  Adapting Lindstr\"om's well-known construction, we consider a wide class of
functions which are generated by flows in a planar acyclic directed graph whose
vertices (or edges) take weights in an arbitrary commutative semiring. We give
a combinatorial description for the set of "universal" quadratic relations
valid for such functions. Their specializations to particular semirings involve
plenty of known quadratic relations for minors of matrices (e.g., Pl\"ucker
relations) and the tropical counterparts of such relations. Also some
applications and related topics are discussed.
