When partially ordered by set inclusion, this family is a lattice.
The partially ordered set is a well partial order.
In a partially ordered set there may be some elements that play a special role.
The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set.
The question whether the structure of partially ordered set would cause similar problems is answered by the following result.
There are 219 partially ordered sets on four labeled elements.
(This is a an example of how any partially ordered set can be considered as a category).
Every partially ordered set is an upper set of itself.
A given partially ordered set may have several different completions.
The identity function on any partially ordered set is always an order automorphism.
