An operator is a function of one variable. An operator P determiness for each value of the variable x a uniquely defined entity P(x). The set X of all values of x for which P(x) is defined is called the domain of P, and the set of all entities y expressible in the form y = P(x) is called its range, Y. An operator P can be regarded as a set of ordered pairs (x, y), in other words as a special type of relation, in which case the statement P(x) = y is equivalent to the statement (x,y)eP. The set of pairs P is a subset of the cartesian product XY. It is sometimes convenient to denote by P(A) the set of all elements of the form P(a) with a in A. Thus in particular Y = P(X).
An operator P is cancellable if P(s) = P(t) implies s = t, for all s and t in X. An operator is one-to-one if every element in its domain is associated with a unique element in its range, and vice versa. These two properties are in fact the same. An operator of this type is called a correspondence. The number of elements in the domain must equal the number in the range. A correspondence P from X to Y has an inverse P` from Y to X such that P`(P(x)) = x for all x in X, in other words P` cancels the effect of P. We also have P(P`(y)) = y for all y in Y.
A closed operator, which we call a transform, has its range contained in its domain, that is YcX. A transform can thus act again on the element it produces by its action, to produce a sequence of values: P(x), P(P(x)), P(P(P(x))), ... in other words the transform can be iterated. The mth iteration can be denoted P^m(x).
On a set with one element there can only be one transform, the identity transform I(x) = x. Such a transform can be defined on any set. It is natural therefore to extend the iteration notation to take P^0 to be the identity transform.
On a set with two elements four transforms are possible: identity I(a) = a, I(b) = b, transposition T(a) = b, T(b) = a, and two absorbers A(a) = A(b) = a and B(a) = B(b) = b.
A transform with the property P(X) = X is conservative, that is its range is not only contained in its domain, but is equal to it. A transform is conservative if and only if it is cancellable (one-to-one). Such a transform we call a permutation. The identity and transposition are permutations. Every permutation has a unique inverse.
It is possible for a permutation to be its own inverse, that is P(P(x)) = x for all x in X in which case we call it a self-inverse correspondence or an involution. The identity and transposition are involutions.
An operator whose range Y is contained in a possibly larger set U is sometimes called a mapping from X to U. In this case there may be elements in U that cannot be expressed in the form P(x).