Axiomatizability and completeness of some classes of \(S\)-polygons
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 \textit{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.
scientific article; zbMATH DE number 220179