Of course, most mathematicians would skip the section on partitions for already knowing about them. That and other misunderstanding will happen when you use established terms with your novel definitions.
Are these correct?
Definition 3.60. A thorning of a point a in a complete lattice L is a subset A, of L\bottom with sup S = a and for all x,y in A, some d in L\bottom with d <= x,y
or do you simply mean A is down directed?
Damn that weird, dysfunctional, squished inward equal sign, notation
Definition 3.61. A weak partition of an element a in a complete lattice L is a subset A, of L\bottom with sup A = a and for all x in A, there's some d in L\bottom with d <= a, sup A\x