  These clauses can be applied recursively to define transformations of sets of ordered tuples in \(D\) (the extensions of two-place predicates), sets of sets of objects in \(D\) (the extensions of unary first-order quantifiers), and so on. (For an introduction to the type theoretic hierachy, see the entry on Type Theory.) Where \(x\) is an item in this hierarchy, we say that \(x\) is invariant under a permutation \(p\) just in case \(p^*(x) = x\). To return to our example above, the set  \(\{\textrm{A}, \textrm{B}, \textrm{C}, \textrm{D}, \textrm{E}\}\) is invariant under all permutations of the letters \(\textrm{A}\) through \(\textrm{E}\): no matter how we switch these letters around, we end up with the same set.  But it is not invariant under all permutations of the entire alphabet.  For example, the permutation that switches the letters \(\textrm{A}\) and \(\textrm{Z}\), mapping all the other letters to themselves, transforms  \(\{\textrm{A}, \textrm{B}, \textrm{C}, \textrm{D}, \textrm{E}\}\) to  \(\{\textrm{Z}, \textrm{B}, \textrm{C}, \textrm{D}, \textrm{E}\}\).  The set containing all the letters, however, is invariant under all permutations of letters.  So is the set of all sets containing at least two letters, and the relation of identity, which holds between each letter and itself.