$\mathcal {P}$-kernel normal system of an additive $\mathcal {P}$-regular semiring

$\mathcal {P}$-regular semigroups are an important class of semigroups. Sen described the congruence on a $\mathcal {P}$-regular semigroup by means of a $\mathcal {P}$-kernel normal system. In this paper we consider the additive $\mathcal {P}$-regular semiring and discuss the $\mathcal {P}$-kernel normal system of the additive $\mathcal {P}$-regular semiring. We prove that every congruence on this kind of semiring can induce a $\mathcal {P}$-kernel normal system, and every $\mathcal {P}$-kernel normal system uniquely determines a congruence. In the end we illustrate a $\mathcal {P}$-kernel normal system of an additive $\mathcal {P}$-regular semiring whose $\overset{+}C$-set is a semi-ideal.