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Recently, we have shown that a semiring $S$ is completely regular if and only if $S$ is a union of skew-rings. In this paper we show that a semiring $S$ satisfying $a^2=na$ can be embedded in a completely regular semiring if and only if $S$ is additive separative.

We show in an additive inverse regular semiring $(S,+,·)$ with $E^{•}\left(S\right)$ as the set of all multiplicative idempotents and $E^{+}\left(S\right)$ as the set of all additive idempotents, the following conditions are equivalent: (i) For all $e,f\in E^{•}\left(S\right)$, $ef\in E^{+}\left(S\right)$ implies $fe\in E^{+}\left(S\right)$. (ii) $(S,·)$ is orthodox. (iii) $(S,·)$ is a semilattice of groups. This result generalizes the corresponding result of regular ring.