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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

Evans, E. (1977), ‘The Boolean ring universal over a meet semilattice’, J. Austral. Math. Soc. Ser. A 23, 402–415.Google Scholar
Mayer, R. D. and Pierce, R. S. (1960), ‘Boolean algebras with ordered bases’, Pacific J. Math. 10, 925–942.CrossRefGoogle Scholar
Mostowski, A. and Tarski, A. (1939), ‘Boolesche Rings mit geordneter Basis’, Fund. Math. 32, 69–86.CrossRefGoogle Scholar
Varlet, J. (1965), ‘Congruences des les demis-lattis’. Bull. Soc. Roy. sci. liège 34, 231–240.Google Scholar