Stability of the Theory of Existentially Closed S-acts over a Right Coherent Monoid S

Let LS denote the language of (right) S-acts over a monoid S and let ΣS be a set of sentences in LS which axiomatises S-acts. A general result of model theory says that ΣS has a model companion, denoted by TS , precisely when the class E of existentially closed S-acts is axiomatisable and in this case, TS axiomatises E . It is known that TS exists if and only if S is right coherent. Moreover, by a result of Ivanov, TS has the model-theoretic property of being stable. In the study of stable first order theories, superstable and totally transcendental theories are of particular interest. These concepts depend upon the notion of type: we describe types over TS algebraically, thus reducing our examination of TS to consideration of the lattice of right congruences of S. We indicate how to use our result to confirm that TS is stable and to prove another result of Ivanov, namely that TS is superstable if and only if S satisfies the maximal condition for right ideals. The situation for total transcendence is more complicated but again we can use our description of types to ascertain for which right coherent monoids S we have that TS is totally transcendental and is such that the U -rank of any type coincides with its Morley rank.