Feels to me like this is required of any 'lattice' (partial ordered set with a unique supremum and infinum for every sub-set). Notably, given any 'cross-sectio

Feels to me like this is required of any 'lattice' (partial ordered set with a unique supremum and infinum for every sub-set). Notably, given any 'cross-section' of a lattice (a set A such that given an x in A, and any x > y then y in A) has a 'surface' (the set of all points x in z such that for all y in A we NOT z > x). That cross section needs to start (1 element) at the cross section containing only the 'infinum' of the entire lattice. Similarly, the surface needs to end small (1 element again) at the cross-section that is the entire lattice. In between, it will generally by larger. I think there might be some 'convexity' results that can be proven about the surfaces of increasing sequences of cross-sections. Given specific kinds of lattices. It feels to me like a cartesian-lattice would always have this convexity porperty. (By convexity I mean that the surface starts of increasing, might be constant for a while, and then must continue decreasing.) Cool subject, makes me wish I was doing a PhD in mathematics.