Relative uniform completeness and order-convex representations of archimedean ℓ-groups and f-rings

Results of Henriksen and Johnson, for archimedean f-rings with identity, and of Aron and Hager, for archimedean ℓ-groups with unit, relating uniform completeness to order-convexity of a representation in a D(X) (the lattice of almost real continuous functions on the space X) are extended to situations without identity or unit. For an archimedean ℓ-group, G, we show: if G admits any representation G⩽D(X) in which G is order-convex, then G is divisible and relatively uniformly complete. A converse to this would seem to require some sort of canonical representation of G, which seems not to exist in the ℓ-group case. But for a reduced archimedean f-ring, A, there is the Johnson representation A⩽D(XA), and we show: A is divisible, relatively uniformly complete and square-dominated if and only if A is order-convex in D(XA) and square-root-closed. Also, we expand on the situation with unit, where we have the Yosida representation, G⩽D(YG): if G is divisible, relatively uniformly complete, and the unit is a near unit, then G is order-convex in D(YG).