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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 $$a_i$$.