Omega-complete partial order

Remarks

Some authors also require a least element. Every directed-complete partial order is an $\omega$-cpo, since the set of terms of an $\omega$-chain is directed. Example: $\N \cup \set{\infty}$ with $n \le \infty$ for all $n$; an $\omega$-chain is either eventually constant, with supremum its final value, or unbounded in $\N$, with supremum $\infty$. Non-example: $\N$ itself, since the chain $0 \le 1 \le 2 \le \cdots$ has no supremum. The converse fails for the set $\omega_1$ of countable ordinals with their usual order. It is an $\omega$-cpo, because a countable set of countable ordinals has a countable supremum. But the directed set $\omega_1$ itself has no supremum in $\omega_1$.