Generator-offset property: Difference between revisions
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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’. Either β’’ = β or β’’ = β’. Assume β’’ = β’. Then γ’’ = γ. The cyclic words Λ<sub>1</sub> = the pattern of α and α’, Λ<sub>2</sub> = the pattern of β and β’, and Λ<sub>3</sub> = the pattern of γ and γ’ must form mosses. By way of illustration, the chain of k-steps might look like this in S: | 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’. Either β’’ = β or β’’ = β’. Assume β’’ = β’. Then γ’’ = γ. The cyclic words Λ<sub>1</sub> = the pattern of α and α’, Λ<sub>2</sub> = the pattern of β and β’, and Λ<sub>3</sub> = the pattern of γ and γ’ must form mosses. By way of illustration, the chain of k-steps might look like this in S: | ||
1 2 3 4 5 6 7 8 9 | 1 2 3 4 5 6 7 8 9 | ||
Σ = Q Q Q R Q Q Q R T | Σ = Q Q Q R Q Q Q R T | ||
Λ<sub>1</sub> = α α α α α α α α α’ | Λ<sub>1</sub> = α α α α α α α α α’ | ||
Λ<sub>2</sub> = β β β β’ β β β β’ β’ | Λ<sub>2</sub> = β β β β’ β β β β’ β’ | ||