Pattern Recognition on Oriented Matroids: Critical Committees and Distance Signals

Abstract

If V(R) is the vertex sequence of a symmetric cycle R in the tope graph of a simple acyclic oriented matroid M on a t-element ground set, then the set min V(R) of minimal elements in the subposet V(R) of the tope poset of M, based at the positive tope, is a critical committee for M that votes for the base tope. We consider the sequence zR of poset ranks of the elements from the vertex sequence of R as a fragment of a signal with period 2t and relate the number of members of the committee min V(R) to the magnitudes of [t/2] components, with odd indices, of the discrete Fourier transform of the distance vector zR.