Partial Conway and Iteration Semirings
Stephen L. Bloom, Zoltán Ésik, Werner Kuich · 2008
A Conway semiring is a semiring S equipped with a unary operation ∗ : S → S, always called ’star’, satisfying the sum star and product star identities. It is known that these identities imply a Kleene type theorem. Some computationally important semirings, such as N or N rat 〈〈Σ ∗ 〉 〉 of rational power series of words on Σ with coefficients in N, cannot have a total star operation satisfying the Conway identities. We introduce here partial Conway semirings, which are semirings S which have a star operation defined only on an ideal of S; when the arguments are appropriate, the operation satisfies the above identities. We develop the general theory of partial Conway semirings and prove a Kleene theorem for this generalization. 1