Recursive structure of MOS scales: Difference between revisions

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Note that the latter two words have at most k s's.

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. Hence it suffices to consider the following two cases, each split into subcases:

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 w2 ∩ w3 contains at least k-1 complete chunks, since assuming otherwise would mean that W₂(L<sup>r+1</sup>s, L<sup>r</sup>s) has at least k+2 chunks, which would contradict the length of W₂(λ, σ).

Hence it suffices to consider the following two cases, each split into subcases:

Case 1

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Case 2.1:

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

~~w3: <L ... sXL ... sXL ... sXL ... s> (complete chunks only)~~

~~⇒ w₂ has k s's and w₃ has (k-1) s's, contradicting the assumption that w is a mos.~~

~~Case 2.2:~~

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

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

⇒ do the same trick as in Case 1.2

Case 2.3:

Case 2.2:

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 2.4:

Case 2.3:

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 2.3 and 2.4 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 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.

=== Preservation of generators ===

@@ Line 415: / Line 415: @@
 Note that the latter two words have at most k s's.
-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. Hence it suffices to consider the following two cases, each split into subcases:
+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 w2 ∩ w3 contains at least k-1 complete chunks, since assuming otherwise would mean that W₂(L<sup>r+1</sup>s, L<sup>r</sup>s) has at least k+2 chunks, which would contradict the length of W₂(λ, σ).
+Hence it suffices to consider the following two cases, each split into subcases:
 Case 1
@@ Line 457: / Line 459: @@
 Case 2.1:
- w2: <L ... sXL ... sXL ... sXL ... sXL ... s> (complete chunks only)
- w3:                 <L ... sXL ... sXL ... sXL ... s> (complete chunks only)
-⇒ w₂ has k s's and w₃ has (k-1) s's, contradicting the assumption that w is a mos.
-Case 2.2:
   w2: <L ... sXL ... sXL ... sXL ... sXL ... s> (complete chunks only)
   w3:               [sXL ... sXL ... sXL ... sXL ... s> (s followed by complete chunks)
 ⇒ do the same trick as in Case 1.2
-Case 2.3:
+Case 2.2:
   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 2.4:
+Case 2.3:
   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 2.3 and 2.4 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 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.
 === Preservation of generators ===