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 and rings. The term identity element is often shortened to identity (as will be done in this article) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
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). These need not be ordinary addition and multiplication, but rather arbitrary operations. The distinction is used most often for sets that support both binary operations, such as rings and fields. The multiplicative identity is often called unity in the latter context (a ring with unity). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. Unity itself is necessarily a unit.
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 | In (identity matrix) |
| m-by-n matrices | ○ (Hadamard product) | Jm, 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) | S2 |
| Two elements, {e, f} | ∗ defined by e ∗ 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
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