On the monoid of lexicographically minimal extensions

TL;DR

This paper studies the structure of lexicographically minimal extensions of binomid indices, providing formulas for their periodicity and showing the monoid of these extensions is an inductive limit of finitely presented monoids.

Contribution

It introduces a formula for the minimal period of lex-minimal extensions and proves the monoid of these extensions is an inductive limit of finitely presented monoids.

Findings

The minimal period of the lex-minimal extension can be explicitly calculated.

The lex-minimal extension is necessarily eventually periodic.

The monoid of lex-minimal extensions is an inductive limit of finitely presented monoids.

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 $\eta$ 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.