In this paper a generalized version (r-naturally fully ordered groupoid) of a naturally fully ordered groupoid with left identity in the sense that only right solvability is permissible is embedded in a concrete groupoid of all non-negative real numbers. First, the introduction of centrality makes the r-naturally fully ordered groupoid with left identity order-isomorphic to the positive cone of a fully ordered central quasigroup. Second, the left Archimedean property enables this ordered groupoid to be embedded in the concrete groupoid.