Essential extensions of partial orders on groups

Abstract:

Let (G, P) be an l-group and $\mathcal {C}(P)$ be the lattice of convex l-subgroups of (G, P). We say that the l-cone Q is essential over P if $\mathcal {C}(Q)$ is contained in $\mathcal {C}(P)$. It is shown that for each nonzero x in G and each Q-value D of x, there is a P-value C of x containing D and no other Q-value of x. We specialize to those essential extensions for which the above C always depends uniquely on x and D; these are called very essential extensions. We show that if (G, P) is a representable l-group then P is the meet of totally ordered very essential extensions of P. Further we investigate connections between the existence of total very essential extensions and both representability and normal valuedness. We also study the role played by the various radicals in the theory. The same two classes of extensions are treated in the context of abelian Riesz groups. Similar questions about existence of such total orders are dealt with. The main result in this connection is that such total extensions always exist for finite valued pseudo lattice groups, and that the original cone is the meet of them.