  As a generalization of weak Bruhat orders on permutations, in 1989 Manin and
Schechtman introduced the notion of a higher Bruhat order on the $d$-element
subsets of a set $[n]=\{1,2,\ldots,n\}$. Among other results in this field,
they proved that the set of such orders for $n,d$ 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.
