Pullbacks and flatness properties of acts. I

Let S be a monoid. A right S-act A is pullback flat (= strongly flat) if th e functor A circle times - (from the category of left S-acts to the categor y of sets) preserves pullbacks. We investigate possible generalizations of this notion, obtained either by restricting attention to certain types of p ullbacks or by weakening the requirement of pullback preservation. We note that it is possible to describe the already familiar notions of flatness, ( principal) weak flatness, and torsion freeness in these terms. Furthermore, a number of new properties arise.