A band generated by two semilattices is regular.
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.