Identity element

From Example Problems
Jump to navigation Jump to search
For other uses, see identity (disambiguation).

In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. This is used for groups and related concepts.

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

Let (S,*) be a set S 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.

Examples

set operation identity
real numbers + (addition) 0
real numbers • (multiplication) 1
n-by-n square matrices + (addition) zero matrix
all functions from a set M to itself function composition identity map
only two elements {e, f} * defined by
e * e = f * e = e and
f * f = e * f = f
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.

See also

ar:عنصر حيادي cs:Neutrální prvek de:Neutrales Element et:Ühikelement es:Elemento neutro fr:Élément neutre hu:Neutrális elem nl:Neutraal element ja:単位元 pl:Element neutralny pt:Elemento neutro sk:Neutrálny prvok sl:Enak element sv:Neutralt element zh:單位元