# Identity element

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, which leaves other elements unchanged when combined with them. This concept is used in algebraic structures such as groups. The term *identity element* is often shortened to *identity* (as will be done in this article) when there is no possibility of confusion.

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**.

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 multiplicative identity (often denoted as 1). The distinction is used most often for sets that support both binary operations, such as rings. The multiplicative identity is often called the **unit** in the latter context, where, though, a unit is often used in a broader sense, to mean an element with a multiplicative inverse.

## Examples

Set | Operation | Identity |
---|---|---|

Real numbers | + (addition) | 0 |

Real numbers | · (multiplication) | 1 |

Positive integers | Least common multiple | 1 |

Non-negative integers | Greatest common divisor | 0 (under most definitions of GCD) |

m-by-n Matrices | + (addition) | Zero matrix |

n-by-n square matrices | Matrix multiplication | I_{n} (identity matrix) |

m-by-n matrices | ∘ (Hadamard product) | J_{m, n} (Matrix of ones) |

All functions from a set, M, to itself | ∘ (function composition) | Identity function |

All distributions on a group, G | ∗ (convolution) | δ (Dirac delta) |

Extended real numbers | Minimum/infimum | +∞ |

Extended real numbers | Maximum/supremum | −∞ |

Subsets of a set M | ∩ (intersection) | M |

Sets | ∪ (union) | ∅ (empty set) |

Strings, lists | Concatenation | Empty string, empty list |

A Boolean algebra | ∧ (logical and) | ⊤ (truth) |

A Boolean algebra | ∨ (logical or) | ⊥ (falsity) |

A Boolean algebra | ⊕ (exclusive or) | ⊥ (falsity) |

Knots | Knot sum | Unknot |

Compact surfaces | # (connected sum) | S^{2} |

Two elements, {e, f} |
∗ defined bye ∗ e = f ∗ e = e and f ∗ f = e ∗ f = f |
Both e and f are left identities, but there is no right identity and no two-sided identity |

## Properties

As the last example (a semigroup) 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. If there were two, e and f, then *e* ∗ *f* would have to be equal to both e and f.

It is also quite possible for (*S*, ∗) to have *no* identity element. A common example of this is the cross product of vectors; in this case, the absence of an identity element is related to the fact that the direction of any nonzero cross product is always orthogonal to any element multiplied – so that it is not possible to obtain a non-zero vector in the same direction as the original. Another example would be the additive semigroup of positive natural numbers.

## See also

- Absorbing element
- Additive inverse
- Inverse element
- Monoid
- Pseudo-ring
- Quasigroup
- Unital (disambiguation)

## References

- M. Kilp, U. Knauer, A.V. Mikhalev,
*Monoids, Acts and Categories with Applications to Wreath Products and Graphs*, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p. 14–15