### $\U0001d504(1,1)$ can be strongly embedded into category of semigroups

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We show that the termination of Mohri's algorithm is decidable for polynomially ambiguous weighted finite automata over the tropical semiring which gives a partial answer to a question by Mohri [29]. The proof relies on an improvement of the notion of the twins property and a Burnside type characterization for the finiteness of the set of states produced by Mohri's algorithm.

We construct an example of a cancellative amenable semigroup which is the ascending union of semigroups, none of which are amenable.