User:Inthar/MV3: Difference between revisions
→MV3 Theorem 1: PWF is too restrictive, LLSLLM is a counterexample to "MV3 implies PMOS except in abcba" |
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Below, assume that the scale word S is not multiperiod. | Below, assume that the scale word S is not multiperiod. | ||
=== Lemma 1: S is | === Lemma 1: S is pairwise MOS (PMOS) except in the case "xyzyx" === | ||
TODO: account for case xyzyx. | TODO: account for case xyzyx. | ||
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''To be continued...'' | ''To be continued...'' | ||
=== | === PMOS implies AG (except in the case xyxzxyx) === | ||
We now prove that except in the case xyxzxyx, if the scale is pairwise well-formed, the scale is generated by two chains of generators, either both of size ''m'' or one with size ''m'' and one of size ''m-1'' (AG) Equivalently, the scale can be built by stacking alternating generators, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3. | We now prove that except in the case xyxzxyx, if the scale is pairwise well-formed, the scale is generated by two chains of generators, either both of size ''m'' or one with size ''m'' and one of size ''m-1'' (AG) Equivalently, the scale can be built by stacking alternating generators, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3. | ||
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Now assume len(S) >= 8. | Now assume len(S) >= 8. | ||
PMOS -> Consider mos temperings | |||
* in x, ξ (ξ = y or z), with gen g1 -> g1g1...g1g1' (g1' = imperfect gen) | * in x, ξ (ξ = y or z), with gen g1 -> g1g1...g1g1' (g1' = imperfect gen) | ||
* in y, η (η = x or z), with gen g2 | * in y, η (η = x or z), with gen g2 | ||