Universal Algebra

Abstract

Abstract In this chapter, we present some very general ideas from the field initiated in its modem form by Garrett Birkhoff and called by him “universal algebra. “ The fundamental notion of universal algebra is that of an algebra. Basically, an algebra is a set together with various operations that take elements of that set and yield elements of that same set. A simple, and familiar, example is the set of natural numbers together with the operations of addition and multiplication. In particular, any two natural numbers can be added (or multiplied), and the result is moreover a natural number. By contrast, the natural numbers together with the operation of subtraction do not form an algebra; for although the difference of any two natural numbers exists, it need not be a natural number. A more dramatic non-example is the set of natural numbers together with the operation of division; in particular, there is no such thing as 1 divided by 0. In the first non-example, the operation can yield a result not in the set; in the second non-example, the operation can yield no result at all.