Partial Conway and iteration semirings

TL;DR

This paper introduces partial Conway semirings with a star operation defined on an ideal, extending the classical theory to include important semirings like natural numbers where total star operations are impossible.

Contribution

It develops the theory of partial Conway semirings and proves a Kleene theorem for this generalized framework.

Findings

Established the concept of partial Conway semirings.

Proved a Kleene theorem for partial Conway semirings.

Extended the applicability of Conway semiring theory to natural number semirings.

Abstract

A Conway semiring is a semiring $S$ equipped with a unary operation $^*:S \to S$, 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 $N^{\rat}\llangle \Sigma^* \rrangle$ of rational power series of words on $\Sigma$ 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.