Example. Ascending Kleene chain [dt-000X]

Given a pointed poset (X,\subseteq ) and a monotonic function f : X \rightarrow X, the ascending Kleene chain is the \omega -chain: \bot \sqsubseteq f(\bot ) \sqsubseteq f(f(\bot )) \sqsubseteq f(f(f(\bot ))) \sqsubseteq \cdots Because \bot is the bottom element, we know that \bot \sqsubseteq f(\bot ). Each subsequent step in the chain exists because of monotonicity.