Neutral element: Difference between revisions

Latest revision as of 09:03, 8 July 2014

This article defines a property of elements or tuples of elements with respect to a binary operation

Definition

Definition with symbols

Given a binary operation $*$ on a set $S$ (i.e., a magma $(S,*)$ ), an element $e$ in $S$ is termed:

left neutral or a left identity if $e*a=a$ for any $a$ in $S$
right neutral or a right identity if $a*e=a$ for any $a$ in $S$
neutral if it is both left and right neutral
middle neutral or a middle identity if $e*e=a$ for any $a$ in $S$

A neutral element is also termed an identity element.

Facts

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 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

Generalizations

Neutral element for a multiary operation

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

@@ Line 5: / Line 5: @@
 ===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>
 * '''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
+* '''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'''.
 ==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===
+* [[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 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
+==Generalizations==
+* [[Neutral element for a multiary operation]]
 ==Relation with other properties==