fix (transformation action)

Let $A$ be a set, and $T:A\to A$ a transformation of thatset. We say that $x\in A$ isfixed 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 $B\subset A$ isfixed 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 identitytransformation on $B$.

The definition generalizes readily to a family of transformations withcommon 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 fixedby all the elements of the family, i.e. whenever

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