On lexicographic representatives in braid monoids

TL;DR

This paper characterizes the automaton recognizing maximal lexicographic representatives in positive braid monoids, analyzes the asymptotic proportion of elements ending with the first generator, and provides a Fibonacci-based formula for automaton size.

Contribution

It offers a detailed description of the minimal automaton for the lexicographic language and derives explicit formulas and bounds related to its structure and element proportions.

Findings

The automaton recognizing the language is finite and explicitly described.

The proportion of elements ending with the first generator converges to at least 1/8.

The automaton size can be calculated using Fibonacci numbers.

Abstract

The language of maximal lexicographic representatives of elements in the positive braid monoid $A_n$ with $n$ generators is a regular language. We describe with great detail the smallest Finite State Automaton accepting such language, and study the proportion of elements of length $k$ whose maximal lexicographic representative finishes with the first generator. This proportion tends to some number $P_{n,1}$, as $k$ tends to infinity, and we show that $P_{n,1}\geq \frac{1}{8}$ for every $n\geq 1$. We also provide an explicit formula, based on the Fibonacci numbers, for the number of states of the automaton.