Axiomatic Thin Position

My goal in this post is to begin defining a system of axioms for thin position that can be applied in a variety of different contexts.  In my last post, I defined the complex of surfaces S(M) for a 3-manifold M, a cell complex with a partial ordering on its vertices.  It turns out that Scharlamann-Thompson thin position can be defined entirely in terms of this cell complex and its ordering, then interpreted topologically.  So one can take this as a model for defining other types of thin positions given a cell complex and an ordering that meet the appropriate axioms.