Ternary scale theorems: Difference between revisions
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No letter around this word can be a '''Z''' because '''YXXY''' exists. None can be a '''Y''' since '''XZX''' exists. Therefore, they have to be two '''X'''s. Then note that the two surrounding letters cannot be '''Z''' (because of the existence of '''XYXXYX''') nor '''X''' (because of the existence of '''YXZ''') so they are '''Y''', then followed by '''X''' (because '''XX''' exists). At this point, we have the sequence | No letter around this word can be a '''Z''' because '''YXXY''' exists. None can be a '''Y''' since '''XZX''' exists. Therefore, they have to be two '''X'''s. Then note that the two surrounding letters cannot be '''Z''' (because of the existence of '''XYXXYX''') nor '''X''' (because of the existence of '''YXZ''') so they are '''Y''', then followed by '''X''' (because '''XX''' exists). At this point, we have the sequence | ||
“_''' | “_'''XYXXYXZXYXXYX'''_”. Both _s are necessarily '''Z'''s. To end the proof, note that we have obtained the configuration around every '''Z''' and this determines the whole sequence. Thus ''W'' = ('''XYXZXYX''')<sup>ω</sup>. | ||
(c) The scale made by taking ''s'' and conflating '''Y''' and '''Z''' into the letter '''W''' must be a MOS. To this scale we may imagine substituting a scale made of an equal amount of '''Y''' and '''Z''' letters into the "slot letters" '''W''' letter by letter. Let ''t''<sub>1</sub> be a length-''k'' subword of the form '''YX'''<sup>''k''-2</sup>'''Y''' under the projection. We may assume that the chunk sizes of the MOS are ''k'' - 2 and ''k'' - 1, or ''k'' - 2 and ''k'' - 3. Either way, there exists some subword with (''k'' - i)-many '''X'''s, i = 1 or 2, and two '''Z'''s. This violates balance because ''t''<sub>1</sub> contains zero '''Z'''s. | (c) The scale made by taking ''s'' and conflating '''Y''' and '''Z''' into the letter '''W''' must be a MOS. To this scale we may imagine substituting a scale made of an equal amount of '''Y''' and '''Z''' letters into the "slot letters" '''W''' letter by letter. Let ''t''<sub>1</sub> be a length-''k'' subword of the form '''YX'''<sup>''k''-2</sup>'''Y''' under the projection. We may assume that the chunk sizes of the MOS are ''k'' - 2 and ''k'' - 1, or ''k'' - 2 and ''k'' - 3. Either way, there exists some subword with (''k'' - i)-many '''X'''s, i = 1 or 2, and two '''Z'''s. This violates balance because ''t''<sub>1</sub> contains zero '''Z'''s. | ||