caa a22 4500 397533187 CHVBK 20180308164657.0 cr unu---uuuuu 161202e199508 xx s 000 0 eng 10.1112/jlms/52.1.35 doi (NATIONALLICENCE)oxford-10.1112/jlms/52.1.35 Brochet J.-M Groupe L.M.D., Université Claude Benard (Lyon 1) 69622 Villeurbanne cedex, France Coherent Products in a Finite Group along a Linear Ordering [Elektronische Daten] [J.-M. Brochet] 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(xo, x1)•θ(x1, x2)•....•θ(xn − 1, xn) ∈ G depends only upon the extremities x0, xn of p, where p = (xo, x1,...,xn) (with 1 ≤ n < ω) is a finite sequence for which (xi, xi + 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 ω1-dense linear ordering, or even a 2ω-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. © The London Mathematical Society Notes and Papers nationallicence Journal of the London Mathematical Society Oxford University Press 52/1(1995-08), 35-47 0024-6107 52:1<35 1995 52 jlms https://doi.org/10.1112/jlms/52.1.35 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1112/jlms/52.1.35 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Brochet J.-M Groupe L.M.D., Université Claude Benard (Lyon 1) 69622 Villeurbanne cedex, France NATIONALLICENCE 773 0- Journal of the London Mathematical Society Oxford University Press 52/1(1995-08), 35-47 0024-6107 52:1<35 1995 52 jlms Metadata rights reserved CC BY-NC-4.0 http://creativecommons.org/licenses/by-nc/4.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-oxford