Freeness in classes without equality

Abstract

AbstractThis 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-calledabstruct classes, which correspond to those closed under isomorphic copies in the presence of equality, and thereduced classes, i.e., those obtained by factoring structures by their largest congruences. The generic problem described in [27] is to investigate under what conditions thisreduction processdoes 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 ofprotoalgebraicity, 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 ofsemialgebraicity, which corresponds to the notion ofequivalential logicof [18]; semialgebraicity turns out to be the property which ensures that freeness is fully preserved by the reduction process.