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

Abstract

A self-similar algebra $\left(\mathfrak{A},\psi \right)$ is an associative algebra $\mathfrak{A}$ with a morphism of algebras $\psi :\mathfrak{A}⟶M_d\left(\mathfrak{A}\right)$, 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 $(\Sigma \times \Sigma {)}^{\ast }$ 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 $L(s)$.

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