Basic Concepts of Algebra

Abstract

An algebra A is a set A together with one or more operations f _i . We may represent an algebra by writing

$${\rm{ }}A = \left\langle {A,{\rm{ }}{f_1},{\rm{ }}{f_2}{\rm{ }}...{\rm{, }}{f_n}} \right\rangle$$

(9-1)

or by using particular symbols for the operations, such as

$${\rm{ }}{\bf{A}}{\rm{ = }}\left\langle {A,{\rm{ + , }} \times } \right\rangle$$

(9-2)

The set A may be finite or infinite, and there may be either a finite or an infinite number of different operations. However, each operation must be finitary, i.e. unary, binary, ternary .... Each n-ary operation must be a well-defined operation, i.e., defined for all n-tuples of elements of A and yielding a unique element of A as a value for each n-tuple (cf. the mapping condition for functions in Section 2.3).