# 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

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