Coherent Products in a Finite Group along a Linear Ordering

Let C = (C, ^ ) be a linear ordering, E a subset of {(x,y):x < y in C} whose transitive closure is the linear ordering C, and let 9:E->G be a map from £ to a finite group G = (G>). We showed with M. Pouzet that, when C is countable, there is F^ E whose transitive closure is still C, and such that 6(p) = 6(xo,x1).d(xvx2),....8(xn_1,xn)eG depends only upon the extremities xo,xn of p, where p = (xo,xv ...,xn) (with 1 < n < co) is a finite sequence for which (x(,xi+1)eFfor all / < n. Here, we show that this property does not hold if C is the real line, but is still true if C does not embed an ojj-dense linear ordering, or even a 2-dense linear ordering when Martin's Axiom holds (it follows in particular that it is independent of ZFC for linear orderings of size ajj). On the other hand, we prove that this property is always valid if E = {(x,y):x < y in C}, regardless of any other condition on C.