A categorical generalization of a theorem of G. Birkhoff on primitive classes of universal algebras
Karel Drbohlav · Czech digital mathematics library · 1965
A primitive class (cf [10]) of universal (or abstract) algebras of some finitary type t is a class which consists exactly of all algebras of type x in which certain equational relations hold true identically.More precisely, let F be any free algebra of type X and let p be any binary relation on F .Let (C (F, p ) be the class of all algebras A of type X such that for any homomorphism y : F'.-•A X j> 10 implies X Cf • y. cf in A .Now, a primitive class P of algebras of type X is simply a class for which there exist some F and p with IP* £CF, p).A wellknown theorem of G. Birkhoff (cf [l]) states that a class P of algebras of type X is primitive if aftf anly if it contains with every algebra A all its subalgebraa and factoralgebras and if it is closed under formation of cartesian products.... Categorical methods seem to be especially convenient for investigating primitive classes of universal algebras and related questions (e.g., cf C7j,[5J,fll]).However, in the present paper we try to find a categorical generalization of the Birkhoff's theorem which would pass over the limits of categories of algebras.Really, there are categories without free joins which our theorem 1,15 does concern.We shall apply