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# fix (transformation action)

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

$T(x)=x.$ |

The subset of fixed elements is called the fixed set of $T$, and is frequently denoted as $A^{T}$.

We say that a subset $B\subset A$ is fixed by $T$ whenever all elements of $B$ are fixed by $T$, i.e.

$B\subset A^{T}.$ |

If this is so, $T$ restricts to the identity transformation on $B$.

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

$T_{i}:A\rightarrow A,\quad i\in I$ |

In this case we say that a subset $B\subset A$ is fixed, if it is fixed by all the elements of the family, i.e. whenever

$B\subset\bigcap_{{i\in I}}A^{{T_{i}}}.$ |

Defines:

fixed set

Related:

Invariant,Transformation, Fix2

Synonym:

fix, fixed, fixes

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

03E20*no label found*

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