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Toric partial orders


Authors: Mike Develin, Matthew Macauley and Victor Reiner
Journal: Trans. Amer. Math. Soc. 368 (2016), 2263-2287
MSC (2010): Primary 06A06, 52C35
DOI: https://doi.org/10.1090/tran/6356
Published electronically: July 9, 2015
MathSciNet review: 3449239
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Abstract: We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders.


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Additional Information

Mike Develin
Affiliation: Department of Audience Research, Facebook Inc., 1601 Willow Road, Menlo Park, California 94025
Email: develin@post.harvard.edu

Matthew Macauley
Affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634-0975
Email: macaule@clemson.edu

Victor Reiner
Affiliation: Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: reiner@math.umn.edu

DOI: https://doi.org/10.1090/tran/6356
Keywords: Braid arrangement, convex geometry, cyclic order, partial order, unimodular, toric arrangement, transitivity, Coxeter element, reflection functor
Received by editor(s): December 5, 2012
Received by editor(s) in revised form: December 19, 2013
Published electronically: July 9, 2015
Additional Notes: The first author was supported by AIM Five-Year Fellowship (2003–2008).
The second author was supported by NSF grant DMS-1211691.
The third author was supported by NSF grant DMS-1001933.
Article copyright: © Copyright 2015 American Mathematical Society