In an earlier post I showed (assuming some knowledge of group theory) that for strings in the two letters 'a' and 'b', allowing negative strings give rise to what mathematicians call the free group on 2 letters, which is an enormous object. If you want the math, look at https://en.wikipedia.org/wiki/Free_group#Construction [Except previously I linked to the wrong part of the page.]

Free groups are a difficult concept, usually introduced at post-graduate level. If you can tell me you understand that concept, I'm happy on that basis to explain how it provides string concatenation with a well-defined inverse.

-- 

Jonathan