On Manin-Schechtman orders related to directed graphs
Vladimir I. Danilov, Alexander V. Karzanov, Gleb A. Koshevoy

TL;DR
This paper generalizes Manin-Schechtman orders by introducing convex orders on path systems in directed graphs, establishing a poset structure with unique minimal and maximal elements, extending classical higher Bruhat order results.
Contribution
It extends the classical theory of higher Bruhat orders to a broader setting involving convex orders on directed graph path systems, with new local transformations and poset properties.
Findings
The set of convex orders forms a poset with unique minimal and maximal elements.
Local transformations, or flips, preserve the poset structure.
The generalization encompasses classical higher Bruhat orders as a special case.
Abstract
As a generalization of weak Bruhat orders on permutations, in 1989 Manin and Schechtman introduced the notion of a higher Bruhat order on the -element subsets of a set . Among other results in this field, they proved that the set of such orders for fixed, endowed with natural local transformations, constitutes a poset with one minimal and one maximal elements. In this paper we consider a wider model, involving the so-called convex order on certain path systems in an acyclic directed graph, introduce local transformations, or flips, on such orders and prove that the resulting structure gives a poset with one minimal and one maximal elements as well, yielding a generalization of the above-mentioned classical result.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Combinatorial Mathematics
