Some consequences of a Fatou property of the tropical semiring

1 Abstract We show that the equatorial semiring Z min = (Zf+1g; min; +) is a Fatou extension of the tropical semiring M = (Nf+1g; min; +). This property allows us to give partial decidability results for the equality problem for rational series with multiplicities in the tropical semiring. We also deduce from it the decidability of the limitedness problem for the equatorial semiring, solving therefore a question of I. Simon. 0 Introduction The tropical semiring is the semiring denoted by M which has support Nf+1g and operations a b = minfa; bg and a b = a + b. It is currently used in the context of cost minimization in operations research. However it appeared that M plays in fact an important role in several problems concerning rational languages (see 8] for a survey of the tropical semiring theory and of its applications). For instance, I. Simon showed that the nite power property for recognizable languages can be reduced to the limitedness problem for the tropical semiring (cf 8]). In the same way, I. Simon used the tropical semiring to study the non-deterministic complexity of a usual nite automaton (cf 9]). An important problem in the tropical semiring theory was to see if it is possible to decide whether two M-rational series are equal or not. We solved recently this question by proving that this equality problem was undecidable (cf 4, 5]). Our proof was strongly based on the introduction of the equatorial semiring Z min which is just the extension of M to arbitrary integers. Indeed it can be shown that the equality problem for Z min is undecidable and that the two above decidability problems are equivalent (cf 4, 5]).