关于满足条件(P_(I))的S-系

On S-acts satisfying Condition(P_(I))

设I是幺半群S的一个理想。首先,研究满足条件(P_(I))的S-系的基本性质,建立S-系的余直积、有向上极限、Rees短正合序列与条件(P_(I))的关系,并利用S-满同态的纯性给出条件(P_(I))的一个充要条件。其次,通过引入I-(局部)循环S-系的概念,给出S-系的直积保持条件(P_(I))的充分必要条件。最后,研究条件(P_(I))的覆盖问题,给出所有S-系具有条件(P_(I))-覆盖的一个新刻画。

Let I be an ideal of a monoid S.Firstly,the basic properties of the S-acts satisfying Condition(P_(I))are investigated.The relations between coproducts,directed colimits,Rees short exact sequences,and Condition(P_(I))are established respectively,and a necessary and sufficient condition of Condition(P_(I))is provided by purity of epimorphisms for S-acts.Then,using the concept of I-(locally)cyclic S-acts,the sufficient and necessary conditions of direct products of S-acts preserving Condition(P_(I))are given.Finally,cover of S-acts with Condition(P_(I))is studied,and a new characterization of monoids over which every S-act has a Condition(P_(I))-cover is presented.