When is a total ordering of a semigroup a well-ordering? (Q920125)

An ordered semigroup (S,\(\leq)\) is a semigroup S with an order \(\leq\) satisfying: for any s, t, and x in S, \(s\leq t\) implies sx\(\leq tx\) and xs\(\leq xt\). The order \(\leq\) is said to be positive if for all a and x in S, \(a\leq ax\) and \(a\leq xa\). This note proves the following theorem: A positive total order \(\leq\) on a semigroup S is a well-order if and only if S has a well-ordered generating set.