On Non-Archimedean Fuzzy Metric Free Topological Groups

We construct the free group over a non-Archimedean fuzzy metric space (X,M,∧) in the sense of George and Veeramani where ∧ is the minimum t-norm. The two main tools used are the concept of a scheme (for every non-empty subset S of N of even cardinal, a permutation φ on S is a scheme for S if it is idempotent, with no fixed points and, additionally, i<j<φ(i)<φ(j) does not hold for every i,j∈S), and the notion of a fuzzy prenorm on a fuzzy topological group. As a consequence of our results, we prove that every non-Archimedean fuzzy metric space (X,M,∧) in the sense of George and Veeramani is isometric to a closed subspace of a non-Archimedean fuzzy metric free (Abelian) group and also that every metric space (X,d) is uniformly isomorphic to a closed subspace of a non-Archimedean fuzzy metric free (Abelian) group. Our results also apply to non-Archimedean fuzzy metric spaces in the sense of Kramosil and Michálek.