Axiomatizability and completeness of some classes of \(S\)-polygons
type: Item
instance of: scholarly article
title: Axiomatizability and completeness of some classes of \(S\)-polygons
zbMATH Open document ID: 0773.03028
DOI: 10.1007/BF01980252
published in: Algebra and Logic
publication date: 1993-06-29T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 08C10
zbMATH DE Number: 220179
zbMATH Keywords: monoid
zbMATH Keywords: flat \(S\)-polygons
zbMATH Keywords: projective \(S\)-polygons
zbMATH Keywords: free \(S\)-polygons
zbMATH Keywords: axiomatizability
zbMATH Keywords: completeness
zbMATH Keywords: model completeness
zbMATH Keywords: categoricity
zbMATH Keywords: ascending chain condition
zbMATH Keywords: ideals
author: Alena Andreevna Stepanova
MaRDI profile type: Publication
cites work: Model-completions and modules
cites work: Rings over which all modules of a given type are almost categorical
cites work: Axiomatisability Problems for S -Systems
cites work: Perfect monoids
cites work: Perfect semigroups
cites work: Q3887676
description: scientific article; zbMATH DE number 220179