STABILITY OF THE THEORY OF EXISTENTIALLY CLOSED S-ACTS OVER A RIGHT COHERENT MONOID S
For Professor L.A. Bokut on the occasion of his 70th birthday.
The authors are grateful to a careful referee for many insightful comments. The second author completed this paper during a visit to C.A.U.L. partially funded by FCT and FEDER, Project ISFL-1-143. She would like to thank C.A.U.L. and Gracinda Gomes for their kind hospitality.
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 
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.
