Another take on the same theme of reachability and cheapest-paths on graphs, and hacker-friendly, can be found in Stephen Dolan's Fun With Semirings ([slides](, [paper]( He describes the adjacency (or edge-cost) matrix of a graph. If it has boolean-valued entries (i.e., in the boolean semiring), the sum of its powers solves reachability. If the values are in the tropical semiring (min, +), the same sum of powers gives the cost of cheapest paths! (and changing (min, +) for (max, +) the most expensive ones). The matrix multiplication in the paper corresponds the one in section 2.5 of the book, where semirings are generalized to quantales ("A unital [quantale]( is an idempotent semiring under join and multiplication", and both the boolean and tropical semiring here are idempotent).

Would it be correct to say that a resuorce theory sprung from a weighted graph as in the top post would have the limitation of single-ingredient, single-product boxes that the meringue pie chart doesn't verify?