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Positive degree and atoms of the presented Artin monoid
Statement
Let be the positive Artin monoid presented by generators and the braid relations associated with a finite set and a Coxeter matrix. A word is a finite string of these generators, and its length is its number of letters. Word length descends to an additive map . It vanishes exactly on the identity, and the atoms (nonidentity elements which are not products of two nonidentity elements) are exactly the generators. Every element factors into atoms, and every factorization of into nonidentity elements has at most factors. In particular there is no infinite chain obtained by repeatedly removing a nonidentity left or right factor.
Statement references
Facts & Assumptions
The presentation of the positive Artin monoid is the quotient of the free monoid on the generators by the congruence generated by the braid relations. Each relation has equally many letters on both sides. (Artin monoid and Artin group presentations, and the canonical monoid-to-group map)
Proof
Given: The objects and hypotheses in the statement.
Goal: Establish all assertions of the statement.
Each defining relation preserves word length, including when applied within a longer word; symmetry, transitivity and adjoining a context preserve that equality. Thus two words representing one element have equal lengths, so is well defined and . A word has length zero exactly when it is empty, proving exactly for .
Every generator has degree one and hence cannot split into two elements of positive degree. A word of length at least two splits after its first letter into two elements of positive degree, so is not an atom. Each word is already a factorization into generators. In a product of nonidentity elements the sum of degrees is at least ; consequently . Repeated removal of a nonidentity factor strictly lowers the nonnegative integer degree, ruling out the stated infinite chains.
Depends on
Used by · 7
- Independent spherical Artin Garside input
- Divisibility of positive lifts and weak order
- Ribbon lcms lift two-generator Brink–Howlett elements
- Reduced Coxeter lifts in the presented positive monoid
- Both-sided divisors of the longest positive lift
- Identifying the presented monoid with its positive image
- Transport of a Garside structure to the presented monoid
Dependency tree · 2 levels
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