I have read the Tarski's fixed point theorem for powersets. It requires an order-preserving/monotonic function. I wonder if there are results (with strengthened conditions) for order-reversing functions.
Let $U$ be a set, then its power set $P(U)$ with inclusion $\subseteq$ forms a partial order. A function $\phi:P(U) \to P(U)$ is monotonic if $$S \subset S' \implies \phi(S) \subset \phi(S')$$ for all $S,S'\in P(U)$.
A function $\psi:P(U) \to P(U)$ is order-reversing if $$S \subset S' \implies \psi(S') \subset \phi(S)$$ for all $S,S'\in P(U)$.
The Tarski's Theorem for minimum fixed points states Let $P(U)$ be a power set and $\phi:P(U) \to P(U)$ be a monotonic function. Define $$m = \bigcap \{S\subseteq U | \phi(S) \subseteq S\},$$ then $m$ is a fixed point of $\phi$.
I think that if we want to extend this result to order-reversing functions, we need additional conditions such as "continuity". I tried to draw the analogy between the fixed point argument for functions that maps from $[0,1]$ to $[0,1]$, but had no luck. We can easily construct counter example for decreasing function without a fixed point on $[0,1]$ through a discontinuity. But I don't know how to define continuity over powersets. I would really appreciate if someone can point me to something.