Kaplansky characterized distributive Birkhoff lattices on which real numbers acted as translations, subjected to certain axioms. A more general setting (unbounded and non-distibutive, but the main results were for the distributive case) was developed in my thesis. Instead of C(X) I had Met(X). The distributive theory is rather simple (while the non-distributive general case awaits for further and more advanced studies).
Włodzimierz Holsztyński
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