Identity element

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Template:Single source 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. It leaves other elements unchanged when combined with them. This is used for groups and related concepts.

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 Template:Mvar with a binary operation ∗ on it (known as a magma). Then an element Template:Mvar of Template:Mvar is called a left identity if ea = a for all Template:Mvar in Template:Mvar, and a right identity if ae = a for all Template:Mvar in Template:Mvar. If Template:Mvar 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
non-negative numbers ab (exponentiation) 1 (right identity only)
integers (to extended rationals)
positive integers least common multiple 1
non-negative integers greatest common divisor 0 (under most definitions of GCD)
Template:Mvar-by-Template:Mvar matrices + (addition) zero matrix
Template:Mvar-by-Template:Mvar square matrices matrix multiplication In (identity matrix)
Template:Mvar-by-Template:Mvar matrices (Hadamard product) Jm, n (Matrix of ones)
all functions from a set Template:Mvar to itself ∘ (function composition) identity function
all distributions on a group Template:Mvar ∗ (convolution) δ (Dirac delta)
extended real numbers minimum/infimum +∞
extended real numbers maximum/supremum −∞
subsets of a set Template:Mvar ∩ (intersection) Template:Mvar
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
only two elements {e, f}  ∗ defined by
ee = fe = e and
ff = ef = f
both Template:Mvar and Template:Mvar 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 Template:Mvar is a left identity and Template:Mvar is a right identity then l = lr = r. In particular, there can never be more than one two-sided identity. If there were two, Template:Mvar and Template:Mvar, then ef would have to be equal to both Template:Mvar and Template:Mvar.

It is also quite possible for (S, ∗) to have no identity element. A common example of this is the cross product of vectors. 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