  We consider three notions of connectivity and their interactions in partially
ordered sets coming from reduced factorizations of an element in a generated
group. While one form of connectivity essentially reflects the connectivity of
the poset diagram, the other two are a bit more involved: Hurwitz-connectivity
has its origins in algebraic geometry, and shellability in topology. We propose
a framework to study these connectivity properties in a uniform way. Our main
tool is a certain linear order of the generators that is compatible with the
chosen element.
