Identity element

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 $\star$ be a binary operation on a set X. An element I of X is an identity for $\star$ if

$I\star x=x=x\star I\,$

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.

See also

• Identity (mathematics)