Generator-offset property: Difference between revisions
| Line 157: | Line 157: | ||
Only two sizes of ''k''-steps of ''S'' can project to P in ''S''<sub>1</sub>, for if there are three sizes of ''k''-steps (α, β, γ), (α, β′, γ′), (α, β′′, γ′′) in ''S'' that project to P, then β, β′ and β′′ are three distinct values. Thus these would project to three different ''k''-steps in ''S''<sub>3</sub>, contradicting the mos property of ''S''<sub>3</sub>. | Only two sizes of ''k''-steps of ''S'' can project to P in ''S''<sub>1</sub>, for if there are three sizes of ''k''-steps (α, β, γ), (α, β′, γ′), (α, β′′, γ′′) in ''S'' that project to P, then β, β′ and β′′ are three distinct values. Thus these would project to three different ''k''-steps in ''S''<sub>3</sub>, contradicting the mos property of ''S''<sub>3</sub>. | ||
===== | ===== ''n'' is odd, etc. ===== | ||
Suppose Q = (α, β, γ) ≠ R = (α, β′, γ′) are the two ''k''-steps in ''S'' that project to P. Then T = (α′, β′′, γ′′) projects to I. Here the values in each component differ by at most 1, and α ≠ α′. Then the cyclic word Λ<sub>1</sub> formed by the a-components of the ''k''-steps in P is α...αα′. Since Σ<sub>2</sub> is a single-period mos pattern of βb + (''n'' − β)(a~c) and β′a + (''n'' − β′)(a~c), the cyclic word Λ<sub>2</sub> = the pattern of β and β′ must be a single-period mos. Similarly, Λ<sub>3</sub> = the pattern of γ and γ′ is a single-period mos. | Suppose Q = (α, β, γ) ≠ R = (α, β′, γ′) are the two ''k''-steps in ''S'' that project to P. Then T = (α′, β′′, γ′′) projects to I. Here the values in each component differ by at most 1, and α ≠ α′. Then the cyclic word Λ<sub>1</sub> formed by the a-components of the ''k''-steps in P is α...αα′. Since Σ<sub>2</sub> is a single-period mos pattern of βb + (''n'' − β)(a~c) and β′a + (''n'' − β′)(a~c), the cyclic word Λ<sub>2</sub> = the pattern of β and β′ must be a single-period mos. Similarly, Λ<sub>3</sub> = the pattern of γ and γ′ is a single-period mos. | ||