Monoids over which all flat cyclic right acts are strongly flat

Abstract

The table at the end of this paper summarizes the definitive results that have so far been obtained on the classification of monoids by properties of their cyclic right acts, where the properties in question belong to the collection (free, projective, strongly flat, condition (P), flat, weakly flat, torsion free). The presence of question marks in the table indicates that an exact description of the class of monoids in question has not yet been determined (although partial results may be known: for example, Theorem 3.6 of the present paper describes theleft PP monoids over which (weakly) flat cyclic right acts have property (P), [16] and [17] contain further such results.)