Recursive structure of MOS scales: Difference between revisions

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If w₂ contains fewer complete chunks (<L...Ls preceded by where < is the left chunk boundary) or at least 2 more than w₃, we are done, since they must automatically have different numbers of s's. It suffices to consider the case where the intersection w₂ ∩ w₃ contains at least k-1 complete chunks, since otherwise W₂(L<sup>r+1</sup>s, L<sup>r</sup>s) would have at least k+1 chunks, which would contradict the length of W₂(λ, σ).

~~Hence it suffices to consider the following two cases:~~

Suppose w₂ and w₃ (which have the same length) contain the same number of complete chunks. Then w₂ ∩ w₃ must contain at most k-2 complete chunks, giving the above contradiction. Thus w₂ has one more complete chunk than w₃.

~~'''Case 1'''~~

Suppose w₂ and w₃ (which have the same length) contain the same number of complete chunks. Then w₂ ∩ w₃ must contain at most k-2 complete chunks, giving the above contradiction.

~~'''Case 2'''~~

~~Suppose~~ w₂ has one more complete chunk than w₃.

(Below X denotes a chunk boundary, < > are chunk boundaries that are also the boundary of the word, [] are non-chunk-boundary word boundaries.)

Case 2.1:

Case 1:

w₂: <L ... sXL ... sXL ... sXL ... sXL ... s> (complete chunks only)

w₃: [sXL ... sXL ... sXL ... sXL ... s> (s followed by complete chunks)

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so this contradicts our original scale being a mos.

Case 2.2:

Case 2:

w₂: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] (complete chunks followed by one or more L's)

w₃: [ ... sXL ... sXL ... sXL ... sXL ... s> (one or more L's followed by an s, followed by complete chunks)

Case 2.3:

Case 3:

w₂: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] (complete chunks followed by one or more L's)

w₃: <L ... sXL ... sXL ... sXL ... s> (one or more L's followed by an s, followed by complete chunks)

Both 2.2 and 2.3 ~~are contradictions~~: The number of letters added to the last incomplete chunk to complete it is at least as long as λ. But since the first chunk is λ, it has to be at least as big as the last chunk.

Both Cases 2 and 3 yield a contradiction: The number of letters added to the last incomplete chunk to complete it is at least as long as λ. But since the first chunk is λ, it has to be at least as big as the last chunk.

=== Preservation of generators ===

@@ Line 417: / Line 417: @@
 If w₂ contains fewer complete chunks (<L...Ls preceded by where < is the left chunk boundary) or at least 2 more than w₃, we are done, since they must automatically have different numbers of s's. It suffices to consider the case where the intersection w₂ ∩ w₃ contains at least k-1 complete chunks, since otherwise W₂(L<sup>r+1</sup>s, L<sup>r</sup>s) would have at least k+1 chunks, which would contradict the length of W₂(λ, σ).
-Hence it suffices to consider the following two cases:
+Suppose w₂ and w₃ (which have the same length) contain the same number of complete chunks. Then w₂ ∩ w₃  must contain at most k-2 complete chunks, giving the above contradiction. Thus w₂ has one more complete chunk than w₃.
-'''Case 1'''
-Suppose w₂ and w₃ (which have the same length) contain the same number of complete chunks. Then w₂ ∩ w₃  must contain at most k-2 complete chunks, giving the above contradiction.
-'''Case 2'''
-Suppose w₂ has one more complete chunk than w₃.
 (Below X denotes a chunk boundary, < > are chunk boundaries that are also the boundary of the word, [] are non-chunk-boundary word boundaries.)
-Case 2.1:
+Case 1:
   w₂: <L ... sXL ... sXL ... sXL ... sXL ... s> (complete chunks only)
   w₃:               [sXL ... sXL ... sXL ... sXL ... s> (s followed by complete chunks)
@@ Line 439: / Line 431: @@
 so this contradicts our original scale being a mos.
-Case 2.2:
+Case 2:
   w₂: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] (complete chunks followed by one or more L's)
   w₃:          [ ... sXL ... sXL ... sXL ... sXL ... s> (one or more L's followed by an s, followed by complete chunks)
-Case 2.3:
+Case 3:
   w₂: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] (complete chunks followed by one or more L's)
   w₃:                 <L ... sXL ... sXL ... sXL ... s> (one or more L's followed by an s, followed by complete chunks)
-Both 2.2 and 2.3 are contradictions: The number of letters added to the last incomplete chunk to complete it is at least as long as λ. But since the first chunk is λ, it has to be at least as big as the last chunk.
+Both Cases 2 and 3 yield a contradiction: The number of letters added to the last incomplete chunk to complete it is at least as long as λ. But since the first chunk is λ, it has to be at least as big as the last chunk.
 === Preservation of generators ===