For reference, what you call a 'cross section' is sometimes called an ideal or a downset. I can't quite make out your definition of 'surface', but, if it is meant to read "the set of all points x in A such that for all y in A we have not y > x", then it is the set of maximal elements of A. I'm not an expert on posets, but, if I've got the terminology right, your notion of convexity sounds like you want to deal with a ranked poset with unimodal h-vector.