On a Generalization of Weak Pullback Flatness

Abstract

In 1998, Laan introduced weakly pullback flat acts and showed that weak pullback flatness is equivalent to the conjunction of Conditions (P) and (E′). In 2013, Golchin introduced Condition (P′), which is a generalization of Condition (P). In this article, we first define Condition (PF″), which lies strictly between weak pullback flatness and Condition (P′), and prove that Condition (PF″) coincides with the conjunction of Conditions (P′) and (E′). Furthermore, we give a classfication of monoids by Condition (PF″) of (cyclic, Rees factor) acts, and find a necessary and sufficient condition under which Condition (PF″) coincides with Condition (P′) (resp., weak pullback flatness, strong flatness) for Rees factor acts. Finally, we investigate Conditions (P′), (E′) and (PF″) using some new types of epimorphisms.

Key Words:

• Condition (PF″)
• Pure epimorphism
• S-act
• Weak pullback flatness

2010 Mathematics Subject Classification:

• 20M30
• 20M50

ACKNOWLEDGEMENT

The authors would like to give many thanks to the referee for reading an early draft of this article, correcting typographical errors and grammatical difficulties, and for his invaluable comments and suggestions, particularly on Theorem 14, and to Professor Husheng Qiao for his helpful suggestions relating to this article. They would also like to thank Professor V. Gould for her useful communications.

Dedicated to Professor Yuqi Guo on the occasion of his seventy-fifth birthday.