SOME CONSEQUENCES OF A FATOU PROPERTY OF THE TROPICAL SEMIRING

We show that the equatorial semiring Z(min) = (Z or { + infinity}, min , + ) is a Fatou extension of the tropical semiring M = (N or { + infi nity}, min, + ). This property allows us to give partial decidability results for the equality problem for rational series with multipliciti es in the tropical semiring. We also deduce from it the decidability o f the limitedness problem for the equatorial semiring, solving therefo re a question of 1. Simon.