  We introduce the poset of mesh patterns, which generalises the permutation
pattern poset. We fully classify the mesh patterns for which the interval
[1^\emptyset,m] is non-pure, where 1^\emptyset is the unshaded singleton mesh
pattern. We present some results on the M\"obius function of the poset, and
show that {\mu}(1^\emptyset,m) is almost always zero. Finally, we introduce a
class of disconnected and non-shellable intervals by generalising the direct
product operation from permutations to mesh patterns.
