invariant (original) (raw)

Let A be a set, and T:A→A a transformation of that set. We say that x∈A is an invariant of T whenever x is fixed by T:

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

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

The definition generalizes readily to a family of transformations with common domain

In this case we say that a subset is invariant, if it is invariant with respect to all elements of the family.