Rational Mathematics

by G. P. Jelliss


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 Pmx.

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 P0 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.