An algebraic context for metric spaces--attractive to me--can be be provided by translation lattices, introduced (in the bounded, distributive case) by Irving Kaplansky, *Lattices of continuous functions II*, Amer. J. Math. 70 (1948), pp. 626-634; see also Włodzimierz Holsztyński, *Lattices with real numbers as additive operators*, Dissertationes Mathematicae LXII, Warszawa 1969.