Lexicographical Order - Generalization

Generalization

Consider the set of functions f from a well-ordered set X to a totally ordered set Y. For two such functions f and g, the order is determined by the values for the smallest x such that f(x) ≠ g(x).

If Y is also well-ordered and X is finite, then the resulting order is a well-order. As already shown above, if X is infinite this is in general not the case.

If X is infinite and Y has more than one element, then the resulting set YX is not a countable set, see also cardinal exponentiation.

Alternatively, consider the functions f from an inversely well-ordered X to a well-ordered Y with minimum 0, restricted to those that are non-zero at only a finite subset of X. The result is well-ordered. Correspondingly we can also consider a well-ordered X and apply lexicographical order where a higher x is a more significant position. This corresponds to exponentiation of ordinal numbers YX. If X and Y are countable then the resulting set is also countable.