加法$\mathcal {P}$-正则半环上的$\mathcal {P}$-核正规系

$\mathcal{P}$-正则半群是一类重要的半群，Sen用核正规系的方法描述了$\mathcal{P}$-正则半群上的同余. 本文考虑加法$\mathcal{P}$-正则半环,在该类半环上引入了$\mathcal {P}$-核正规系,证明了该类半环上的每个同余都可以获得一个$\mathcal{P}$-核正规系,并且$\mathcal {P}$-核正规系惟一地确定了一个同余. 最后对$\overset{+} C$-集是半理想的加法$\mathcal {P}$-正则半环刻画了$\mathcal {P}$-核正规系.

$\mathcal {P}$-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.