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## Distances on Free Semigroups and Their Applications

Authors: M. M. Choban, I. A. Budanaev

### Abstract

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 8.1). 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.

M. M. Choban
Tiraspol State University, Republic of Moldova
str. Iablochkin 5, Chisinau, Moldova
E-mail:

I. A. Budanaev
Institute of Mathematics and Computer Sciences of ASM
str. Academiei, 3/2, MD-2028, Chisinau, Moldova
E-mail:

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