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Garside combinatorics for Thompson’s monoid $$F^+$$ and a hybrid with the braid monoid $$B_{\infty }^{+}$$. (English) Zbl 1422.05106
Artin’s braid group $$B_n$$ is known to be a group of fractions for the monoid $$B^+_n$$ of positive $$n$$-strand braids. In its study, the Garside elements $$\Delta_n$$ play an important role. On this model of simple braids, defined to be the left divisors of Garside’s elements $$\Delta_n$$ in the monoid $$B^+_\infty$$, the authors investigate simple elements in Thompson’s monoid $$F^+$$ and in an hybrid of this monoid with $$B^+_\infty$$. In both cases, we count how many simple elements left divide the right lcm of the first $$n-1$$ atoms, and characterize their normal forms in terms of forbidden factors. In the case of $$H^+$$, a generalized Pascal triangle appears. Moreover, the authors state very interesting open questions.
##### MSC:
 05E15 Combinatorial aspects of groups and algebras (MSC2010) 20M05 Free semigroups, generators and relations, word problems 20E22 Extensions, wreath products, and other compositions of groups 20F36 Braid groups; Artin groups 68Q42 Grammars and rewriting systems
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