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Essential completions of distributive lattices

  • Gerhard Gierz (a1) and Albert R. Stralka (a1)

Abstract

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.

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Copyright

References

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[1]Ball, R. “Distributive Cauchy lattices”, Algebra Universalis (to appear).
[2]Banaschewski, B. and Bruns, G., “Injective hulls in the category of distributive lattices”, J. reine und angewandte Math. 232 (1968), 102103.
[3]Birkhoff, G.Lattice Theory, (Amer. Math. Soc. Colloq. Publications, Providence, Rhode Island 1967).
[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).
[5]Gierz, G. and Stralka, A.R., “Essential extension and congruence extension,” Quarterly J. Math. Oxford (2), 35 (1984), 2536.
[6]Gierz, G. and Stralka, A.R., “The Zariski topology on distributive lattices, (Preprint (1983).
[7]Lawson, J.D., “The duality of continuous posets”, Houston J. Math 5 (1979), 357394.
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Essential completions of distributive lattices

  • Gerhard Gierz (a1) and Albert R. Stralka (a1)

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