Editing Lexicographic order (section)

== Functions over a well-ordered set==

The functions from a [[well-ordered set]] <math>X</math> to a [[totally ordered set]] <math>Y</math> may be identified with sequences indexed by <math>X</math> of elements of <math>Y.</math> They can thus be ordered by the lexicographical order, and for two such functions <math>f</math> and <math>g,</math> the lexicographical order is thus determined by their values for the smallest <math>x</math> such that <math>f(x) \neq g(x).</math>

If <math>Y</math> is also well-ordered and <math>X</math> is finite, then the resulting order is a well-order. As shown above, if <math>X</math> is infinite this is not the case.