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 ω, ω², or ω^ω. It is further shown that a familiar construction gives rise to continuum many such orderings of order type ω. A new family of such orderings of order type ω is constructed, and it is shown that there are only four such orderings of order type ω^ω, the two familiar recursive path orderings and two closely related orderings. It is shown that any total well-founded ordering on Nⁿ that is preserved under vector addition must have order type ω^λ for some 1⩽k⩽n; if k<n, there are continuum many such orderings, and if k=n, there are only n-factorial, namely the n-factorial lexicographic orderings