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Abstract
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If S is a monoid, the set S × S equipped with componentwise S-action is
called the diagonal act of S and is denoted by D(S). This gives an
answer to a conjecture posed by Bulman-Fleming and Gilmour (Semigroup Forum
79:298–314, 2009). Besides, we present a fair characterization of monoids S over which the diagonal act D(S) is (principally) weakly flat and finally, we impose a condition on D(S) in order to make S a left PSF monoid.
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