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

Ursula Martin, Elizabeth Scott

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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 ω, ω<sup>2</sup> or ω<sup>ω</sup>. We show that a familiar construction gives rise to continuum many such orderings of order type ω. We construct a new family of such orderings of order type ω<sup>2</sup>, and show that there are continuum many of these. We show that there are only four such orderings of order type ω<sup>ω</sup>, the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings on N<sup>n</sup> which are preserved under vector addition. We show that any such ordering must have order type ω<sup>k</sup> for some 1 ≤ k ≤ n. We show that if <latex>$k < n$</latex> there are continuum many such orderings, and if k = n there are only n!, the n! lexicographic orderings.
Original languageEnglish
Pages (from-to)624-635
JournalJournal of Symbolic Logic
Issue number2
Publication statusPublished - 1997

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