An **operator** is a function of one variable.
An operator P determiness for each value of the variable x a uniquely defined entity Px.
The set X of all values of x for which Px is defined is called the **domain** of P,
and the set of all entities y expressible in the form y = Px 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 Px = y is equivalent to the statement (x, y) ∈ P. The set of pairs P is a subset of the cartesian product ×(X, Y), i.e. the set of all pairs (x, y) with x in X and y in Y. It is sometimes convenient to denote by PA the set of all elements of the form Pa with a in A. Thus in particular Y = PX.

An operator P is **cancellable** if Ps = Pt 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^Px = x for all x in X,
in other words P^ cancels the effect of P. We also have PP^y = y for all y in Y.

A **closed** operator, which we call a **transform**, has its range contained in its domain, that is Y ⊆ X.
A transform can thus act again on the element it produces by its action, to produce a sequence of values: Px, PPx, PPPx, ...
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 Ix = 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 Ia = a, Ib = b,
**transposition** Ta = b, Tb = a, and two **absorbers** Aa = Ab = a and Ba = Bb = b.

A transform with the property PX = 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). [This assumes the set is finite, it is not true for infinite sets.]
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 PPx = 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 Px.