Because the data a and b are unconstrained, the ordering O has to be what is known as a total ordering, so that for any two values a and b, at least one of O(a, b) or O(b, a) produces true. The alternative is called a partial order, in which both of O(a, b) or O(b, a) produces false.
The ≤ ordering over integers is a total ordering. The < ordering over integers is a partial ordering; if a = b, then neither a < b nor b < a is true (< is a total ordering in the presence of a collection of unique integers).
This page last modified on 24 January 2006.