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

Ursula Martin and Elizabeth Scott
The Journal of Symbolic Logic
Vol. 62, No. 2 (Jun., 1997), pp. 624-635
DOI: 10.2307/2275551
Stable URL: http://www.jstor.org/stable/2275551
Page Count: 12
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The Order Types of Termination Orderings on Monadic Terms, Strings and Monadic Terms, Strings and Multisets
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Abstract

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, ω2 or ωω. 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 ω2, and show that there are continuum many of these. We show that there are only four such orderings of order type ωω, the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings on Nn which are preserved under vector addition. We show that any such ordering must have order type ωk for some 1 ≤ k ≤ n. We show that if $k < n$ there are continuum many such orderings, and if k = n there are only n!, the n! lexicographic orderings.

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