Associative Property - Non-associativity

Non-associativity

A binary operation on a set S that does not satisfy the associative law is called non-associative. Symbolically,

For such an operation the order of evaluation does matter. For example:

  • Subtraction
(5-3)-2 \, \ne \, 5-(3-2)
  • Division
(4/2)/2 \, \ne \, 4/(2/2)
  • Exponentiation
2^{(1^2)} \, \ne \, (2^1)^2

Also note that infinite sums are not generally associative, for example:

(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+\dots \, = \, 0

whereas

1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+\dots \, = \, 1

The study of non-associative structures arises from reasons somewhat different from the mainstream of classical algebra. One area within non-associative algebra that has grown very large is that of Lie algebras. There the associative law is replaced by the Jacobi identity. Lie algebras abstract the essential nature of infinitesimal transformations, and have become ubiquitous in mathematics. They are an example of non-associative algebras.

There are other specific types of non-associative structures that have been studied in depth. They tend to come from some specific applications. Some of these arise in combinatorial mathematics. Other examples: Quasigroup, Quasifield, Nonassociative ring.

Read more about this topic:  Associative Property