Monoids over which all flat cyclic right acts are strongly flat

Flatness properties of monocyclic acts

“…In fact, n is positive since wt -r t. In more detail, assume (using Lemma 2. We point out that this theorem generalizes the result used in [1] that S/p(x 2, x) is flat if and only if x is regular, and also the results in [2] that Sip(e, f) (e, f idempotents with ef = e) and S/p(xe, e) (e idempotent, x ~ S, ex = x) are both flat. Note also that if S is a monoid having 1 as its only right cancellable element, then for any non-regular element xeS the act S/p(x2,x) is torsion-free but not principally weakly flat.…”

“…(Exact references for most of these results are contained in [2].) In this lemma, if2 is a left congruence on S and ifueS, then uZ denotes the left congruence on S defined by x(uZ)y if and only if (xu))v(yu).…”

“…During the course of that investigation it became clear that flatness properties of what are here termed monocyclic acts (cyclic acts S/p in which p = p(s, t) is the smallest right congruence on S identifying elements s and t of S) are worthy of independent study. Many of the early results on flatness, for example the fact that for all right S-acts to be flat S must necessarily be regular, are involved essentially with monocyclic acts only, as are a number of the findings in [2]. (Incidentally, the authors are indebted to the referee for suggesting that the monoid {N u {0}, + ) might be such that all of its cyclic acts are monocyclic.…”

“…Our paper [2] is concerned with the classification of monoids by properties of cyclic acts. We first note (upon examination of the original proofs) that consideration of monocyclic acts is enough to give each set of necessary conditions presented there except the condition FP1/x FP2 for all strongly flat cyclic acts to be projective: in fact, we have already seen in Proposition 3.13 of the present paper that for monocyclic acts these two concepts coincide.…”

“…Several results in the literature will also appear as corollaries. For comparison purposes, as well as for ease of reading the present paper, the reader may find it helpful to have reference [2] at hand.…”