Ternary parallelogram scales are MOS substitution: Difference between revisions
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We thus wish to constrain ways of labeling {{nowrap|[0 : ''m''] × [0 : ''n'']}} with {{nowrap|ℤ/''mn''ℤ}} elements such that | We thus wish to constrain ways of labeling {{nowrap|[0 : ''m''] × [0 : ''n'']}} with {{nowrap|ℤ/''mn''ℤ}} elements such that | ||
* {{nowrap|'''v''' {{=}} (1, 0)}} is consistently the π-image of a ''k''<sub>'''v'''</sub>-step interval of ''w'', {{nowrap|0 < ''k''<sub>'''v'''</sub> < ''mn''}} | * {{nowrap|'''v''' {{=}} (1, 0)}} is consistently the π-image of a ''k''<sub>'''v'''</sub>-step interval of ''w'', {{nowrap|0 < ''k''<sub>'''v'''</sub> < ''mn'',}} so traveling one step east always shifts the label by ''k''<sub>'''v'''</sub> | ||
* {{nowrap|'''w''' {{=}} (0, 1)}} is consistently the π-image of a ''k''<sub>'''w'''</sub>-step interval, {{nowrap|0 < ''k''<sub>'''w'''</sub> < ''mn''}} | * {{nowrap|'''w''' {{=}} (0, 1)}} is consistently the π-image of a ''k''<sub>'''w'''</sub>-step interval, {{nowrap|0 < ''k''<sub>'''w'''</sub> < ''mn'',}} so traveling one step north always shifts the label by ''k''<sub>'''v'''</sub> | ||
* every element of {{nowrap|ℤ/''mn''ℤ}} is used exactly once in the labeling. | * every element of {{nowrap|ℤ/''mn''ℤ}} is used exactly once in the labeling. | ||
After rotating ''w'', we may assume that (0, 0) is labeled 0. The labeling now extends to a surjective homomorphism <math>\varphi: \mathbb{Z}^2\langle \mathbf{v},\mathbf{w}\rangle \to \mathbb{Z}/mn\mathbb{Z},</math> where {{nowrap|φ('''v''') {{=}} ''k''<sub>'''v'''</sub>}} and {{nowrap|φ('''w''') {{=}} ''k''<sub>'''w'''</sub>.}} φ has {{nowrap|[0 : ''m''] × [0 : ''n'']}} as a fundamental domain. | After rotating ''w'', we may assume that (0, 0) is labeled 0. The labeling now extends to a surjective homomorphism <math>\varphi: \mathbb{Z}^2\langle \mathbf{v},\mathbf{w}\rangle \to \mathbb{Z}/mn\mathbb{Z},</math> where {{nowrap|φ('''v''') {{=}} ''k''<sub>'''v'''</sub>}} and {{nowrap|φ('''w''') {{=}} ''k''<sub>'''w'''</sub>.}} φ has {{nowrap|[0 : ''m''] × [0 : ''n'']}} as a fundamental domain. | ||