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Model-theoretic properties of free, projective, and flat \(S\)-acts. (English. Russian original) Zbl 1288.03026

J. Math. Sci., New York 164, No. 2, 195-227 (2010); translation from Fundam. Prikl. Mat. 14, No. 7, 63-110 (2008).
Summary: 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 [A. V. Mikhalev et al., Fundam. Prikl. Mat. 10, No. 4, 107–157 (2004); translation in J. Math. Sci., New York 140, No. 2, 250–285 (2007; Zbl 1073.03020)] 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.

MSC:

03C60 Model-theoretic algebra
20M30 Representation of semigroups; actions of semigroups on sets

Citations:

Zbl 1073.03020
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References:

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