Generator-offset property: Difference between revisions
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S<sub>1</sub>: L ... L s | S<sub>1</sub>: L ... L s | ||
S<sub>2</sub>: L' ... L' s' | |||
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. This shows that the maximum variety of S must be at least 3, and that P has at least two preimages in S. | 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. This shows that the maximum variety of S must be at least 3, and that P has at least two preimages in S. | ||