Description of Normal Bases of Boundary Algebras and Factor Languages of Slow Growth

For an algebra $A$, denote by $V_A(n)$ the dimension of the vector space spanned by the monomials whose length does not exceed $n$. Let $T_A(n)=V_A(n)-V_A(n-1)$. An algebra is said to be boundary if $T_A(n)-n<\mathrm{const}$. In the paper, the normal bases are described for algebras of slow growth or for boundary algebras. Let $\mathscr L$ be a factor language over a finite alphabet $\mathscr A$. The growth function $T_{\mathscr L}(n)$ is the number of subwords of length $n$ in $\mathscr L$. We also describe the factor languages such that $T_{\mathscr L}(n)\le n+\mathrm{const}$.