Date: 02 Jun 2005

Scott's conjecture is true, position sensitive weights

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The classification of total reduction orderings for strings over a 2-letter alphabet w.r.t. monoid presentations with 2 generators was published by U. Martin, see [9], and used the hypothetical truth of Scott's conjecture, which was 3 years old in 1996.

Now the results due to Ursula Martin and Elizabeth Scott are completed with the truth of Scott's conjecture. The final proof is simple, but we had difficulties. E. Scott proved the case, when some invariant takes the value either 0, or a positive rational or ∞, see [15]. Later we proved the case of positive reals which are well approximable to arbitrary order, see [11], and then the case of \({}^n\sqrt k\) and the case of λ where both of λ and λ−1 are algebraic integers, like √5−1/2.

It is a challenging problem, whether there is a reasonably small subset G \(\subseteq\) a,b *x a,b * such that each total reduction ordering ≻ of {a, b}* is uniquely determined by its restriction to G.

This work was supported by SERC/SBCC grant GR/J31230 and Hungarian National Science Foundation, grant nos. 7442 and 16432. The authors would also like to thank the referees for helpful suggestions.