Date: 02 Jun 2005

Scott's conjecture is true, position sensitive weights

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Abstract

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.