Identity element


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 (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 (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)
positive integers least common multiple 1
nonnegative integers greatest common divisor 0 (under most definitions of GCD)
m-by-n matrices + (addition) matrix of all zeroes
n-by-n square matrices · (multiplication) In (matrix with 1 on diagonal
and 0 elsewhere
)
all functions from a set M to itself ∘ (function composition) identity function
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)
boolean logic ⊕ (Exclusive 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 (S, *) to have no identity element. The most 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 3110152487, p. 14-15

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Look at other dictionaries:

  • identity element — n. an element of a mathematical system that does not change the other elements in the system when it operates on them: zero is the identity element for addition (x + 0 = x) and one is the identity element for multiplication (y × 1 = y) …   English World dictionary

  • identity element — noun an operator that leaves unchanged the element on which it operates the identity under numerical multiplication is 1 • Syn: ↑identity, ↑identity operator • Hypernyms: ↑operator * * * noun see identity 8 * * …   Useful english dictionary

  • identity element — loginis ekvivalentumo elementas statusas T sritis automatika atitikmenys: angl. biconditional gate; equality gate; equivalence element; equivalence gate; equivalent to element; exclusive NOR gate; identity element; material equivalence element… …   Automatikos terminų žodynas

  • identity element — noun A member of a structure which, when applied to any other element via a binary operation induces an identity mapping; more specifically, given an operation ,, an element I is a left identity if I , x = x for any x in the structure See Also:… …   Wiktionary

  • identity element — noun Date: 1902 an element (as 0 in the set of all integers under addition or 1 in the set of positive integers under multiplication) that leaves any element of the set to which it belongs unchanged when combined with it by a specified operation …   New Collegiate Dictionary

  • identity element — Math. identity (def. 9b). [1900 05] * * * …   Universalium

  • Identity — may refer to:Philosophy* Identity (philosophy), the sameness of two things * Identity theory of mind, in the philosophy of mind, holds that the mind is identical to the brain * Personal identity (philosophy) * Identity (social science) * Identity …   Wikipedia

  • Identity (mathematics) — For other uses, see Identity (disambiguation). In mathematics, the term identity has several different important meanings: An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer… …   Wikipedia

  • Identity function — In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument. In terms of equations, the function is given by f ( x ) = x… …   Wikipedia

  • identity — /uy den ti tee, i den /, n., pl. identities. 1. the state or fact of remaining the same one or ones, as under varying aspects or conditions: The identity of the fingerprints on the gun with those on file provided evidence that he was the killer.… …   Universalium


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