Theorem. Bédard's figure is generated by a finite automaton.
Proof :
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The language of reduced words (as well as that of ShortLex words) for a Coxeter group is
regular (Brink & Howlett).
ShortLex tree for affine A2
and its minimal version
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The set of words containing one side of a braid relation
(a braid word) is regular.
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The set of reduced words with a unique reduced expression is that
of reduced words containing no braid words (Tits).
-
[ unique reduced words ]
= [ reduced words ] AND NOT [ containing braid word ]
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