On the monoid of lexicographically minimal extensions
Jonathan Caalim, Yu-ichi Tanaka
TL;DR
This work studies lex-minimal infinite extensions of finite binomid indices, reframing the construction as an unbounded knapsack problem to obtain structural and periodicity results. It provides a precise formula for the minimal period $p=\gcd\{i\le m : A_i=A_{\max}\}$, where $A_i=S_i/i$, and an upper bound on the preperiod $q=km(m+1)/2$ (with potential zero preperiod in the purely periodic case). The paper also develops a monoid-theoretic framework by introducing atomic binomid indices, showing $\mathbb{L}$ is the inductive limit of finitely presented monoids and that $\mathbf{Atom}\cap\mathbb{L}$ forms a unique basis. Together, these results connect combinatorial index theory with UKP and monoid structure, offering both computational methods and structural insight for lex-minimal binomid 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.