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

A self-similar algebra $(\mathfrak{A},\psi )$ is an associative algebra $\mathfrak{A}$ with a morphism of algebras $\psi: \mathfrak{A} \longrightarrow M_d \left( \mathfrak{A}\right)$,where$M_d \left( \mathfrak{A}\right)$isthesetof$d\times d$matriceswithcoefficientsfrom$\mathfrak{A}$. 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 $(\mathrm{\Sigma }\times \mathrm{\Sigma }{)}^{\ast }$ where $\mathrm{\Sigma }=\{0,1\}$ 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 $L(s)$.