Flatness properties of diagonal acts over monoids

If S is a monoid, the right S-act S×S, equipped with componentwise S-action, is called the diagonal act of S. The question of when this act is cyclic or finitely generated has been a subject of interest for many years, but so far there has been no explicit work devoted to flatness properties of diagonal acts. Considered as a right S-act, the monoid S is free, and thus is also projective, flat, weakly flat, and so on. In 1991, Bulman-Fleming gave conditions on S under which all right acts S ^I (for I a non-empty set) are projective (or, equivalently, when all products of projective right S-acts are projective). At approximately the same time, Victoria Gould solved the corresponding problem for strong flatness. Implicitly, Gould’s result also answers the question for condition (P) and condition (E). For products of flats, weakly flats, etc. to again have the same property, there are some published results as well. The specific questions of when S×S has certain flatness properties have so far not been considered. In this paper, we will address these problems. S. Bulman-Fleming research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494. Some of the results in this article are contained in the M.Math. thesis of A. Gilmour, University of Waterloo (2007).