Identity element: Difference between revisions
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:<span class="dablink">''For other uses, see [[identity (disambiguation)]].''</span> |
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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. |
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. |
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The term ''identity element'' is often shortened to ''identity'' when there is no possibility of confusion; we do so in this article. |
The term ''identity element'' is often shortened to ''identity'' when there is no possibility of confusion; we do so in this article. |
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Let |
Let ''S'' be a set with a binary operation * on it. 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'''. |
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For example, if ( |
For example, if (''S'',*) denotes the [[real number]]s with addition, then [[0 (number)|0]] is an identity. If (''S'',*) denotes the real numbers with multiplication, then [[1 (number)|1]] is an identity. If (''S'',*) denotes the ''n''-by-''n'' square [[matrix (mathematics)|matrices]] with addition, then the zero matrix is an identity. If (''S'',*) denotes the ''n''-by-''n'' matrices with multiplication, then the [[identity matrix]] is an identity. If (''S'',*) denotes the set of all [[function (mathematics)|functions]] from a set ''M'' to itself, with function composition as operation, then the [[identity map]] is an identity. If ''S'' has only two elements, ''e'' and ''f'', and the operation * is defined by ''e'' * ''e'' = ''f'' * ''e'' = ''e'' and ''f'' * ''f'' = ''e'' * ''f'' = ''f'', then both ''e'' and ''f'' are left identities, but there is no right or two-sided identity. |
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As the last example shows, it is possible for ( |
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. |
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See also: [[inverse element]], [[additive inverse]], [[group (mathematics)|group]], [[monoid]], [[quasigroup]]. |
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See [[identity (disambiguation)]] for other usages of this term. |
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==See Also== |
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*[[Inverse element]] |
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*[[Additive inverse]] |
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*[[Group (mathematics)|Group]] |
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*[[Monoid]] |
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*[[Quasigroup]] |
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[[Category:Abstract algebra]] |
[[Category:Abstract algebra]] |
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[[Category:Algebra]] |
[[Category:Algebra]] |
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[[de:Neutrales Element]] |
[[de:Neutrales Element]] |
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[[es:Elemento neutro]] |
[[es:Elemento neutro]] |
Revision as of 03:48, 24 February 2005
- For other uses, see identity (disambiguation).
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.
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 with a binary operation * on it. 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.
For example, if (S,*) denotes the real numbers with addition, then 0 is an identity. If (S,*) denotes the real numbers with multiplication, then 1 is an identity. If (S,*) denotes the n-by-n square matrices with addition, then the zero matrix is an identity. If (S,*) denotes the n-by-n matrices with multiplication, then the identity matrix is an identity. If (S,*) denotes the set of all functions from a set M to itself, with function composition as operation, then the identity map is an identity. If S has only two elements, e and f, and the operation * is defined by e * e = f * e = e and f * f = e * f = f, then both e and f are left identities, but there is no right or two-sided identity.
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.