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1·Bipartite graphs with P-regular endomorphism monoids are characterized. P-regularity of the endomorphism monoid of lexicographic product of graphs is discussed.

刻划了具有P -正则自同态幺半群的二分图，讨论了字典序积图的自同态幺半群的P -正则性。

2·In this article, we showed the algebraic structure of syntactic categories based on monoid and defined categorial equation whose solutions are described by consistency and correlation .

本文给出了建立在含幺半群基础上的范畴语法的代数结构 ，定义了范畴方程和它的解并对范畴方程的解作了分类 ：相容性的相关性。

3·In this paper, we propose a reduced linear representation of such distribution based on the formal power series on free monoid which can be easily used to calculate the probability distribution.

该文利用自由幺半群上的有理形式幂级数理论，构造出该概率分布的约化线性表示，从而完全解决了噪声变量的概率分布计算问题。