By Louis Auslander

Lawsuits of the yankee Mathematical Society
Vol. sixteen, No. 6 (Dec., 1965), pp. 1230-1236
Published through: American Mathematical Society
DOI: 10.2307/2035904
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Page count number: 7

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The maximality of the choice of s1 , s2 , ... 5), and the existence of the expected decomposition follows. As for uniqueness, there is a subtle point, which is crucial for applications, in particular in terms of the existence of an automatic structure. , whence all of the entries si at a time. 5), then s1 must be the greatest common divisor of s1 s2 and ∆. 5), then s1 must be the greatest common divisor of s1 s2 s3 and ∆. Proving this requires to use most of the properties of the Garside element ∆, in particular the assumption that the left- and right-divisors of ∆ coincide and the assumption that any two left-divisors of ∆ admit a least upper bound with respect to left-divisibility.

2 (quasi-Garside monoid). 1, except possibly the last one (finiteness of the number of divisors of ∆). We recall that a group G is said to be a group of left-fractions for a monoid M if M is included in G and every element of G can be expressed as f −1 g with f, g in M . 3 (Garside and quasi-Garside group). A group G is said to be a Garside group (resp. quasi-Garside group) if there exists a Garside (resp. quasi-Garside) monoid (M, ∆) such that G is a group of left-fractions for M . In the above context, the terminology “(quasi)-Garside monoid” is frequently used for the monoid alone.

The Klein bottle monoid cannot provide a Garside monoid: a generating subset of K+ must contain at least one element g satisfying |g|a = 1, and every such element has infinitely many left-divisors and, moreover, infinitely many left-divisors making an increasing sequence for left-divisibility. However, checking the following result is easy. 3 (normal decomposition). 4) ∀g ∈ K+ \{1} ( g si+1 ⇒ si g ∆ ). 3 can be connected with the fact that the leftdivisibility relation on K+ is a lattice order, actually a quite simple one as it is a linear order.

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