It eventually occurred to me that the problem might have something to do with the theory of posets and functions which preserve order in posets. I already knew something about posets (partially ordered sets). So I looked them up on the Internet and found that the problem actually had something to do with complete lattices. A complete lattice is a poset in which every subset has a supremum and an infimum. The power set

with the subset order is certainly a complete lattice because if is a collection of subsets ofI presently found out about the **KnasterTarski theorem**: If *L* is a complete lattice and f:*L*→*L* is an order-preserving function, then the set of all fixed points of f is also a complete lattice.

This doesnt solve the original problem directly, but I sensed that I was on the right track. And then I found this: http://www.cas.mcmaster.ca/~forressa/ac … 1-talk.pdf

So, in the original problem, taking

will do the trick.

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