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.

If you forget about property (3), which I think will come from going to the tight algebra, non-zero elements of you inverse semigroup can be represented by directed paths in your graph together with a distinguished in vertex of the path and out vertex of the path (they don't have to be the first or last vertex and they can be the same). You can multiply if you can line up the out vertex of the first path with the in vertex of the second and take the union and get a valid birooted path.

I believe that the empty path at a vertex gets identified with your two sums in the with quotient although maybe if there are sources or sinks one has to be a little careful.

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.

If you forget about property (3), which I think will come from going to the tight algebra, non-zero elements of you inverse semigroup can be represented by directed paths in your graph together with a distinguished in vertex of the path and out vertex of the path (they don't have to be the first or last vertex and they can be the same). You can multiply if you can line up the out vertex of the first path with the in vertex of the second and take the union and get a valid birooted path.

I believe that the empty path at a vertex gets identified with your two sums in the with tight algebra although maybe if there are sources or sinks one has to be a little careful.

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.

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.

If you forget about property (3), which I think will come from going to the tight algebra, non-zero elements of you inverse semigroup can be represented by directed paths in your graph together with a distinguished in vertex of the path and out vertex of the path (they don't have to be the first or last vertex and they can be the same). You can multiply if you can line up the out vertex of the first path with the in vertex of the second and take the union and get a valid birooted path.

I believe that the empty path at a vertex gets identified with your two sums in the with quotient although maybe if there are sources or sinks one has to be a little careful.