Lattice ideal

Remarks

For $a \in L$, the set $\downarrow a = \set{x \in L : x \le a}$ is an ideal, the principal ideal generated by $a$. In a finite lattice every ideal is principal, generated by the join of all its elements. Example: in the power set of $\N$ ordered by $\subseteq$, the finite subsets form an ideal. It is nonempty, every subset of a finite set is finite, and the union of two finite sets is finite. It is not principal, since no finite set contains all finite sets. Non-example: in the power set of $\set{1, 2}$, the set $\set{\emptyset, \set{1}, \set{2}}$ satisfies clauses 1 and 2 but not clause 3, since $\set{1} \cup \set{2} = \set{1, 2}$ is missing. The same notion makes sense in any partially ordered set, with clause 3 replaced by: any two elements of $I$ have an upper bound in $I$. In a lattice the two versions agree.