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Fragments of the word \(\Delta\) in a braid group. (English. Russian original) Zbl 0599.20044
Math. Notes 36, 505-510 (1984); translation from Mat. Zametki 36, No. 1, 25-34 (1984).
The braid group \(B_{n+1}\) is defined by generators \(a_1,\ldots,a_n\) and relations \(a_ia_{i+1}a_i = a_{i+1}a_ia_{i+1}\), \(i=1,\ldots,n-1\), \(a _ia_j = a_ja_i\), \(i<j-1\). Let \(\Pi_{n+1}\) be the semigroup with the same presentation. For \(1\le i\le j\le n)\) let \(A_{i,j}\) and \(B_{i,j}\) denote the words \(a_ia_{i+1}\cdots a_j\) and \(a_ja_{j-1}\cdots a_i\), respectively: \(A_{i,i-1}\) and \(B_{i,i+1}\) are empty. Let \(\Delta\) denote the so-called fundamental word \(A_{1,n}A_{1,n-1}\cdots A_{1,1}\). Words of the form \(A_{\gamma_n,n}A_{\gamma_{n-1},n-1}\cdots A_{\gamma_1,1}\), where \(\gamma_i\le i+1\) for all \(i\), are called right fragments of \(\Delta\). Words of the form \(B_{1,\gamma_1}B_{2,\gamma_2}\cdots B_{n,\gamma_n}\) are called left fragments of \(\Delta\). It is proved that a word in the alphabet \(a_1,\ldots,a_n\) is a divisor of \(\Delta\) in \(\Pi_{n+1}\) iff it is equal in \(\Pi_{n+1}\) to some fragment of \(\Delta\). F. Garside [Matematika, Moskva 14, No. 4, 116–132 (1970; Zbl 0211.34103)] proved that in \(B_{n+1}\) every word is uniquely representable in the form \(\Delta^ mX\), where \(X\) is a positive word that is not divisible by \(\Delta\) in \(\Pi_{n+1}\). Using the concept of fragment the author introduces the notions of left and right normal form and proves that in \(B_{n+1}\) every word is uniquely representable in left normal form and right normal form. A criterion is found for fragments of \(\Delta\) to be divisible in \(\Pi_{n+1}\) by a given letter \(a_i\).
Reviewer: Oleg V. Belegradek

MSC:
20F36 Braid groups; Artin groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
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