NOTES ON EFFECTIVE DESCENT AND PROJECTIVITY IN QUASIVARIETIES OF UNIVERSAL ALGEBRAS Dedicated to Walter Tholen on the occasion of his sixtieth birthday

We present sufficient conditions under which effective descent morphisms in a quasivariety of universal algebras are the same as regular epimorphisms and examples for which they are the same as regular epimorphisms satisfying projectivity. 1. Preliminaries A variety is a full subcategory of the category of structures for a first order (one sorted) language, closed under substructures, products and homomorphic images. It is a regular category not necessarily exact for which effective descent morphisms are exactly the regular epimorphisms (strong surjective homomorphisms). The same is true of "prevarieties" (full subcategories of the category of structures, closed under substructures, products and strong homomorphic images) [4]. Any quasivariety is the subcategory of a variety orthogonal to a set of epimorphisms which are either strong surjective homomorphisms or bijective homomorphisms. Projectivity of the domain of such a bijective homomorphism w.r.t. a regular epimorphism p was shown in [2] to be a necessary and sufficient condition for p to be an effective descent morphism in models of Preorder. In the case of universal algebras varieties are exact categories and consequently their effective descent morphisms are the regular epimorphisms i.e., the surjective homomorphisms. A quasivariety of universal algebras as the subcategory of a variety of universal algebras orthogonal to a set of regular epimorphisms is a regular category whose regular epimorphisms are again the surjective homomorphisms. A quasivariety of universal algebras is (a full subcategory of the category of structures for a first order (one sorted) algebraic language) axiomatizable by quasi-identities [3], that is, by sentences of the form