Abstract:
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
$T_{\mathscr L}(n)\le n+\mathrm{const}$.
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