JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (8): 6-12.doi: 10.6040/j.issn.1671-9352.0.2022.437

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On S-acts satisfying Condition (PI)

Xingliang LIANG1,2(),Yun DANG1,Xueying CHEN1   

  1. 1. School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi?an 710021, Shaanxi, China
    2. School of Mathematics, Northwest University, Xi?an 710127, Shaanxi, China
  • Received:2022-08-19 Online:2023-08-20 Published:2023-07-28

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Abstract:

Let I be an ideal of a monoid S. Firstly, the basic properties of the S-acts satisfying Condition (PI) are investigated. The relations between coproducts, directed colimits, Rees short exact sequences, and Condition (PI) are established respectively, and a necessary and sufficient condition of Condition (PI) 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 (PI) are given. Finally, cover of S-acts with Condition (PI) is studied, and a new characterization of monoids over which every S-act has a Condition (PI)-cover is presented.

Key words: monoid, ideal, S-act, Condition (PI), I-left reversible, I-locally cyclic

CLC Number: 

  • O152.7
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