Your inverse monoid without the partial sum relations seems like a variation of McAlister's monoid here but for paths in a graph instead of words over a set.
The main difference is that the McAlister monoid is the case the graph is a bouquet of loops at a vertex and satisfies just 1,2. I suspect if you take the tight algebra in the sense of Exel of the McAlister monoid it would impose your relations (3). One should check that though.