Describtion of normal basis of boundary algebras and factor languages of small growth

TL;DR

This paper characterizes the structure of boundary algebras with small growth and describes factor languages with linear growth, providing a detailed understanding of their normal bases and subword complexity.

Contribution

It introduces a description of normal bases for boundary algebras and characterizes factor languages with linear growth, advancing the understanding of small growth algebraic structures.

Findings

Normal basis description for boundary algebras

Characterization of factor languages with linear growth

Bound on growth function T_L(n) as n + constant

Abstract

Let $A$ be an algebra with fixed set of generators $a_1,\dots,a_s$. $V_A(n)$ be dimension of the space, generated by worlds of length $\le n$ over $a_i$, $T_A(n)=V_A(n)-V_A(n-1)$. If $T_A(n)<\mbox{Const}$, algebra $A$ is a {\it boundary algebra}. We describe a normal basis of boundary algebras, i.e. algebras with small growth. Let $\cal L$ be a factor language over alphabet $\cal A$. {\it Growth function} $T_{\cal L}(n)$ is number of subwords $\cal L$ of degree $n$. We describe factor languages of small growth such that $T_{\cal L}(n)\le n+\mbox{const}$.