Rank-3 scale theorems: Difference between revisions
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=== Definitions and theorems === | === Definitions and theorems === | ||
Throughout, let ''S'' be a scale word in steps ''x'', ''y'', ''z'' (and assume all three of these letters are used). | Throughout, let ''S'' be a scale word in steps ''x'', ''y'', ''z'' (and assume all three of these letters are used). | ||
==== Definition: Unconditionally MV3 ==== | |||
''S'' is ''unconditinoally MV3'' if ''S'' is MV3 for all possible choices of step ratio x:y:z. | |||
==== Definition: EMOS ==== | ==== Definition: EMOS ==== | ||
''S'' is ''elimination-MOS'' (EMOS) if the result of removing (all instances of) any one of the step sizes is a MOS. | ''S'' is ''elimination-MOS'' (EMOS) if the result of removing (all instances of) any one of the step sizes is a MOS. | ||
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==== MV3 Theorem 1 (WIP) ==== | ==== MV3 Theorem 1 (WIP) ==== | ||
''The following are equivalent for a non-multiperiod scale word S with steps x, y, z:'' | ''The following are equivalent for a non-multiperiod scale word S with steps x, y, z:'' | ||
# ''S is MV3 | # ''S is unconditionally MV3.'' | ||
# ''(a) S is PMOS, or (b) S is of the form x'y'z'y'x', or (c) S has signature nx ny nz, n ≥ 2.'' | # ''(a) S is PMOS, or (b) S is of the form x'y'z'y'x', or (c) S has signature nx ny nz, n ≥ 2.'' | ||
# ''(a) S is AG *and* is of the form ax by bz, or (b) S is of the form x'y'x'z'x'y'x', or (c) S is one of the exceptions to PMOS in statement 2.'' | # ''(a) S is AG *and* is of the form ax by bz, or (b) S is of the form x'y'x'z'x'y'x', or (c) S is one of the exceptions to PMOS in statement 2.'' | ||