User:Dfeuer/Definition:Complete Join Subsemilattice

Definition

Let $(S, \preceq)$ be an ordered set.

Let $C \subseteq S$.

Then $C$ is a complete join subsemilattice (of $S$) iff:

For each $D \subseteq C$:

$D$ has a supremum in $S$

$\sup D \in C$