Generator-offset property: Difference between revisions

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==== Proof ====

===== If the generator of a projection of ''S'' is a ''k''-step, the word of stacked ''k''-steps in ''S'' is pairwise well-formed =====

Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to single-period mosses ''S''1 (via identifying b with c), ''S''2 (via identifying a with c) and ''S''3 (via identifying a with b). Suppose ''S''1's generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''2 and ''S''3, respectively. These binary words, which we call Σ2 and Σ3, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''1 and ''S''2, which come in at most two sizes. Since ''S''1 is a single-period mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ2 and Σ3 are single-period mosses.

Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to single-period mosses ''S''1 (via identifying b with c), ''S''2 (via identifying a with c) and ''S''3 (via identifying a with b). Suppose ''S''1's generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''2 and ''S''3, respectively. These binary words, which we call Σ2 and Σ3, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''1 and ''S''2, which come in at most two sizes. Since ''S''1 is a single-period mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ2 and Σ3 are single-period mosses with period ''k''e.

index: 1 2 3 4 ... ''n''

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 ==== Proof ====
 ===== If the generator of a projection of ''S'' is a ''k''-step, the word of stacked ''k''-steps in ''S'' is pairwise well-formed =====
-Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to single-period mosses ''S''<sub>1</sub> (via identifying b with c), ''S''<sub>2</sub> (via identifying a with c) and ''S''<sub>3</sub> (via identifying a with b). Suppose ''S''<sub>1</sub>'s generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''<sub>2</sub> and ''S''<sub>3</sub>, respectively. These binary words, which we call Σ<sub>2</sub> and Σ<sub>3</sub>, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''<sub>1</sub> and ''S''<sub>2</sub>, which come in at most two sizes. Since ''S''<sub>1</sub> is a single-period mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ<sub>2</sub> and Σ<sub>3</sub> are single-period mosses.
+Suppose ''S'' has ''n'' notes (after dealing with small cases, we may assume ''n'' ≥ 7) and ''S'' projects to single-period mosses ''S''<sub>1</sub> (via identifying b with c), ''S''<sub>2</sub> (via identifying a with c) and ''S''<sub>3</sub> (via identifying a with b). Suppose ''S''<sub>1</sub>'s generator is a ''k''-step, which comes in two sizes: P, the perfect ''k''-step, and I, the imperfect ''k''-step. By stacking ''k''-steps, we get two words of length ''n'' of ''k''-steps of ''S''<sub>2</sub> and ''S''<sub>3</sub>, respectively. These binary words, which we call Σ<sub>2</sub> and Σ<sub>3</sub>, must be mosses, since ''m''-steps in the new words correspond to ''mk''-steps in the mos words ''S''<sub>1</sub> and ''S''<sub>2</sub>, which come in at most two sizes. Since ''S''<sub>1</sub> is a single-period mos, gcd(''k'', ''n'') = 1. Hence when 0 < ''m'' < ''n'', ''mk'' is ''not'' divisible by ''n'' and ''mk''-steps come in ''exactly'' two sizes; hence both Σ<sub>2</sub> and Σ<sub>3</sub> are single-period mosses with period ''k''e.
   index: 1 2 3 4 ...   ''n''