Flatness properties of acts over semigroups

Flatness properties of acts over monoids have been studied in considerable detail. Much less has been done for acts over semigroups, partially because some results carry over directly and partially because the situation is more complex. Recall that flatness of an act A means the preservation of all diagrams of a certain kind under tensoring with A. We consider various conditions for describing different types of flatness for (unitary) acts: pullback flatness, equalizer flatness, finite limit flatness, limit flatness; the preservation of said flatness by certain constructions, and the interrelations between the various subclasses of acts that arise during this process. We also study pure epimorphisms of acts and describe pullback flatness of acts in terms of pure epimorphisms, avoiding non-categorical notions. In most cases, the notion of firmness from the Morita theory of semigroups plays an important role, either by being a necessity or a way to lessen the number of diagrams that need to be considered.