Garside combinatorics for Thompson's monoid $F^+$ and a hybrid with the braid monoid $B\_\infty^+$

TL;DR

This paper explores Garside combinatorics in Thompson's monoid $F^+$ and a hybrid monoid $H^+$, analyzing simple elements, their normal forms, and counting divisors, revealing combinatorial structures like a generalized Pascal triangle.

Contribution

It introduces a Garside-theoretic framework for Thompson's monoid and a hybrid monoid, characterizing simple elements and their normal forms with combinatorial insights.

Findings

Counted simple elements dividing the right lcm of initial atoms.

Characterized normal forms using forbidden factors.

Discovered a generalized Pascal triangle structure in $H^+$.

Abstract

On the model of simple braids, defined to be the left divisors of Garside's elements $\Delta\_n$ in the monoid $B\_\infty^+$ , we investigate simple elements in Thompson's monoid $F^+$ and in a larger monoid $H^+$ that is a hybrid of $B\_\infty^+$ and $F^+$ : 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.