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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.

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, where the unbounded and non-distributive translation lattices are included (they all are called d-lattices there to stress the importance of the distance $d$).

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.

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.

Kaplansky characterized distributive Birkhoff lattices on which real numbers acted as translations, subjected to certain axioms. A more general setting (unbounded and non-distributive, 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).

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.