Neutral element - Revision history

CJKG: /* Definition */ Definition 1.4 in: http://mathematics.ceu.hu/sites/mathematics.ceu.hu/files/attachment/basicpage/39/victor.pdf

2014-07-08T09:03:36Z

Definition: Definition 1.4 in: http://mathematics.ceu.hu/sites/mathematics.ceu.hu/files/attachment/basicpage/39/victor.pdf

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	* '''right neutral''' or a '''right identity''' if <math>a * e = a</math> for any <math>a</math> in <math>S</math>		* '''right neutral''' or a '''right identity''' if <math>a * e = a</math> for any <math>a</math> in <math>S</math>
	* '''neutral''' if it is both left and right neutral		* '''neutral''' if it is both left and right neutral
			* '''middle neutral''' or a '''middle identity''' if <math>e * e = a</math> for any <math>a</math> in <math>S</math>

	A neutral element is also termed an '''identity element'''.		A neutral element is also termed an '''identity element'''.

Vipul: /* Facts */

2012-06-18T23:41:41Z

Facts

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	==Facts==		==Facts==

	~~===Any left neutral and right neutral element are equal===~~

	The proof of this fact goes as follows: let <math>e_1</math> be a left neutral element and <math>e_2</math> be a right neutral element. Then, the product <math>e_1 * e_2</math> is equal to <math>e_1</math> (because <math>e_2</math> is right neutral) and is also equal to <math>e_2</math> (because <math>e_1</math> is left neutral). Hence, <math>e_1 = e_2</math>.

	~~{{proofat\|[[Equality of left and right neutral element]]}}~~

	~~===Some easy corollaries===~~

			* [[Equality of left and right neutral element]]: This states that any ''left'' neutral element (if it exists) must equal any ''right'' neutral element (if it exists). The idea is to consider the product of these elements and show that it equals both of them.
			* [[Binary operation on magma determines neutral element]]: There can exist at most one two-sided neutral element. Thus, if a neutral element exists, it is unique. This is a corollary of the equality of left and right neutral element.
	* If there exists a left neutral element, there can exist at most one right neutral element; moreover, if it exists, then it is the same as the left neutral element and is hence a neutral element		* If there exists a left neutral element, there can exist at most one right neutral element; moreover, if it exists, then it is the same as the left neutral element and is hence a neutral element
	* If there exists a right neutral element, there can exist at most one left neutral element; moreover, if it exists, then it is the same as the right neutral element and is hence a neutral element		* If there exists a right neutral element, there can exist at most one left neutral element; moreover, if it exists, then it is the same as the right neutral element and is hence a neutral element
	* [[Binary operation on magma determines neutral element]]: There can exist at most one two-sided neutral element. Thus, if a neutral element exists, it is unique

	==Generalizations==		==Generalizations==

Vipul: /* Relation with other properties */

2012-06-18T23:14:27Z

Relation with other properties

← Older revision		Revision as of 23:14, 18 June 2012
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	* [[Binary operation on magma determines neutral element]]: There can exist at most one two-sided neutral element. Thus, if a neutral element exists, it is unique		* [[Binary operation on magma determines neutral element]]: There can exist at most one two-sided neutral element. Thus, if a neutral element exists, it is unique

			==Generalizations==

			* [[Neutral element for a multiary operation]]
	==Relation with other properties==		==Relation with other properties==

Vipul: /* Facts */

2012-06-18T23:14:07Z

Facts

← Older revision		Revision as of 23:14, 18 June 2012
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	* If there exists a left neutral element, there can exist at most one right neutral element; moreover, if it exists, then it is the same as the left neutral element and is hence a neutral element		* If there exists a left neutral element, there can exist at most one right neutral element; moreover, if it exists, then it is the same as the left neutral element and is hence a neutral element
	* If there exists a right neutral element, there can exist at most one left neutral element; moreover, if it exists, then it is the same as the right neutral element and is hence a neutral element		* If there exists a right neutral element, there can exist at most one left neutral element; moreover, if it exists, then it is the same as the right neutral element and is hence a neutral element
	* There can exist at most one neutral element. Thus, if a neutral element exists, it is unique		* [[Binary operation on magma determines neutral element]]: There can exist at most one two-sided neutral element. Thus, if a neutral element exists, it is unique

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Definition */

2010-04-09T23:52:56Z

Definition

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	===Definition with symbols===		===Definition with symbols===

	Given a binary operation <math>*</math> on a set <math>S</math>, an element <math>e</math> in <math>S</math> is termed:		Given a binary operation <math></math> on a set <math>S</math> (i.e., a [[defining ingredient::magma]] <math>(S,)</math>), an element <math>e</math> in <math>S</math> is termed:

	* '''left neutral''' or a '''left identity''' if <math>e * a = a</math> for any <math>a</math> in <math>S</math>		* '''left neutral''' or a '''left identity''' if <math>e * a = a</math> for any <math>a</math> in <math>S</math>

Vipul: /* Facts */

2008-07-31T20:28:31Z

Facts

← Older revision		Revision as of 20:28, 31 July 2008
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	The proof of this fact goes as follows: let <math>e_1</math> be a left neutral element and <math>e_2</math> be a right neutral element. Then, the product <math>e_1 * e_2</math> is equal to <math>e_1</math> (because <math>e_2</math> is right neutral) and is also equal to <math>e_2</math> (because <math>e_1</math> is left neutral). Hence, <math>e_1 = e_2</math>.		The proof of this fact goes as follows: let <math>e_1</math> be a left neutral element and <math>e_2</math> be a right neutral element. Then, the product <math>e_1 * e_2</math> is equal to <math>e_1</math> (because <math>e_2</math> is right neutral) and is also equal to <math>e_2</math> (because <math>e_1</math> is left neutral). Hence, <math>e_1 = e_2</math>.

			{{proofat\|[[Equality of left and right neutral element]]}}

	===Some easy corollaries===		===Some easy corollaries===

Vipul: 2 revisions

2008-05-07T23:51:44Z

2 revisions

← Older revision	Revision as of 23:51, 7 May 2008
(No difference)

Vipul: /* Definition */

2008-02-12T17:46:46Z

Definition

← Older revision		Revision as of 17:46, 12 February 2008
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	* '''right neutral''' or a '''right identity''' if <math>a * e = a</math> for any <math>a</math> in <math>S</math>		* '''right neutral''' or a '''right identity''' if <math>a * e = a</math> for any <math>a</math> in <math>S</math>
	* '''neutral''' if it is both left and right neutral		* '''neutral''' if it is both left and right neutral

			A neutral element is also termed an '''identity element'''.

	==Facts==		==Facts==

Vipul at 12:10, 8 March 2007

2007-03-08T12:10:36Z

New page

{{magma-element property}}

==Definition==

===Definition with symbols===

Given a binary operation <math>*</math> on a set <math>S</math>, an element <math>e</math> in <math>S</math> is termed:

* '''left neutral''' or a '''left identity''' if <math>e * a = a</math> for any <math>a</math> in <math>S</math>
* '''right neutral''' or a '''right identity''' if <math>a * e = a</math> for any <math>a</math> in <math>S</math>
* '''neutral''' if it is both left and right neutral

==Facts==

===Any left neutral and right neutral element are equal===

The proof of this fact goes as follows: let <math>e_1</math> be a left neutral element and <math>e_2</math> be a right neutral element. Then, the product <math>e_1 * e_2</math> is equal to <math>e_1</math> (because <math>e_2</math> is right neutral) and is also equal to <math>e_2</math> (because <math>e_1</math> is left neutral). Hence, <math>e_1 = e_2</math>.

===Some easy corollaries===

* If there exists a left neutral element, there can exist at most one right neutral element; moreover, if it exists, then it is the same as the left neutral element and is hence a neutral element
* If there exists a right neutral element, there can exist at most one left neutral element; moreover, if it exists, then it is the same as the right neutral element and is hence a neutral element
* There can exist at most one neutral element. Thus, if a neutral element exists, it is unique

==Relation with other properties==

===Weaker properties===

* [[Idempotent element]]
* [[Cancellative element]]: Any left neutral element is left cancellative, and any right neutral element is right cancellative. Hence, any neutral element is cancellative