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 ${\sum }_{i=n-k+1}^n{e}_i\ge {\sum }_{i=1}^k{e}_i$ 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 $\eta$ can be extended to a unique lexicographically minimal infinite binomid index $\tilde{\eta }$. This lex-minimal extension $\tilde{\eta }$ is necessarily eventually periodic. In this research, we give a formula for the minimal period and provide an upper bound for the preperiod of $\tilde{\eta }$. We also show that the monoid of lex-minimal extensions is an inductive limit of finitely presented monoids.

11 pages