Algebraic theory of quasivarieties of heterogeneous partial algebras

Abstract

Based on existence equations, quasivarieties of heterogeneous partial algebras have the same algebraic description as those of total algebras. Because of the restriction of the valuations to the free variables of a formula -the usual reference to the needed variables e.g. for identities (in order to get useful and manageable results) is essentially replaced here by the use of the "logical Craig projections" -already varieties of heterogeneous partial algebras behave to some extent rather like quasivarieties than having the properties known from varieties of total homogeneous algebras. It is one of the main aims of this note to make this more explicit. On the other hand we want to list several results known for quasivarieties of heterogeneous partial algebras -and adopt them to the extended signature -after having recalled the language and the main concepts necessary for the understanding of the results.