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On transitive subrelations of binary relations

Published online by Cambridge University Press:  12 March 2014

Christopher S. Hardin
Christopher S. Hardin*
Affiliation:
Department of Mathematics, Union College,807 Union Street, Schenectady, NY 12308, USA, E-mail: [email protected]
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Abstract

The transitive closure of a binary relation R can be thought of as the best possible approximation of R “from above” by a transitive relation. We consider the question of approximating a relation from below by transitive relations. Our main result is that every thick relation (a relation whose complement contains no infinite chain) on a countable set has a transitive thick subrelation. This allows for a solution to a problem arising from previous work by the author and Alan Taylor. We also exhibit a thick relation on an uncountable set with no transitive thick subrelation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1429 - 1440
Copyright
Copyright © Association for Symbolic Logic 2011

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