Posets related to the connectivity set of Coxeter groups

We define the notion of connectivity set for elements of any finitely generated Coxeter group. Then we define an order related to this new statistic and show that the poset is graded and each interval is a shellable lattice. This implies that any interval is Cohen–Macauley. We also give a Galois connection between intervals in this poset and a boolean poset. This allows us to compute the Möbius function for any interval.