ABSTRACT
In Section 1.2. we defined semigroups as algebras (G;.) of type T = (2) where the associative law
x • (y • z) (x • y) • z is satisfied. Satisfaction of this law means that for all elements x, y, z E G, the equation x • (y • z) = (x • y) • z holds. To write this equation, we need the symbols x, y and z. But these symbols are not themselves elements of G; they are only symbols for which elements from G may be substituted. Such symbols are called variables. To write identities or laws of an algebra we need a language, which must include such variables as well as symbols to represent the operations. In the associative law above we did not distinguish between the operation and the symbol used to denote it, using • for both, but in our new formal language we shall usually make such a distinction. That is, we will have formal operation symbols distinct from concrete operations on a set.
