## Some consequences of a Fatou property of the tropical semiring.(English)Zbl 0806.68083

The tropical semiring $$\mathcal M$$ is the semiring $$\mathbb{N} \cup \{+ \infty\}$$ with sum $$\min(a,b)$$ and product $$a + b$$. It is shown that $$\mathbb{Z}_{\min} = \mathbb{Z} \cup \{+ \infty\}$$ (with the same operations) is a Fatou extension of $$\mathcal M$$; that is, each noncommutative rational series over $$\mathbb{Z}_{\min}$$, which has coefficients in $$\mathcal M$$, is rational over $$\mathcal M$$. The equality problem for rational series over $$\mathcal M$$ or $$\mathbb{Z}_{\min}$$ is undecidable. However, it is decidable for a series which is both $$\mathbb{Z}_{\min}$$- and $$\mathbb{Z}_{\max}$$-rational. An application to the limitedness problem is also given.

### MSC:

 68Q70 Algebraic theory of languages and automata

### Keywords:

automata series; tropical semiring; limitedness problem
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### References:

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