|
|
| Line 11: |
Line 11: |
| == Unproven Conjectures == | | == Unproven Conjectures == |
| * Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes. | | * Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes. |
| == MV3 proofs ==
| |
| Under construction
| |
| === Definitions and theorems ===
| |
| Throughout, let ''S'' be a scale word in steps ''x'', ''y'', ''z'' (and assume all three of these letters are used).
| |
| ==== Definition: Unconditionally MV3 ====
| |
| An abstract scale word ''S'' is ''MV3'', ''unconditionally MV3'' or ''abstractly MV3'' if ''S'' is MV3 for all possible choices of step ratio x:y:z.
| |
|
| |
| ==== Definition: EMOS ====
| |
| ''S'' is ''elimination-MOS'' (EMOS) if the result of removing (all instances of) any one of the step sizes is a MOS.
| |
| ==== Definition: PMOS ====
| |
| ''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS.
| |
| ==== Definition: GO ====
| |
| ''S'' satisfies the ''generator-offset property'' (GO) if it satisfies the following equivalent properties:
| |
| # ''S'' can be built by stacking a single chain of alternating generators g1 and g2, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3.
| |
| # ''S'' is generated by two chains of generators separated by a fixed interval; either both chains are of size ''m'', or one chain has size ''m'' and the second has size ''m-1''.
| |
|
| |
| These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2.
| |
|
| |
| For theorems relating to the GO property, see [[generator-offset property]].
| |
|
| |
|
| [[Category:Fokker block]] | | [[Category:Fokker block]] |