Principal Weak Flatness and Regularity of Diagonal Acts

Abstract

For a monoid S, the set S × S equipped with the componentwise right S-action is called the diagonal act of S and is denoted by D(S). A monoid S is a left PP (left PSF) monoid if every principal left ideal of S is projective (strongly flat). We shall call a monoid S left P(P) if all principal left ideals of S satisfy condition (P). We shall call a monoid S weakly left P(P) monoid if the equalities as = bs, xb = yb in S imply the existence of r ∈ S such that xar = yar, rs = s. In this article, we prove that a monoid S is left PSF if and only if S is (weakly) left P(P) and D(S) is principally weakly flat. We provide examples showing that the implications left PSF ⇒ left P(P) ⇒ weakly left P(P) are strict. Finally, we investigate regularity of diagonal acts D(S), and we prove that for a right PP monoid S the diagonal act D(S) is regular if and only if every finite product of regular acts is regular. Furthermore, we prove that for a full transformation monoid S = 𝒯 _X , D(S) is regular.

Key Words:

• Diagonal act
• Direct product
• Left PP monoid
• Left PSF monoid
• (Principally) weakly flat
• Regular act

2010 Mathematics Subject Classification:

• Primary 20M30
• Secondary 20M50

ACKNOWLEDGMENTS

The authors M. Sedaghatjoo and M. Ershad would like to thank the Ministry of Science, Research, and Technology of Iran and Shiraz University for providing sabbatical leave financial support. The author M. Ershad would also like to thank Professor Norman Reilly for having stimulating discussion whilst visiting SFU. Research of V. Laan was supported by the Estonian Science Foundation grant no. 8394 and Estonian Targeted Financing Project SF0180039s08. The authors are grateful to the referee for carefully reading the article and helpful comments and suggestions.

Notes

Communicated by V. Gould.