교환법칙,commutativity

교환법칙,commutativity (rev. 1.62)

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commutative property, commutative law, commutativity

이항 연산자로 이항연산,binary_operation을 할 때, 순서를 바꾸어도 결과가 같은 성질.

집합 $S$ 에 연산 $\circ$ 가 정의되어 있을 때, $\forall a, b \in S$ 에 대해
$a \circ b = b \circ a$
가 성립하면 교환법칙이 성립한다고 한다. // -> 가환 commutative adj.


commutative property 교환성질, commutative law 교환법칙, commutativity 교환성 이 가장 일대일에 가까운 번역일 듯.

commutativity : 영어 표현을 보면 교환성 대신 가환성이라고 번역해도 무방한 그런 거? - commutative_diagram을 wk에선 WpKo:가환_그림으로 번역했던데.
{
WpSp:Commutative_diagram
WpKo:가환_그림
}



[edit]

1. 교환법칙이 성립하는 예


교환법칙을 만족하면 가환,commutative
결합한 결과가 결합하는 순서,order와 무관하면 가환,commutative이며 교환법칙이 성립한다고 함.

... 페이지를 저렇게 형용사형으로 따로 만들기보다는, 이 pagename을 변경하거나 필요하면 둘로 나눌...? '가환성 commutativity' and '교환법칙 commutative_law' 이게 사실 번역이 더 정확함.


[edit]

2. 교환법칙이 성립하지 않는 예

QQQ 비가환의 번역? anti-인가 non-인가?
주의: 보통 비가환이면 noncommutative 같은데...kms에선 anticommutative가 비가환. 저 아래쪽 참조.
혹시 같은거?

,anticommutativity
tmp ref. WpEn:Anticommutative_property
"anticommutativity is a specific property of some non-commutative operations."
그럼 non-comm. 중의 일부가 anti-comm. ?

사원수,quaternion 보니 <비가환,noncommutative 반가환,anticommutative?>이라고 적혀있는게 있는데 source는 모르겠네

비가환,noncommutative
비가환 non-commutative
한편으로는 wpko에서 noncommutative=비가환.
ex. 이렇게 interwiki 연결되어있음
WpKo:비가환_원환면 WpEn:Noncommutative_torus
noncommutative_geometry
{
WpKo:분류:비가환_기하학 -> WpEn:Category:Noncommutative_geometry
WpKo:비가환_기하학 -> WpEn:Noncommutative_geometry

https://everything2.com/title/Noncommutative Geometry
QM에서 영향을 받았다고...
}


비가환군, // noncommutative_group or non-commutative_group ? 둘중에 TBD
[https]수학백과: 비가환군
{
가환군,이 아닌 군.
AKA 비아벨군


비가환환, // noncommutative_ring or non-commutative_ring ?? 둘중에 TBD
{
noncommutative ring

환 R의 곱셈 연산이 교환법칙을 만족하지 않을 때 비가환환이라 함.
[https]수학백과: 비가환환
여기서 나눗셈환,division_ring 의체 비가환체 skew_field 언급.

mkl skew_field = division_ring



[edit]

2.1. anticommutativity

AKA anticommutative property
adj. anticommutative

대한수학회 번역은 '비교환성, 비가환성'
anticommutative law = 비교환법칙

반교환법칙? 이란 표현도 봤는데.

A­×B=-B×A (외적)
CHK

[edit]

3. subcommutative

[edit]

4. 기타

교환자,commutator
{
대충, 군론과 환론의 교환자가 있고 둘 다 교환법칙 실패여부 측정??관련이고, (가환이면 0이 나오고 비가환이면 nonzero가 나오는?) 둘이 정의가 약간 다르고, 환론의 교환자는 양자역학에서 쓰이는 듯. 환론의 교환자는 anti-commutator도 있는 듯. chk

// Libre:
이항연산자(이항연산,binary_operation 연산자,operator)가 교환법칙,commutativity이 성립하는지에 대해 확인하는 연산자.
군,group에서와 환,ring에서의 정의가 다르다.

// wpko:
두 원소 사이의 교환 법칙의 실패를 측정하는 이항 연산.





이항연산,binary_operation에 대해 교환법칙이 성립하는 군,group은 가환군,commutative_group.

(물리 > QM > 불확정성원리,uncertainty_principle)
운동량 연산자가 행렬로 표현될 수 있기 때문에 운동량과 위치의 교환불가성이 설명 가능.
matrix_multiplication에서 교환법칙이 성립하지 않는 것(AB≠BA)과 마찬가지 원리.[1]
$\hat{x}\hat{p}-\hat{p}\hat{x}=0$ 이 아니고,
$\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar I$ 이다.[2]





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