# fix (transformation action)

Let $A$ be a set, and $T:A\to A$ 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

$${T}_{i}:A\to A,i\in I$$ |

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

$$B\subset \bigcap _{i\in I}{A}^{{T}_{i}}.$$ |

Title | fix (transformation action) |

Canonical name | FixtransformationAction |

Date of creation | 2013-03-22 12:26:12 |

Last modified on | 2013-03-22 12:26:12 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 15 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 03E20 |

Synonym | fix |

Synonym | fixed |

Synonym | fixes |

Related topic | Invariant |

Related topic | Transformation |

Related topic | Fix2 |

Defines | fixed set |