Before revision:  For instance, norm closures of bounded continuous functions  as considered in the G-framework  lead to spaces which are lattice-isomorphic to sublattices of a classical L^1-space lacking, however, Dedekind \sigma-completeness .
After revision:  For instance, norm closures of bounded continuous functions as considered in the G-framework lead to spaces which are lattice-isomorphic to sublattices of a classical L^1-space lacking, however, any form of order completeness .