Crowns,cutsets, and valuable partial orders |
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Authors: | David G. Wagner |
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Affiliation: | (1) Department of Combinatorics and Optimization, University of Waterloo, N2L 3G1 Waterloo, Ontario, Canada |
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Abstract: | We consider the problem of recognizing those partial orders which admit a valuation: this is a linear-algebraic condition which arises naturally in an algebraic/geometric context. We show that a partial order has at most one valuation (which is integer-valued) and present various structural conditions which are either necessary or sufficient for a partial order to be valuable. The first main result is a reduction theorem which allows us to restrict attention to those partial orders which do not have a bounded cutset. We use this and a theorem of Kelly and Rival to prove the second main result: that every contraction of a bounded partial order is fibre-valuable if and only if the partial order is a dismantlable lattice. This result has a geometric interpretation.This research was supported by the Natural Sciences and Engineering Research Council of Canada under operating grant OGP0105392. |
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Keywords: | 06A07 05E99 05C75 |
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