Varieties, Equational Classes, and Free Algebras

Abstract

In this Chapter, only certain classes of algebras of the same type shall be regarded.

First we introduce so-called varieties as classes of algebras, which are closed in respect to formation of subalgebras, homomorphic images, and direct products. We then come to a method for constructing algebra classes that strongly differs from the first method at first sight: Starting from certain equations from variables and operation symbols of a certain type τ, we form the class of all algebras of type τ that fulfill these equations. The result is an equational class. We will see, however, that there is a close connection between the two methods of algebra class construction: A class of algebras is equationally definable if and only if it is a variety.

Free algebras are “the most general” algebras within a variety or an equational class (or an equationally definable class). In the section on equational classes, we will also address such concepts such as conclusion of an equational set. In addition, we investigate methods to receive such conclusions.