Neutral element: Difference between revisions

Revision as of 20:28, 31 July 2008

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 $S$ , 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

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 $e_{1}$ be a left neutral element and $e_{2}$ be a right neutral element. Then, the product $e_{1} * e_{2}$ is equal to $e_{1}$ (because $e_{2}$ is right neutral) and is also equal to $e_{2}$ (because $e_{1}$ is left neutral). Hence, $e_{1} = e_{2}$ .

For full proof, refer: Equality of left and right neutral element

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

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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>.
+{{proofat|[[Equality of left and right neutral element]]}}
 ===Some easy corollaries===