In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
Definitions
Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an .
An identity with respect to addition is called an Additive identity| (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit. The symbol was chosen because it is the first letter in Einheit, the German for "unit" or "unity."
Examples
{| class="wikitable"
! Set !! Operation !! Identity
|-
| Real numbers, complex numbers|| + (addition) || 0
|-
| Real numbers, complex numbers, excluding 0||· (multiplication) || 1
|-
| Positive integers || Least common multiple || 1
|-
| Non-negative integers || Greatest common divisor || 0 (under most definitions of GCD)
|-
| rowspan = "2"| Vectors || Vector addition
| Zero vector
|-
|Scalar multiplication || 1
|-
<!-- ||' R<sup>n</sup> || · (multiplication) || 1 -->
| -by- matrices || Matrix addition
| Zero matrix
|-
| -by- square matrices || Matrix multiplication
| I<sub>n</sub> (identity matrix)
|-
| -by- matrices || ○ (Hadamard product)
| (matrix of ones)
|-
| All functions from a set, , to itself || ∘ (function composition) || Identity function
|-
| All distributions on a group, <!-- a crap refurbished --> || ∗ (convolution) || (Dirac delta)
|-
| rowspan = "2" | Extended real numbers || Minimum/infimum || +∞
|-
|| Maximum/supremum || −∞
|-
| rowspan = "2" | Subsets of a set || ∩ (intersection) ||
|-
|| ∪ (union) || ∅ (empty set)
|-
| Strings, lists || Concatenation || Empty string, empty list
|-
| rowspan = "4" | A Boolean algebra || <math display="inline">\and</math> (conjunction) || <math display="inline">\top</math> (truth)
|-
|| <math display="inline">\leftrightarrow</math> (equivalence) || <math display="inline">\top</math> (truth)
|-
|| <math display="inline">\vee</math> (disjunction) || <math display="inline">\bot</math> (falsity)
|-
|| <math display="inline">\nleftrightarrow</math> (nonequivalence) || <math display="inline">\bot</math> (falsity)
|-
| Knots || Knot sum || Unknot
|-
| Compact surfaces || # (connected sum) || S<sup>2</sup>
|-
| Abstract groups || Direct product || Trivial group
|-
| Two elements,
| ∗ defined by<br> and <br>
| Both and are left identities,<br> but there is no right identity<br> and no two-sided identity
|-
| Homogeneous relations on a set X || Relative product || Identity relation
|-
| Relational algebra || Natural join (⨝) || The unique relation degree zero and cardinality one
|-
| A unital magma || Its operation || Its identity element
|}
Properties
In the example S = {e,f} with the equalities given, S is a semigroup. It demonstrates the possibility for to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in a single two-sided identity.
To see this, note that if is a left identity and is a right identity, then . In particular, there can never be more than one two-sided identity: if there were two, say and , then would have to be equal to both and .
It is also quite possible for to have no identity element, such as the case of even integers under the multiplication operation.