, G. and Stralka, A.R., “Essential extension and congruence extension,” Quarterly J. Math. Oxford (2), 35 (1984), 2536.CrossRefGoogle Scholar
[6]Gierz, G. and Stralka, A.R., “The Zariski topology on distributive lattices, (Preprint (1983).Google Scholar
[7]Lawson, J.D., “The duality of continuous posets”, Houston J. Math 5 (1979), 357394.Google Scholar