Self-Similar Algebras with connections to Run-length Encoding and Rational Languages

TL;DR

This paper explores the properties of self-similar algebras, revealing connections to run-length encoding and rational languages, and establishes new results on eigenvalues and language rationality.

Contribution

It introduces a novel link between self-similar algebras, eigenvalues, and rational languages, including proving the rationality of certain word sets and their asymptotic behavior.

Findings

Eigenvalues relate to smooth words over a 2-letter alphabet.

The language of words making an algebra element a unit is rational.

Asymptotic formula for the number of such words of given length.

Abstract

A self-similar algebra $\left(\mathfrak{A}, \psi \right)$ is an associative algebra $\mathfrak{A}$ with a morphism of algebras $\psi: \mathfrak{A} \longrightarrow M_d \left( \mathfrak{A}\right)$, where $M_d \left( \mathfrak{A}\right)$ is the set of $d\times d$ matrices with coefficients from $\mathfrak{A}$. We study the connection between self-similar algebras with run-length encoding and rational languages. In particular, we provide a curious relationship between the eigenvalues of a sequence of matrices related to a specific self-similar algebra and the smooth words over a 2-letter alphabet. We also consider the language $L(s)$ of words $u$ in $(\Sigma\times \Sigma)^*$ where $\Sigma=\{0,1\}$ such that $s\cdot u$ is a unit in $\mathfrak{A}$. We prove that $L(s)$ is rational and provide an asymptotic formula for the number of words of a given length in $L(s)$.