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Freeness in classes without equality

Published online by Cambridge University Press:  12 March 2014

Raimon Elgueta
Raimon Elgueta*
Affiliation:
Departament de Matemètica Aplicada II, Universitat Politécnica de Catalunya, Pau Gargallo 5, 08028 Barcelona, Spain E-mail: elgueta@ma2.upc.es
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Abstract

This paper is a continuation of [27], where we provide the background and the basic tools for studying the structural properties of classes of models over languages without equality. In the context of such languages, it is natural to make distinction between two kinds of classes, the so-called abstruct classes, which correspond to those closed under isomorphic copies in the presence of equality, and the reduced classes, i.e., those obtained by factoring structures by their largest congruences. The generic problem described in [27] is to investigate under what conditions this reduction process does not alter the metatheory of a class.

Here we focus our attention on a concrete aspect of this generic problem that we import from universal algebra, namely the existence and description of free models. As in [27], we can find here again the basic notion of protoalgebraicity, which was originally introduced in [7] as the weakest condition to guarantee that the reduction process behaves reasonably well from an algebraic point of view. Our concern, however, takes us to handle a further notion, that of semialgebraicity, which corresponds to the notion of equivalential logic of [18]; semialgebraicity turns out to be the property which ensures that freeness is fully preserved by the reduction process.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 1159 - 1194
Copyright
Copyright © Association for Symbolic Logic 1999

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