Definitions

multiplicative identity

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. 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 when there is no possibility of confusion; we do so in this article.

Let (S,*) be a set S with a binary operation * on it (known as a magma). 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, unfortunately, a unit is also sometimes used to mean an element with a multiplicative inverse.

Examples

set operation identity
real numbers + (addition) 0
real numbers • (multiplication) 1
real numbers ab (exponentiation) 1 (right identity only)
m-by-n matrices + (addition) matrix of all zeroes
n-by-n square matrices • (multiplication) matrix with 1 on diagonal
and 0 elsewhere
all functions from a set M to itself ∘ (function composition) identity map
all functions from a set M to itself * (convolution) δ (Dirac delta)
character strings, lists concatenation empty string, empty list
extended real numbers minimum/infimum +∞
extended real numbers maximum/supremum −∞
subsets of a set M ∩ (intersection) M
sets ∪ (union) { } (empty set)
boolean logic ∧ (logical and) ⊤ (truth)
boolean logic ∨ (logical or) ⊥ (falsity)
compact surfaces # (connected sum)
only 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 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 an algebra to have no identity element. The most common examples of this are the dot product and cross product of vectors. In the former case the lack of an identity element is related to the fact that the elements multiplied are vectors but the product is a scalar. With cross products 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 vector in the same direction as the original.

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 3110152487, p. 14-15

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