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\newtheorem{proposition}{Proposition}Let $A$ be a set, and $T:A\rightarrow A$ a transformation of that
set. We say that $x\in A$ is
\emph{fixed} by $T$, or that $T$ \emph{fixes} $x$, whenever
$$T(x)=x.$$
The subset of fixed elements is called {\em 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
$$B\subset \bigcap_{i\in I} A^{T_i}.$$