Generator-offset property: Difference between revisions

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(Note: We say “assume” as shorthand for either “assume without loss of generality” or “assume after excluding other possibilities”.)

~~Suppose for sake~~ of ~~contradiction that P has only one preimage~~ in S~~. Assume that S2's generator is also a k-step~~, and that Σ2's imperfect generator is located at index n. Then S1 and S2 are the same mos pattern (up to knowing which step size is the bigger one) and even the same mode. Assume the L of S1 (it could be s, ~~but it doesn’t matter~~) ~~is the result of identifying b and c, and all instances of s in S1 come from a. Then~~ the ~~steps of S2 corresponding to the L of S1 must be either all b’s or all a~c’s, thus these steps are all b’s in S~~ (~~otherwise they would be identified with the~~ a, ~~against the assumption that S1 and S2 are the same mos pattern and mode). So S has only two step sizes (a and~~ b)~~, contradicting the assumption that S has exactly three step sizes~~.

We can write sizes of intervals in S as vectors (p, q, r) using the basis (a, b, c).

~~We can write sizes~~ of ~~intervals~~ in S ~~as vectors~~ (p, q, r) ~~using~~ the ~~basis~~ (a, b, c). Only two k-steps of S can project to P in S1, for if P has three preimages (α, β, γ), (α, β’, γ’), (α, β’’, γ’’) in S, then β, β’ and β’’ are three distinct values. Thus these would project to three different k-steps in S3, contradicting the mos property of S3.

Suppose for sake of contradiction that P has only one preimage (A, B, C) in S. Then S2's generator is (A+C)*(a~c) + Bb, thus is also a k-step, and Σ2's imperfect generator is also located at index n, like Σ1's imperfect generator. Then S1 and S2 are the same mos pattern (up to knowing which step size is the bigger one) and even the same mode. Assume the L of S1 (it could be s, but it doesn’t matter) is the result of identifying b and c, and all instances of s in S1 come from a. Then the steps of S2 corresponding to the L of S1 must be either all b’s or all a~c’s, thus these steps are all b’s in S (otherwise they would be identified with the a, against the assumption that S1 and S2 are the same mos pattern and mode). So S has only two step sizes (a and b), contradicting the assumption that S has exactly three step sizes.

Only two k-steps of S can project to P in S1, for if P has three preimages (α, β, γ), (α, β’, γ’), (α, β’’, γ’’) in S, then β, β’ and β’’ are three distinct values. Thus these would project to three different k-steps in S3, contradicting the mos property of S3.

So suppose Q = (α, β, γ) ≠ R = (α, β’, γ’) are the two preimages of P in S. Then I has preimage (α’, β’’, γ’’), which we denote T. 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 Λ1 = the pattern of α and α’, Λ2 = the pattern of β and β’, and Λ3 = the pattern of γ and γ must form mosses. By way of illustration, the chain of k-steps might look like this in S:

@@ Line 125: / Line 125: @@
 (Note: We say “assume” as shorthand for either “assume without loss of generality” or “assume after excluding other possibilities”.)
-Suppose for sake of contradiction that P has only one preimage in S. Assume that S<sub>2</sub>'s generator is also a k-step, and that Σ<sub>2</sub>'s imperfect generator is located at index n. Then S<sub>1</sub> and S<sub>2</sub> are the same mos pattern (up to knowing which step size is the bigger one) and even the same mode. Assume the L of S<sub>1</sub> (it could be s, but it doesn’t matter) is the result of identifying b and c, and all instances of s in S<sub>1</sub> come from a. Then the steps of S<sub>2</sub> corresponding to the L of S<sub>1</sub> must be either all b’s or all a~c’s, thus these steps are all b’s in S (otherwise they would be identified with the a, against the assumption that S<sub>1</sub> and S<sub>2</sub> are the same mos pattern and mode). So S has only two step sizes (a and b), contradicting the assumption that S has exactly three step sizes.
+We can write sizes of intervals in S as vectors (p, q, r) using the basis (a, b, c).
-We can write sizes of intervals in S as vectors (p, q, r) using the basis (a, b, c). Only two k-steps of S can project to P in S<sub>1</sub>, for if P has three preimages (α, β, γ), (α, β’, γ’), (α, β’’, γ’’) in S, 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>.
+Suppose for sake of contradiction that P has only one preimage (A, B, C) in S. Then S<sub>2</sub>'s generator is (A+C)*(a~c) + Bb, thus is also a k-step, and Σ<sub>2</sub>'s imperfect generator is also located at index n, like Σ<sub>1</sub>'s imperfect generator. Then S<sub>1</sub> and S<sub>2</sub> are the same mos pattern (up to knowing which step size is the bigger one) and even the same mode. Assume the L of S<sub>1</sub> (it could be s, but it doesn’t matter) is the result of identifying b and c, and all instances of s in S<sub>1</sub> come from a. Then the steps of S<sub>2</sub> corresponding to the L of S<sub>1</sub> must be either all b’s or all a~c’s, thus these steps are all b’s in S (otherwise they would be identified with the a, against the assumption that S<sub>1</sub> and S<sub>2</sub> are the same mos pattern and mode). So S has only two step sizes (a and b), contradicting the assumption that S has exactly three step sizes.
+Only two k-steps of S can project to P in S<sub>1</sub>, for if P has three preimages (α, β, γ), (α, β’, γ’), (α, β’’, γ’’) in S, 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>.
 So suppose Q = (α, β, γ) ≠ R = (α, β’, γ’) are the two preimages of P in S. Then I has preimage (α’, β’’, γ’’), which we denote T. 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: