The order types of termination orderings on monadic terms, strings and multisets

Total well-founded orderings on monadic terms satisfying the replacement and full invariance properties are considered. It is shown that any such ordering on monadic terms in one variable and two unary function symbols must have order type omega , omega /sup 2/, or omega /sup omega /. It is further shown that a familiar construction gives rise to continuum many such orderings of order type omega . A new family of such orderings of order type omega is constructed, and it is shown that there are only four such orderings of order type omega /sup omega /, the two familiar recursive path orderings and two closely related orderings. It is shown that any total well-founded ordering on N/sup n/ that is preserved under vector addition must have order type omega /sup lambda / for some 1>