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.
