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 \(P_{n,1}\ge \frac{3}{16}=0.1875\) for every \(n\ge 1\). We also provide an explicit formula, based on the Fibonacci numbers, for the number of states of the automaton. Finally, we present the pseudocode of an algorithm which computes the adjacency matrix of the finite-state automaton.

具有n个生成器的正辫子mono半体\（A_n \）中元素的最大词典代表语言是一种常规语言。我们非常详细地描述了接受这种语言的最小有限状态自动机，并研究了长度k的元素的比例，其最大词典序代表以第一个生成器结尾。这个比例趋向于一定数目\（P_ {N，1} \） ，作为ķ趋于无穷大，我们表明，\（P_ {N，1} \ GE \压裂{3} {16} = 0.1875 \）为每\（n \ ge 1 \）。我们还基于斐波纳契数为自动机的状态数提供了一个明确的公式。最后，我们提出一种算法的伪代码，该算法可计算有限状态自动机的邻接矩阵。