Identity element

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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.

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.

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 an inverse of y.

See also: Group, Monoid, Quasigroup.