invariant

Let $A$ be a set, and $T:A\to A$ a transformation of thatset. We say that $x\in A$ is an invariant of $T$ whenever $x$ isfixed by $T$:

$$T(x)=x.$$

We say that a subset $B\subset A$ isinvariant with respect to $T$ whenever

$$T(B)\subset B.$$

If this is so, the restriction of $T$is a well-defined transformation of the invariant subset:

$${T|}_B:B\to B.$$

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 is invariant, if it is invariantwith respect to all elements of the family.