Interpreting a graph like a relation allows to compose such relation more and more, as in my snippet (the interesting part is that multiplying two matrices is like composing two relations).

So, x A y means there's an arc from x to y. And x AA y (i.e. x A² y) means: there is some z such that x A z and z A y, so there's a path from x to y through z. When I learned that, I said "hey so those tricky rules for multiplying matrices make sense!".

Another interesting point is that working on algebric structures one could use different operators than sum and product, on condition that the underlying structure is preserved (this is the meaning of my Tolkien quote). Try to reimplement + and * in the matrix product algorithm with min and +.