  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 $L(s)$.
