Ternary scale theorems: Difference between revisions
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=== Proof === | === Proof === | ||
The result of substituting '''Y''' with '''X''' (let us call this map ''p'') is the MOS ''M'' = 2''a'''''X'''2''c'''''Z''', which has exactly 2 periods since gcd(''a'', ''c'') = 1. ''M'' thus consists of two generator chains separated by the period of ''M'', which has ''a'' + ''c'' = len(''s'') steps. It thus suffices for there to exist ''k'', 0 < ''k'' < ''a'' + ''c'', such that every perfect ''k''-step generator has the same preimage in ''s'', which will be our desired generator. Suppose that the perfect ''k''-step of ''M'' is ''i'''''W''' + ''j'''''Z''' where 0 < ''i'' < ''a''. Since ''a'' is odd, possibly after taking the period-complement we may assume that ''i'' is even. Hence each subword ''w'' of ''s'' such that its projection ''p''(''w'') subtends a perfect ''k''-step satisfies |''w''|<sub>'''X'''</sub> = |''w''|<sub>'''Y'''</sub> (= ''i''/2). It plainly follows that every such ''w'' satisfies |''w''|<sub>'''X'''</sub> = |''w''|<sub>'''Y'''</sub> = ''i''/2 and |''w''|<sub>'''Z'''</sub> = ''j''. {{qed}} | The result of substituting '''Y''' with '''X''' (let us call this map ''p'') is the MOS ''M'' = 2''a'''''X'''2''c'''''Z''', which has exactly 2 periods since gcd(''a'', ''c'') = 1. ''M'' thus consists of two generator chains separated by the period of ''M'', which has ''a'' + ''c'' = len(''s'') steps. It thus suffices for there to exist ''k'', 0 < ''k'' < ''a'' + ''c'', such that every perfect ''k''-step generator has the same preimage in ''s'', which will be our desired generator. Suppose that the perfect ''k''-step of ''M'' is ''i'''''W''' + ''j'''''Z''' where 0 < ''i'' < ''a''. Since ''a'' is odd, possibly after taking the period-complement we may assume that ''i'' is even. Hence each subword ''w'' of ''s'' such that its projection ''p''(''w'') subtends a perfect ''k''-step satisfies |''w''|<sub>'''X'''</sub> = |''w''|<sub>'''Y'''</sub> (= ''i''/2). It plainly follows that every such ''w'' satisfies |''w''|<sub>'''X'''</sub> = |''w''|<sub>'''Y'''</sub> = ''i''/2 and |''w''|<sub>'''Z'''</sub> = ''j''. The proof that ''s'' is balanced can be adapted from the proof that PWF scales are balanced, because ''s'' is pairwise-MOS. {{qed}} | ||
== Theorem 7 (Classification of MV3 scales) == | == Theorem 7 (Classification of MV3 scales) == | ||