Varieties defined by basic equations have the amalgamation property

Abstract

A variety is a class of algebraic structures axiomatized by a set of equations. An equation is basic if there is at most one occurrence of a nonnullary operation symbol on each side. We show that a variety axiomatized by basic equations has the strong amalgamation property. Suppose further that the language has no constant symbol and, for each equation, either one side is operation-free, or exactly the same variables appear on both sides. Then also the joint embedding property holds, hence if the language is finite we get the existence of Fraïssé limits for finite algebras. Examples include virtually all the varieties defining classical Maltsev conditions. In a few special cases, the above properties are preserved when further unary operation symbols appear in the equations.