# Identity element

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In algebra, an **identity element** or **neutral element** with respect to a binary operation is an element which leaves the other operand unchanged, generalising the concept of zero with respect to addition or one with respect to multiplication.

Formally, let be a binary operation on a set *X*. An element *I* of *X* is an identity for if

holds for all *x* in *X*. An identity element, if it exists, is unique.

## Examples

- Existence of an identity element is one of the properties of a group or monoid.
- An identity matrix is the identity element for matrix multiplication; a zero matrix is the identity element for matrix addition.
- The empty set is the identity element for set union.