On the monoid of lexicographically minimal extensions

Abstract

A sequence $(e_i)_{i \le m}$ of nonnegative integers $e_i$, where $m \in \mathbb{N}$ or $m =\infty$, is called a binomid index if $\sum_{i=n-k+1}^{n} e_i\geq \sum_{i=1}^ke_i$ for all $k, n \in \mathbb{N}$ such that $ 1\le k \le n < m$. Infinite binomid indices give rise to binomid sequences (also known as Raney sequences) and generalized binomial coefficients. A finite binomid index $η$ can be extended to a unique lexicographically minimal infinite binomid index $\tildeη$. This lex-minimal extension $\tildeη$ is necessarily eventually periodic. In this research, we give a formula for the minimal period and provide an upper bound for the preperiod of $\tildeη$. We also show that the monoid of lex-minimal extensions is an inductive limit of finitely presented monoids.