Comment by aweinstock

The post "A Very General Method of Computing Shortest Paths" (https://r6.ca/blog/20110808T035622Z.html) shows that the Gauss-Jordan algorithm for solving matrix equations, the Floyd-Warshall algorithm for all-pairs shortest paths, and Kleene's algorithm for converting DFAs to regular expressions, are all the same algorithm on matrices of elements from a star-semiring.