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}}.