Encyclopedia > Identity element

  Article Content

Identity element

In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set.

The term identity element is often shortened to identity when there is no possibility of confusion, and we will do so in this article.

Let S be a set with a binary operation * on it. Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

For example, if (S,*) denotes the real numbers with addition, then 0 is an identity. If (S,*) denotes the real numbers with multiplication, then 1 is an identity. If (S,*) denotes the n-by-n square matrices with addition, then the zero matrix is an identity. If (S,*) denotes the n-by-n matrices with multiplication, then the identity matrix is an identity. If (S,*) denotes the set of all functions from a set M to itself, with function composition as operation, then the identity map is an identity. If S has only two elements, e and f, and the operation * is defined by e * e = f * e = e and f * f = e * f = f, then both e and f are left identities, but there is no right or two-sided identity.

As the last example shows, it is possible for (S,*) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if l is a left identity and r is a right identity then l = l * r = r. In particular, there can never be more than one two-sided identity.

If e is an identity of (S,*) and a * b = e, then a is called a left inverse of b and b is called a right inverse of a. If an element x is both a left inverse and a right inverse of y, then x is called a two-sided inverse, or simply an inverse, of y.

As with identities, it is possible for an element y to have several left inverses or several right inverses. y can even have several left inverses and several right inverses. However if the operation * is associative, then if y has both a left inverse and a right inverse, then they are equal.

See also: Group, Monoid, Quasigroup.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
North Lindenhurst, New York

... more races. 11.66% of the population are Hispanic or Latino of any race. There are 3,808 households out of which 37.3% have children under the age of 18 living with them, ...

This page was created in 47.7 ms