Editing Identity element

{{Short description|Specific element of an algebraic structure}}
In [[mathematics]], an '''identity element''' or '''neutral element''' of a [[binary operation]] is an element that leaves unchanged every element when the operation is applied.<ref>{{Cite web |url = http://mathworld.wolfram.com/IdentityElement.html |title = Identity Element |last = Weisstein |first = Eric W. |authorlink = Eric W. Weisstein|website = mathworld.wolfram.com |language = en |access-date = 2019-12-01 }}</ref><ref>{{Cite web |url = https://www.merriam-webster.com/dictionary/identity+element |title = Definition of IDENTITY ELEMENT |website = www.merriam-webster.com |access-date = 2019-12-01 }}</ref> For example, 0 is an identity element of the [[addition]] of [[real number]]s. This concept is used in [[algebraic structure]]s such as [[group (mathematics)|group]]s and [[ring (mathematics)|ring]]s. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity)<ref name=":0">{{Cite web |url = https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/identity-element |title = Identity Element |website = www.encyclopedia.com |access-date = 2019-12-01}}</ref> when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.

==Definitions==
Let {{math|(''S'', ∗)}} be a set&nbsp;{{mvar|S}} equipped with a [[binary operation]]&nbsp;∗. Then an element&nbsp;{{mvar|e}} of&nbsp;{{mvar|S}} is called a {{visible anchor|left identity element|text='''[[left and right (algebra)|left]] identity'''}} if {{math|1=''e'' ∗ ''s'' = ''s''}} for all&nbsp;{{mvar|s}} in&nbsp;{{mvar|S}}, and a {{visible anchor|right identity element|text='''[[left and right (algebra)|right]] identity'''}} if {{math|1=''s'' ∗ ''e'' = ''s''}} for all&nbsp;{{mvar|s}} in&nbsp;{{mvar|S}}.<ref>{{harvtxt|Fraleigh|1976|p=21}}</ref> If {{mvar|e}} is both a left identity and a right identity, then it is called a '''{{visible anchor|two-sided identity}}''', or simply an '''{{visible anchor|identity}}'''.<ref>{{harvtxt|Beauregard|Fraleigh|1973|p=96}}</ref><ref>{{harvtxt|Fraleigh|1976|p=18}}</ref><ref>{{harvtxt|Herstein|1964|p=26}}</ref><ref>{{harvtxt|McCoy|1973|p=17}}</ref><ref>{{Cite web|url=https://brilliant.org/wiki/identity-element/|title=Identity Element {{!}} Brilliant Math & Science Wiki|website=brilliant.org|language=en-us|access-date=2019-12-01}}</ref>

An identity with respect to addition is called an [[Additive identity|{{visible anchor|additive identity}}]] (often denoted as&nbsp;0) and an identity with respect to multiplication is called a '''{{visible anchor|multiplicative identity}}''' (often denoted as&nbsp;1).<ref name=":0" /> These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. In the case of a [[Group (mathematics)|group]] for example, the identity element is sometimes simply denoted by the symbol <math>e</math>. The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as [[ring (mathematics)|ring]]s, [[integral domain]]s, and [[field (mathematics)|field]]s. The multiplicative identity is often called '''{{visible anchor|unity}}''' in the latter context (a ring with unity).<ref>{{harvtxt|Beauregard|Fraleigh|1973|p=135}}</ref><ref>{{harvtxt|Fraleigh|1976|p=198}}</ref><ref>{{harvtxt|McCoy|1973|p=22}}</ref> This should not be confused with a [[unit (ring theory)|unit]] in ring theory, which is any element having a [[multiplicative inverse]]. By its own definition, unity itself is necessarily a unit.<ref>{{harvtxt|Fraleigh|1976|pp=198,266}}</ref><ref>{{harvtxt|Herstein|1964|p=106}}</ref>

==Examples==
{| class="wikitable"
! Set !! Operation !! Identity
|-
| rowspan = "2"| [[Real number]]s, [[complex number]]s|| + ([[addition]]) || [[0 (number)|0]]
|-
| · ([[multiplication]]) || [[1 (number)|1]]
|-
| [[Positive integer]]s || [[Least common multiple]] || 1
|-
| [[Non-negative integer]]s || [[Greatest common divisor]] || 0 (under most definitions of GCD)
|-
| rowspan = "2"| [[Vector (mathematics and physics)|Vectors]] || [[Vector addition]]
| [[Zero vector]]
|-
|[[Scalar multiplication]] || [[1 (number)|1]]
|-
<!-- ||' ''R'''<sup>''n''</sup> || · (multiplication) || [[1 (number)|1]] -->
| {{mvar|m}}-by-{{mvar|n}} [[matrix (mathematics)|matrices]] || [[Matrix addition]]
| [[Zero matrix]]
|-
| {{mvar|n}}-by-{{mvar|n}} square matrices || [[Matrix multiplication]]
| ''I''<sub>''n''</sub> ([[identity matrix]])
|-
| {{mvar|m}}-by-{{mvar|n}} matrices || ○ ([[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]]) || {{math|''δ''}} ([[Dirac delta]])
|-
| rowspan = "2" | [[Extended real number]]s || [[Minimum]]/infimum || +∞
|-
|| [[Maximum]]/supremum || −∞
|-
| rowspan = "2" | Subsets of a [[Set (mathematics)|set]]&nbsp;{{mvar|M}} || ∩ ([[set intersection|intersection]]) || {{mvar|M}}
|-
|| ∪ ([[set union|union]]) || ∅ ([[empty set]])
|-
| [[string (computer science)|Strings]], [[tuple|lists]] || [[Concatenation]] || [[Empty string]], empty list
|-
| rowspan = "4" | A [[Boolean algebra (structure)|Boolean algebra]] || <math display="inline">\and</math> ([[logical conjunction|conjuction]]) || <math display="inline">\top</math> ([[Tautology (logic)|truth]])
|-
|| <math display="inline">\leftrightarrow</math> ([[logical biconditional|equivalence]]) || <math display="inline">\top</math> ([[Tautology (logic)|truth]])
|-
||  <math display="inline">\vee</math> ([[Logical disjunction|disjunction]]) || <math display="inline">\bot</math> ([[Contradiction|falsity]])
|-
|| <math display="inline">\nleftrightarrow</math> ([[Exclusive or|nonequivalence]]) || <math display="inline">\bot</math> ([[Contradiction|falsity]])
|-
| [[knot (mathematics)|Knots]] || [[Knot sum]] || [[Unknot]]
|-
| [[Compact surfaces]] || # ([[connected sum]]) || [[sphere|''S''<sup>2</sup>]]
|-
| [[Group (mathematics)|Groups]] || [[Direct product]] || [[Trivial group]]
|-
| 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
|-
| [[Homogeneous relation]]s on a set ''X'' || [[Relative product]] || [[Identity relation]]
|-
| [[Relational algebra]] || [[Natural join]] (⨝) || The unique relation [[relation of degree zero|degree zero]] and cardinality one
|}

==Properties==
In the example ''S'' = {''e,f''} with the equalities given, ''S'' is a [[semigroup]]. It demonstrates the possibility for {{math|(''S'', ∗)}} to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in 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, say {{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,<ref>{{harvtxt|McCoy|1973|p=22}}</ref> such as the case of even integers under the multiplication operation.<ref name=":0" /> Another common example is the [[cross product]] of [[Euclidean vector|vectors]], where 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. That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of structure without identity element involves the additive [[semigroup]] of [[Positive number|positive]] [[natural number]]s.

==See also==

* [[Absorbing element]]
* [[Additive inverse]]
* [[Generalized inverse]]
* [[Identity (mathematics)|Identity (equation)]]
* [[Identity function]]
* [[Inverse element]]
* [[Monoid]]
* [[Pseudo-ring #Properties weaker than having an identity|Pseudo-ring]]
* [[Quasigroup]]
* [[Unital (disambiguation)]]

==Notes and references==

{{reflist}}

==Bibliography==

* {{citation | last1 = Beauregard | first1 = Raymond A. | last2 = Fraleigh | first2 = John B. | title = A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields | location = Boston | publisher = [[Houghton Mifflin Company]] | year = 1973 | isbn = 0-395-14017-X | url-access = registration | url = https://archive.org/details/firstcourseinlin0000beau }}
* {{ citation | last1 = Fraleigh | first1 = John B. | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}
* {{ citation | last1 = Herstein | first1 = I. N. |authorlink = I. N. Herstein| title = Topics In Algebra | location = Waltham | publisher = [[Blaisdell Publishing Company]] | year = 1964 | isbn = 978-1114541016 }}
* {{ citation | last1 = McCoy | first1 = Neal H. | title = Introduction To Modern Algebra, Revised Edition | location = Boston | publisher = [[Allyn and Bacon]] | year = 1973 | lccn = 68015225 }}

==Further reading==

* 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:Algebraic properties of elements]]
[[Category:Binary operations|*Identity element]]
[[Category:Properties of binary operations]]
[[Category:1 (number)]]