Rank-3 scale theorems: Difference between revisions
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===== LQ is equivalent to floor-LQ in case of 2-step scales (WIP) ===== | ===== LQ is equivalent to floor-LQ in case of 2-step scales (WIP) ===== | ||
==== 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 unconditionally 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.''ppp | ||
# ''(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.'' | ||
====== MV3 implies LQ except in the case "xyzyx" (WIP) ====== | ====== MV3 implies LQ except in the case "xyzyx" (WIP) ====== | ||
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====== PMOS implies AG (except in the case xyxzxyx) (WIP) ====== | ====== PMOS implies AG (except in the case xyxzxyx) (WIP) ====== | ||
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==== AG scale is unconditionally MV3 implies "ax by bz" and that the scale has odd size ==== | |||
AG by itself does ''not'' imply "ax by bz"; [[blackdye]] (LSLMLSLMLS) is a counterexample. | AG by itself does ''not'' imply "ax by bz"; [[blackdye]] (LSLMLSLMLS) is a counterexample. | ||