Planar flows and Pl\"ucker's type quadratic relations over semirings

It is well known, due to Lindstr\"om, that the minors of a (real or complex) matrix can be expressed in terms of weights of flows in a planar directed graph. Another classical fact is that there are plenty of homogeneous quadratic relations involving flag minors, or Pl\"ucker coordinates of the corresponding flag manifold. Generalizing and unifying these facts and their tropical counterparts, we consider a wide class of functions on $2^{[n]}$ that are generated by flows in a planar graph and take values in an arbitrary commutative semiring, where $[n]=\{1,2,\ldots,n\}$. We show that the ``universal'' homogeneous quadratic relations fulfilled by such functions can be described in terms of certain matchings, and as a consequence, give combinatorial necessary and sufficient conditions on the collections of subsets of $[n]$ determining these relations.