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@techreport { 11303_15594,
author = {Rambau, Jörg},
title = {Triangulations of Cyclic Polytopes and higher Bruhat Orders},
institution = {Technische Universität Berlin},
year = {1996},
doi = {10.14279/depositonce-14367},
url = {http://dx.doi.org/10.14279/depositonce-14367},
keywords = {bruhat order, cyclic polytopes, edelman reiner, several new functorial construction, maximal chain, similar method, structure theorem, well-studied tamari lattice, kapranov voevod-sky, general case, order-preserving map, one-to-one correspondence, manin schechtman, ordered n-categories, certain equivalence class, poset structure, cyclic d-polytope},
abstract = {Recently Edelman & Reiner} suggested two poset structures S}1(n,d) and S}2(n,d) on the set of all triangulations of the cyclic d-polytope C(n,d) with n vertices. Both posets are generalizations of the well-studied Tamari lattice. While S}2(n,d) is bounded by definition, the same is not obvious for S}1(n,d). In the paper by Edelman & Reiner} the bounds of S}2(n,d) were also confirmed for S}1(n,d) whenever d \le 5, leaving the general case as a conjecture. In this paper their conjecture is answered in the affirmative for all~d, using several new functorial constructions. Moreover, a structure theorem is presented, stating that the elements of S}1(n,d+1) are in one-to-one correspondence to certain equivalence classes of maximal chains in S}1(n,d). In order to clarify the connection between S}1(n,d) and the higher Bruhat order B}(n-2,d-1) of Manin & Schechtman}, we define an order-preserving map from B}(n-2,d-1) to S}1(n,d), thereby concretizing a result by Kapranov & Voevodsky} in the theory of ordered n-categories.}
}