Projectivity of Universal Algebras

Abstract

The basic relations between universal algebras are homomorphisms and their special cases – monomorphisms, epimorphisms, and isomorphisms (cf., for example, (Kurosh, 1963; Cohn, 1965)). In this paper, we introduce and study a novel relation between universal algebras, which is called projectivity and allows obtaining properties of universal algebras from properties of those universal algebras to which they are weakly or finally projective. The key property of projectivity relations is that they transfer operations from one prearithmetic to another. In addition, given a universal algebra, the projectivity relation allows building universal algebras of the same type on arbitrary sets.