Identity element

Identity element (たんいげん British: identity element) or middle Tatemoto (ちゅうりつげん British: neutral element) is the special cause of the meeting with the binary operation, and, in mathematics particularly abstract algebra, no other causes are not affected by the combination with the identity element by the binary operation.

Table of contents

Definition

I assume (M,∗) the magma which meeting M and upper binary operation ∗ make. Former e of M about ∗ to be an identity element (both sides); for all former a of M

${\displaystyle a*e=e*a=a}$ 

When I satisfy を, I say. Furthermore, when I satisfy a∗e = a finely for any former a of M, it is said with the right identity element, and it is said with the left identity element when I meet e∗a = a. An identity element is the left identity element and the right identity element. When operation is commutation, there is not the distinction of right and left. Magma, semigroup, the ring having an identity element are called the magma of the unit, the semigroup (monoid) of the unit, the ring of the unit each.

I call additive identity with an addition identity element (I often express it with 0) to distinguish whether it is a concept about which operation and, in the algebraic system having the addition such as rings and two operation of the multiplication, say an identity element about the multiplication with multiplicative identity (I often express it with 1).

Example

Identity element of various kinds of algebraic systems

A stand gathers	Operation	Identity element
Whole real number R	The sum +	0
Whole real number R	The product •	1
Whole real number R	冪乗 ab	The right identity element: 1
Whole positive integer N	Least common multiple LCM	1
Whole non-negative integer Z≥0	Greatest common divisor GCD	0 (depending on a definition)
The whole m-line n- line line	Sum + of the line	Null matrix O
n-next square matrix	The product of the line •	n-next identity matrix In
Whole representation MM from meeting M to M oneself	Composition ∘	恒等写像
Whole representation MM from meeting M to M oneself	Enfold ∗	D rack delta δ
The whole character string	Combination of the character string	Empty character string
Whole expansion real number R	A minimum or the lower limit ∧	Regular infinite +∞
Whole expansion real number R	Maximum or the upper limit ∨	Negative infinity −∞
Whole subset 2M of meeting M	Acquaintance ∩	Universal set M
Whole Sets of the small meeting	End ∪	Empty set {}
The Bour logic	Logical product ∧	The truth ⊤
The Bour logic	Logical sum ∨	False ⊥
The Bour logic	EX-OR xor	False ⊥
Closed surface	Connection sum #	Spherical surface S2
2 yuan meeting {e, f}	∗: e∗e = f∗e = e f∗f = e∗f = f	The left identity element: e, f The right identity element: Unavailable Both sides identity element: Unavailable

Property

As for the left identity element and the right identity element, there can be a plural number in one algebraic system. However, they agree, and they become one (both sides) mere identity element of the algebraic system if magma (M,∗) has the left identity element and the right identity element. If, as for this, left identity element e₂ is the right identity element real e₁,

${\displaystyle e_{1}=e_{1}*e_{2}=e_{2}}$ 

But, I understand it because it is established. There is in particular the both sides identity element only in one at most.

Magma (S,∗) may have no identity element. Well known examples include the cross product of the space vector. I know that an identity element about the cross product does not exist from a fact to have the direction where cross product of two non-zero vector is at right angles to two both original vectors. Another example which does not have an identity element includes additive semigroup (N, +) which the whole natural number makes (the plus).

Addition of the identity element

The meeting M1 which added new former 1 that was different from all causes of M to M when magma (M,∗) was given with = M ∪ {1}:

It is a * 1 = 1 * a = a for any a ∈ M¹

I can assume cause 1 an identity element about ∗ of the M¹ by determining と, and extending operation ∗ of M on the M¹. This (M¹,∗) is called 1-addition of (M,∗).

e is not an identity element about ∗ on M¹ anymore even if originally M had identity element e about ∗.

References

• Mathematics 〉, 1972 of the Takayuki Tamura "half theory of groups" Kyoritsu publication 〈 Kyoritsu lecture present age.
• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3110152487, p. 14-15

Allied item

• Origin of absorption
• Inverse element
• Addition inverse element
• Unit mark
• Monoid (semigroup of the unit)

Outside link

• identity element - PlanetMath.
• left identity and right identity - PlanetMath. (English)
• Weisstein, Eric W., "Identity element" - MathWorld.

It is acquired by "https://ja.wikipedia.org/w/index.php?title= identity element &oldid=55375608"

This article is taken from the Japanese Wikipedia Identity element

This article is distributed by cc-by-sa or GFDL license in accordance with the provisions of Wikipedia.

Wikipedia and Tranpedia does not guarantee the accuracy of this document. See our disclaimer for more information.

In addition, Tranpedia is simply not responsible for any show is only by translating the writings of foreign licenses that are compatible with CC-BY-SA license information.