La logique des parties fractionnaires de nombres réels

The group of decimals with addition modulo 1 and the natural order, which is not an ordered group in the usual sense, is used to enhance the efficiency of timed automata of Bouchy, Finkel and Leroux (Decomposition of decidable first-order logics over integers and reals, in: Temporal Representation and Reasoning, Proceedings of the 15th Symposium TIME 2008, IEEE Computer Society Press, 2008, pp. 147-155). In this note, we axiomatize this structure and show that it admits quantifier elimination.