Garside combinatorics for Thompson's monoid \(F^+\) and a hybrid with the braid monoid \(B_{\infty }^{+}\) (Q2316682)

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.