Suppose A = {a, b} and a precedes b in the alphabet.

Recall that A* (the closure of A), is just all possible strings over A.

The lexicographic ordering on A* is defined as follows: x ≺ y iff either x is a proper prefix of y or x and y have a longest common proper prefix u such that x = uv, y = uw, and head(v) precedes head(w) in A. (This mathematical definition simply describes what we think of as dictionary ordering of strings. For example: a ≺ b, aa ≺ ab, etc.)

Is the lexicographic ordering on A* well-ordered?