Rank-3 scale theorems: Difference between revisions

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==== Definition: PMOS ====

''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS.

==== Definition: AG ====

==== Definition: GO ====

''S'' satisfies the ''~~alternating~~ generator property'' (AG) if it satisfies the following equivalent properties:

''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''.

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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 AG property, see [[AG]].

For theorems relating to the GO property, see [[generator-offset property]].

==== Definitions: Billiard scale ====

Let n = a_1 + ... + a_r be the scale size, w a scale word with signature a_1 X_1, ..., a_r X_r, let L be a line of the form L(t) = (a_1, ..., a_r)t + v_0, where v_0 is a constant vector in R^r. We say that L is ''in generic position'' if L contains a point (0, α_1, α_2, ... α_{r-1}) where α_i and α_i/α_j for i != j are irrational.

@@ Line 21: / Line 21: @@
 ==== Definition: PMOS ====
 ''S'' is ''pairwise MOS'' (PMOS) if the result of equating any two of the step sizes is a MOS.
-==== Definition: AG ====
+==== Definition: GO ====
-''S'' satisfies the ''alternating generator property'' (AG) if it satisfies the following equivalent properties:
+''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''.
@@ Line 28: / Line 28: @@
 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 AG property, see [[AG]].
+For theorems relating to the GO property, see [[generator-offset property]].
 ==== Definitions: Billiard scale ====
 Let n = a_1 + ... + a_r be the scale size, w a scale word with signature a_1 X_1, ..., a_r X_r, let L be a line of the form L(t) = (a_1, ..., a_r)t + v_0, where v_0 is a constant vector in R^r. We say that L is ''in generic position'' if L contains a point (0, α_1, α_2, ... α_{r-1}) where α_i and α_i/α_j for i != j are irrational.