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 θ: E → G be a map from E 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 θ̃( p ) = θ( x o , x 1 )•θ( x 1 , x 2 )•….•θ( x n − 1 , x n ) ∈ G depends only upon the extremities x 0 , x n of p , where p = ( x o , x 1 ,…, x n ) (with 1 ⩽ n < ω) is a finite sequence for which ( x i , x i + 1 ) ∈ F for all i < 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 ω 1 ‐dense linear ordering, or even a 2 ω ‐dense linear ordering when Martin's Axiom holds (it follows in particular that it is independent of ZFC for linear orderings of size ω). 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.