Quasi-Equational Logic for Partial Algebras

In the line of introducing a manageable model theoretic approach to partial algebras, here such classes of partial algebras are to be considered in which free algebras still exist (in a categorical language: which are epireflective). This note is to be understood as one among others introducing this kind of model theory (in another one, see [4], varieties of partial algebras are considered). We want to make an end to the widely spread opinion that there are several equational theories and several notions of validity around for partial algebras. At the same time we want to provide for all those who might use partial algebras such tools that they hopefully can really work with. We just use the usual first order formulas of a model theoretic language with terms but substituting the notion of “equation” by that of “existenceequation”, and we intend to give a procedure how to interpret their satisfaction and their validity in partial algebras. In this note we shall restrict ourselves to existence-equations and quasi-existence-equations (the latter comparable to the notion of quasi-equation in [8] §11.1., i.e. essentially: universally quantified Horn-formulas).