Proper Semirings and Proper Convex Functors.

Esik and Maletti introduced the notion of a proper semiring and proved that some important (classes of) semirings – Noetherian semirings, natural numbers – are proper. Properness matters as the equivalence problem for weighted automata over a semiring which is proper and finitely and effectively presented is decidable. Milius generalised the notion of properness from a semiring to a functor. As a consequence, a semiring is proper if and only if its associated “cubic functor” is proper. Moreover, properness of a functor renders soundness and completeness proofs for axiomatizations of equivalent behaviour.