Ternary scale theorems: Difference between revisions
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Consider the two generators in the GS of ''s'', which are detemperings of the generator {{nowrap|''i'''''X''' + ''j'''''W'''}} of ''T''('''X''', '''W'''), where {{nowrap|gcd(''j'', 2''k'') {{=}} 1}}. Assume, possibly after inverting the generator, that the imperfect generator of ''T'' has {{nowrap|''j'' − 1}} '''W'''s and the perfect generator has ''j'' '''W'''s. | Consider the two generators in the GS of ''s'', which are detemperings of the generator {{nowrap|''i'''''X''' + ''j'''''W'''}} of ''T''('''X''', '''W'''), where {{nowrap|gcd(''j'', 2''k'') {{=}} 1}}. Assume, possibly after inverting the generator, that the imperfect generator of ''T'' has {{nowrap|''j'' − 1}} '''W'''s and the perfect generator has ''j'' '''W'''s. | ||
'''Claim 1''': Deleting '''X'''s from the generator subwords of ''s'' gives every ''j''-step subword in the scale ''E''<sub>X</sub>(''s'')('''Y''', '''Z'''), the scale word obtained by deleting all '''X''''s from ''s''. These ''j''-step subwords are adjacent and alternating. | '''Claim 1''': Deleting '''X'''s from the generator subwords of ''s'' gives every ''j''-step subword in the scale ''E''<sub>X</sub>(''s'')('''Y''', '''Z'''), the scale word obtained by deleting all '''X''''s from ''s''. These ''j''-step subwords are adjacent and alternating under the ordering induced by the AGS stack. | ||
Proof: | Proof: | ||