The Order Types of Termination Orderings on Monadic Terms, Strings and Multisets

We consider total well-founded orderings on monadic terms satisfying the replacement and full invariance properties. We show that any such ordering on monadic terms in one variable and two unary function symbols must have order type omega, omega(2) or omega(omega). We show that a familiar construction gives rise to continuum many such orderings of order type omega. We construct a new family of such orderings of order type omega(2), and show that there are continuum many of these. We show that there are only four such orderings of order type omega(omega), the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings on N-n which are preserved under vector addition. we show: that any such ordering must have order type omega(k) for some 1 less than or equal to k less than or equal to n. We show that if k < tr there are continuum many such orderings, and if k = n there are only n!, the n! lexicographic orderings.