Distances on free semigroups and their applications

In this article it is proved that for any quasimetric $d$ on a set $X$ with a base-point $p_X$ there exists a maximal invariant extension $\hat\rho$ on the free monoid $F^a(X,\mathcal V)$ in a non-Burnside quasi-variety $\mathcal V$ of topological monoids (Theorem 6.1). This fact permits to prove that for any non-Burnside quasi-variety $\mathcal V$ of topological monoids and any $T_0$-space $X$ the free topological monoid $F(X,\mathcal V)$ exists and is abstract free (Theorem 7.1). Corollary 10.2 affirms that $F(X,\mathcal V)$, where $\mathcal V$ is a non-trivial complete non-Burnside quasi-variety of topological monoids, is a topological digital space if and only if $X$ is a topological digital space.