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Slim trees

Part of: Ordered sets

Published online by Cambridge University Press:  09 April 2009

Elliott Evans
Elliott Evans
Affiliation:
Department of Computer Science, University of Tennessee, Knoxville, Tennessee 37916, U.S.A.
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Abstract

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A semilattice tree T with 0 is slim if there is a chain C with 0 so that the lattices θ (T) and θ(C) of semilattice congruences are isomorphic. This paper establishes elementary consequences of slimness and uses simple constructive techniques to show certain small trees slim. If T is the union of at most countably many branches, each of which has a maximum or a countable cotinal subset, then T is slim. For trees with enough maximals slimness is equivalent with not having any uncountable anti-chains. If a tree T has a countable cofinal subset then T is slim. Thus finitary trees are slim.

Keywords

treeslim treeuniversal Boolean ring of a semilatticeanti-chainbranchfinitary tree.

MSC classification

Secondary: 06A12: Semilattices
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 2 , December 1980 , pp. 201 - 214
Copyright
Copyright © Australian Mathematical Society 1980

References

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