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On the relation of a distributive lattice to its lattice of ideals

Published online by Cambridge University Press:  17 April 2009

Herbert S. Gaskill
Herbert S. Gaskill
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, British Columbla, Canada.
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Abstract

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In this note we examine the relationship of a distributive lattice to its lattice of ideals. Our main result is that a distributive lattice and its lattice of ideals share exactly the same collection of finite sublattices. In addition we give a related result characterizing those finite distributive lattices L which can be embedded in a lattice L′ whenever they can be embedded in its lattice of ideals T(L′).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 7 , Issue 3 , December 1972 , pp. 377 - 385
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Birkhoff, Garrett, Lattice theory, 3rd ed. (Colloquium Publ. 25, Amer. Math. Soc., Providence, Rhode Island, 1967).Google Scholar
[2]Galvin, Fred and Jónsson, Bjarni, “Distributive sublattices of a free lattice”, Canad. J. Math. 13 (1961), 265–272.CrossRefGoogle Scholar
[3]Grätzer, George, Universal algebra (Van Nostrand, Princeton, New Jersey; Toronto; London; Melbourne; 1968).Google Scholar
[4]Grätzer, George, “universal algebra” Trends in lattice theory (Sympos., US Naval Academy, Annapolis, Md., 1966, 173–210. Van Nostrand Reinhold Co., New York; Toronto, Ontario; London; 1970).Google Scholar