Geodesic
Model-theoretic properties of free, projective, and flat $S$-acts
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 7, pp. 63-110.

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This is the second in a series of articles surveying the body of work on the model theory of $S$-acts over a monoid $S$. The first concentrated on the theory of regular $S$-acts. Here we review the material on model-theoretic properties of free, projective, and (strongly, weakly) flat $S$-acts. We consider questions of axiomatizability, completeness, model completeness, and stability for these classes. Most but not all of the results have already appeared; we remark that the description of those monoids $S$ such that the class of free left $S$-acts is axiomatizable, is new. 
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V. Gould; A. V. Mikhalev; E. A. Palyutin; A. A. Stepanova. Model-theoretic properties of free, projective, and flat $S$-acts. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 7, pp. 63-110. http://geodesic.mathdoc.fr/item/FPM_2008_14_7_a6/

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