  Iteration semirings are Conway semirings satisfying Conway's group
identities. We show that the semirings $\N^{\rat}\llangle \Sigma^* \rrangle$ of
rational power series with coefficients in the semiring $\N$ of natural numbers
are the free partial iteration semirings. Moreover, we characterize the
semirings $\N_\infty^{\rat}\llangle \Sigma^* \rrangle$ as the free semirings in
the variety of iteration semirings defined by three additional simple
identities, where $\N_\infty$ is the completion of $\N$ obtained by adding a
point of infinity. We also show that this latter variety coincides with the
variety generated by the complete, or continuous semirings. As a consequence of
these results, we obtain that the semirings $\N_\infty^{\rat}\llangle \Sigma^*
\rrangle$, equipped with the sum order, are free in the class of symmetric
inductive $^*$-semirings. This characterization corresponds to Kozen's
axiomatization of regular languages.
