Generator-offset property: Difference between revisions

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In aX bY bZ, consider (i+j)-steps (representing the generator) with: (1) the maximum possible number of Y’s (at least 2 more than the # of Z's), (2) the maximum number of Z’s (at least 2 more than the # of Y's), or (3) an intermediate number of Y’s and Z’s between the two or (4) (the preimage of) the imperfect generator. Since a + 2b >= 5, there are at least 4 perfect generators, so there must be at least one of each of (1), (2), and (3), giving a contradiction to SV3. [Whenever the boundaries of the (i+j)-step are moved within S, the numbers of Y's and Z's change one at a time and reach a maximum at some choice of the boundary and a minimum with another choice, guaranteeing that intermediate values are reached.]

Any generator of aX 2bW must have an odd number of W steps; if a generator had an even number of W steps, it would be generated by stacking the generator of the mos aX bW' with W' = 2W, a contradiction since the generator of aX bW' can be generated by a generator of aX 2bW. This with (4) immediately gives (5).

Any generator of aX 2bW must have an odd number of W steps. This with (4) immediately gives (5).

For (6), odd-numbered SGA scales are [[Fokker block]]s (in the 2-dimensional lattice generated by the generator and the offset). To see this, consider the following lattice depiction of such a scale:

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 In aX bY bZ, consider (i+j)-steps (representing the generator) with: (1) the maximum possible number of Y’s (at least 2 more than the # of Z's), (2) the maximum number of Z’s (at least 2 more than the # of Y's), or (3) an intermediate number of Y’s and Z’s between the two or (4) (the preimage of) the imperfect generator. Since a + 2b >= 5, there are at least 4 perfect generators, so there must be at least one of each of (1), (2), and (3), giving a contradiction to SV3. [Whenever the boundaries of the (i+j)-step are moved within S, the numbers of Y's and Z's change one at a time and reach a maximum at some choice of the boundary and a minimum with another choice, guaranteeing that intermediate values are reached.]
-Any generator of aX 2bW must have an odd number of W steps; if a generator had an even number of W steps, it would be generated by stacking the generator of the mos aX bW' with W' = 2W, a contradiction since the generator of aX bW' can be generated by a generator of aX 2bW. This with (4) immediately gives (5).
+Any generator of aX 2bW must have an odd number of W steps. This with (4) immediately gives (5).
 For (6), odd-numbered SGA scales are [[Fokker block]]s (in the 2-dimensional lattice generated by the generator and the offset). To see this, consider the following lattice depiction of such a scale: