 Greatest fixed point

A fixed point of a function on a partially ordered set that is greater than or equal to every other fixed point. For the order-preserving function $H(X)=A\cap \{s:\mathrm{Post}(s)\subseteq X\}$ on the power set of a finite transition system with no terminal states, its greatest fixed point is the set of states from which every infinite path in a transition system remains in $A$. Iteration from the whole state space decreases to this set; every strict decrease removes at least one state.