Abstract root systems and intervals in weak Bruhat order

We present a new axiomatization of what is known as the “root system” of a Coxeter group that does not involve vector spaces or Coxeter groups. We use this to combinatorially characterize the structure of intervals in weak Bruhat order in much the same way that finite distributive lattices are characterized as the lattices of order ideals of partially ordered sets. In fact, the result for distributive lattices follows as a special case.