How prove the greatest post-fixed point of monotonic function is a fixed point of function

Question

I have monotonic function $f : \mathcal{P}(M) → \mathcal{P}(M)$ on $(\mathcal{P}(M),\subseteq)$

Is it possible to prove that greatest post-fixed point of $f$ is a fixed point of $f$ not using Knaster/Tarski's theorem? If yes, could anyone please provide a full explanation to this problem? Thanks in advance.

Edited: M is a set and CCPO $(\mathcal{P}(M),\subseteq)$

Answer 1

I didn't know this notion but I found that a postfixpoint of $f$ is any $P$ such that $f(P)\subseteq P$.

Let $M$ be a set and let $Q$ be its proper subset. Consider $f\colon \mathcal{P}(M)\to \mathcal{P}(M)$ defined by $f(A)=A\cap Q$. Then each set is a postfixpoint but $f(M)=Q\neq M$. The hpothesis is false.