Partial Conway and iteration semirings

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.

💡 Research Summary

The paper addresses a fundamental limitation of Conway semirings: the star operation must be total, which excludes many semirings of practical interest such as the natural numbers ℕ or the semiring of rational power series ℕ^rat⟨⟨Σ*⟩⟩. In these structures a total star satisfying the Conway identities cannot exist, and in some cases they cannot even be embedded into a Conway semiring. To overcome this, the authors introduce partial Conway semirings, where the star operation is defined only on an ideal I of the underlying semiring S. Within this domain the classic Conway identities—sum‑star ((a+b)^* = a^(ba^)^) and product‑star ((ab)^ = 1 + a(ba)^*b)—are required to hold, but only when the arguments belong to I (for the sum‑star) or when at least one factor belongs to I (for the product‑star). This definition naturally generalizes ordinary Conway semirings (the case I = S) while allowing many previously excluded semirings to be treated.

The authors develop a parallel theory for matrices. Given a semiring S, the matrix theory Mat S consists of all n × p matrices with the usual addition and multiplication. An ideal of matrices M(I) consists of all matrices whose entries lie in the ideal I ⊆ S. The paper defines a star operation on square matrices belonging to M(I) by a block‑matrix construction that mirrors the classical construction for Conway semirings. The resulting partial Conway matrix theory satisfies matrix versions of the sum‑star and product‑star identities: \