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Consider partial maps from the free monoid into the field of real numberswith a rational domain. We show that two families of such series are actuallythe same: the unambiguous rational series on the one hand, and the max-plus andmin-plus rational series on the other hand. The decidability of equality wasknown to hold in both families with different proofs, so the above unifies thepicture. We give an effective procedure to build an unambiguous automaton froma max-plus automaton and a min-plus one that recognize the same series.

Series which are both max-plus and min-plus rational are unambiguous