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In [[mathematics]], an '''identity element''' (or '''neutral element''') is a special type of element of a [[Set (mathematics)|set]] with respect to a [[binary operation]] on that set. It leaves other elements unchanged when combined with them. This is used for [[group (mathematics)|group]]s 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.
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</ul>


Let {{math|(''S'', ∗)}} be a set&nbsp;{{mvar|S}} with a binary operation&nbsp;∗ on it (known as a [[magma (algebra)|magma]]). Then an element&nbsp;{{mvar|e}} of&nbsp;{{mvar|S}} is called a '''[[left and right (algebra)|left]] identity''' if {{math|1=''e'' ∗ ''a'' = ''a''}} for all&nbsp;{{mvar|a}} in&nbsp;{{mvar|S}}, and a '''[[left and right (algebra)|right]] identity''' if {{math|1=''a'' ∗ ''e'' = ''a''}} for all&nbsp;{{mvar|a}} in&nbsp;{{mvar|S}}. If {{mvar|e}} is both a left identity and a right identity, then it is called a '''two-sided identity''', or simply an '''identity'''.
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An identity with respect to addition is called an [[additive identity]] (often denoted as&nbsp;0) and an identity with respect to multiplication is called a [[multiplicative identity]] (often denoted as&nbsp;1). The distinction is used most often for sets that support both binary operations, such as [[ring (mathematics)|ring]]s. The multiplicative identity is often called the '''unit''' in the latter context, where, though, a [[unit (ring theory)|unit]] is often used in a broader sense, to mean an element with a multiplicative inverse.
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==Examples==
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{| class="wikitable" style="margin: 1em auto 1em auto"
 
!set!!operation!!identity
  <li>[http://itaob.com/news/html/?43039.html http://itaob.com/news/html/?43039.html]</li>
|-
 
|[[real number]]s||+ ([[addition]])||[[0 (number)|0]]
  <li>[http://enseignement-lsf.com/spip.php?article64#forum18173160 http://enseignement-lsf.com/spip.php?article64#forum18173160]</li>
|-
 
|real numbers||· ([[multiplication]])||[[1 (number)|1]]
  <li>[http://freedomtorock.com/activity/p/14719/ http://freedomtorock.com/activity/p/14719/]</li>
|-
 
|[[non-negative number]]s
</ul>
| rowspan=2 |{{math|''a''<sup>''b''</sup>}} ([[exponentiation]])
| rowspan=2 |1 (right identity only)
|-
|[[integer]]s (to [[projective line|extended]] [[rational number|rationals]])
|-
|[[positive integer]]s||[[least common multiple]]||1
|-
|[[non-negative integer]]s||[[greatest common divisor]]||0 (under most definitions of GCD)
|-
<!-- ||'''R'''<sup>''n''</sup>  ||· (multiplication)|[[1 (number)|1]] -->
|{{mvar|m}}-by-{{mvar|n}} [[matrix (mathematics)|matrices]]|| + (addition)
|[[zero matrix]]
|-
|{{mvar|n}}-by-{{mvar|n}} square matrices|| [[matrix multiplication]]
|''I''<sub>''n''</sub> ([[identity matrix]])
|-
|{{mvar|m}}-by-{{mvar|n}} matrices|| <math>\circ</math> ([[Hadamard product (matrices)|Hadamard product]])
| {{math|''J''<sub>''m'', ''n''</sub>}} ([[Matrix of ones]])
|-
|all [[function (mathematics)|functions]] from a set&nbsp;{{mvar|M}} to itself||∘ ([[function composition]])||[[identity function]]
|-
|all [[distribution (mathematics)|distributions]] on a [[group (mathematics)|group]]&nbsp;{{mvar|G}}<!-- a crap refurbished -->||∗ ([[convolution]])||δ ([[Dirac delta]])
|-
|[[extended real number]]s || [[minimum]]/infimum || +∞
|-
|extended real numbers || [[maximum]]/supremum || −∞
|-
|subsets of a [[Set (mathematics)|set]]&nbsp;{{mvar|M}} || ∩ ([[set intersection|intersection]]) || {{mvar|M}}
|-
|sets || ∪ ([[set union|union]]) || ∅ ([[empty set]])
|-
|[[string (computer science)|strings]], [[tuple|lists]]|| [[concatenation]] || [[empty string]], empty list
|-
|a [[boolean algebra (structure)|boolean algebra]] || ∧ ([[logical and]]) || ⊤ (truth)
|-
|a boolean algebra || ∨ ([[logical or]]) || ⊥ (falsity)
|-
|a boolean algebra || ⊕ ([[Exclusive or]]) || ⊥ (falsity)
|-
|[[knot (mathematics)|knots]] || [[knot sum]] || [[unknot]]
|-
|[[compact surfaces]] || # ([[connected sum]]) || [[sphere|''S''<sup>2</sup>]]
|-
|only two elements {{math|{''e'', ''f''} }}
|∗ defined by<br> {{math|1=''e'' ∗ ''e'' = ''f'' ∗ ''e'' = ''e''}} and <br> {{math|1=''f'' ∗ ''f'' = ''e'' ∗ ''f'' = ''f''}}
|both {{mvar|e}} and {{mvar|f}} are left identities,<br> but there is no right identity<br> and no two-sided identity
|}
 
==Properties==
As the last example shows, it is possible for {{math|(''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 {{mvar|l}} is a left identity and {{mvar|r}} is a right identity then {{math|1=''l'' = ''l'' ∗ ''r'' = ''r''}}. In particular, there can never be more than one two-sided identity.  If there were two, {{mvar|e}} and {{mvar|f}}, then {{math|''e'' ∗ ''f''}} would have to be equal to both {{mvar|e}} and {{mvar|f}}.
 
It is also quite possible for {{math|(''S'', ∗)}} to have ''no'' identity element. A common example of this is the [[cross product]] of [[Euclidean vector|vectors]]. The absence of an identity element is related to the fact that the [[Direction (geometry)|direction]] of any nonzero cross product is always [[orthogonal]] to any element multiplied&nbsp;– 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 number|positive]] [[natural number]]s.
 
==See also==
*[[Absorbing element]]
*[[Inverse element]]
*[[Additive inverse]]
*[[Monoid]]
*[[Unital (disambiguation)]]
*[[Quasigroup]]
*[[Pseudo-ring #Properties weaker than having an identity|Pseudo-ring]]
 
==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.&nbsp;29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p.&nbsp;14–15
 
[[Category:Abstract algebra]]
[[Category:Algebra]]
[[Category:Binary operations|*Identity element]]
[[Category:One]]
 
__NOTOC__

Revision as of 19:37, 14 February 2014

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