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Axiomatizability and completeness of some classes of $$S$$-polygons. (English. Russian original) Zbl 0773.03028
Algebra Logic 30, No. 5, 379-388 (1991); translation from Algebra Logika 30, No. 5, 583-594 (1991).
Let $$K$$ be a class of algebraic systems of a signature $$\Sigma$$. Recall that a class $$K$$ is said to be axiomatizable if there exists a set $$Z$$ of propositions of signature $$\Sigma$$ such that the class $$K$$ contains precisely those systems on which are propositions in $$Z$$ are true. A class $$K$$ is said to be complete (model complete) if the theory of the class $$K_ \infty$$ of all infinite systems in $$K$$ is complete (model complete). A class $$K$$ is said to be categorical if it is categorical in some uncountable cardinality.
Let $$S$$ be a monoid; let $$K$$ be the class of flat $$S$$-polygons, or the class of projective $$S$$-polygons, or the class of free $$S$$-polygons; let a property $$P$$ of the class $$K$$ be axiomatizability, or completeness, or model completeness, or categoricity. It is natural to raise this qestion: What conditions should the monoid $$S$$ satisfy in order for the class $$K$$ to possess the property $$P$$? Necessary and sufficient conditions that one has to impose on a monoid $$S$$ in order for the class of flat $$S$$-polygons to be axiomatizable are stated by V. Gould [J. Lond. Math. Soc., II. Ser. 35, 193-201 (1987; Zbl 0637.03029)]. That paper also proves that for a monoid $$S$$ satisfying the ascending chain condition $$M^ L$$ for principal left ideals the axiomatizability of the class of projective $$S$$-polygons is equivalent to the axiomatizability of the class of flat $$S$$-polygons and to the monoid $$S$$ being perfect. Theorem 1 of the present article generalizes this result to the case of an arbitrary monoid. Theorem 2 provides a description of a monoid $$S$$with finitely many right ideals for which the class of free $$S$$-polygons is axiomatizable. Theorems 3 and 4 prove that for a (commutative) monoid $$S$$, completeness, model completeness, and categoricity of the class $${\mathcal P}({\mathcal F})$$ are equivalent to $$S$$ being a group.

##### MSC:
 03C60 Model-theoretic algebra 08C10 Axiomatic model classes
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##### References:
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