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Abstract
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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 PP if all principal left ideals of S satisfy condition (P). We shall call
a monoid S weakly left PP 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 PP and D(S) is principally weakly flat. We provide examples showing that the implications left PSF ? left PP?
weakly left PP 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.
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