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In this article we show that there are invariant distances on the monoid L(A) of all strings closely related to Levenshtein’s distance. We will use a distinct definition of the distance on L(A), based on the Markov - Graev method, proposed by him for free groups. As result we will show that for any quasimetric d on alphabet A in union with the empty string there exists a maximal invariant extension d* on the free monoid L(A). This new approach allows the introduction of parallel and semiparallel decompositions of two strings. In virtue of Theorem 3.1, they offer various applications of distances on monoids of strings in solving problems from distinct scientific fields. The discussion covers topics in fuzzy strings, string pattern search, DNA sequence matching etc. [ABSTRACT FROM AUTHOR]