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.
Włodzimierz Holsztyński
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