Distances on free semigroups and their applications

M. M. Choban ; I. A. Budanaev

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2018), pp. 92-119.

Résumé

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.

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