ABSTRACT
Our study of polynomial identities has led us to introduce some variants of PI, namely superidentities, trace identities, and LGIs, which we have handled in an ad hoc fashion in order to push forward to our main objective, Kemer's solution of Specht's problem in characteristic 0, and the surrounding theory. Now that we can pause for breath, let us reformulate identities from the point of view of universal algebra (using the Pi-theory for associative algebras as our point of departure). At the same time, we shall introduce varieties and note some of the parallels with algebraic geometry.
11.1 Identities in Universal Algebra A good exposition of universal algebra can be found in [Jac80, Chapter 2]; we only review some basic properties. An (abstract) algebraic system is a collection of sets Ai, A2, . . . endowed with a set fi of various ra-ary operations for m € N, each typically notated as
