Algebraic lattice

Remarks

Example: the lattice of subgroups of a group $G$, whose compact elements are the finitely generated subgroups. Each subgroup $H$ is the join of the cyclic subgroups $\langle h \rangle$ for $h \in H$. Non-example: the closed subsets of $[0, 1]$, ordered by inclusion, form a complete lattice in which the join of a family is the closure of its union. A nonempty closed set $F$ is not compact: pick $p \in F$; the closed sets $C_n = [0, 1] \setminus (p - 1/n, p + 1/n)$ form a directed family whose join is $[0, 1] \supseteq F$, yet $p \notin C_n$ for every $n$. So the only compact element is $\varnothing$, and no nonempty closed set is a join of compact elements. Every algebraic lattice is a continuous lattice. Some authors do not require completeness and speak of algebraic posets or algebraic dcpos instead.