Completeness in order theory

Definition

Let $(P, \le)$ be a partially ordered set and let $\mathcal{C}$ be a class of subsets of $P$. The poset $P$ is complete for $\mathcal{C}$, or has all $\mathcal{C}$-joins, if every $S \in \mathcal{C}$ has a supremum in $P$. Dually, $P$ has all $\mathcal{C}$-meets if every $S \in \mathcal{C}$ has an infimum in $P$. The standard choices of $\mathcal{C}$ are:

1. all subsets, giving a complete lattice;
2. all finite nonempty subsets, giving a join-semilattice, and with meets as well, a lattice;
3. all nonempty subsets with an upper bound, giving (with the dual condition on lattices) a conditionally complete lattice;
4. all directed subsets, giving a directed-complete partial order;
5. all chains, giving a chain-complete poset.