PlanetMath: invariant

(more info)

Math for the people, by the people.

Main Menu

invariant

(Definition)

Let be a set, and $T:A\rightarrow A$ a transformation of that set. We say that $x\in A$ is an invariant of whenever is fixed by :

$\displaystyle T(x)=x.$

We say that a subset $B\subset A$ is invariant with respect to whenever

$\displaystyle T(B)\subset B.$

If this is so, the restriction of

is a well-defined transformation of the invariant subset:

$\displaystyle T\Big\vert _B : B\rightarrow B.$

The definition generalizes readily to a family of transformations with common domain

$\displaystyle T_i : A\rightarrow A,\quad i\in I$

In this case we say that a subset is invariant, if it is invariant with respect to all elements of the family.

"invariant" is owned by rmilson.

(view preamble)

Log in to rate this entry.
(view current ratings)

Cross-references: domain, invariant subset, well-defined, restriction, subset, fixed, transformation
There are 108 references to this entry.

This is version 5 of invariant, born on 2002-02-22, modified 2002-02-22.
Object id is 2504, canonical name is Invariant.
Accessed 11376 times total.

Classification:

03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact