Lattices, distributive lattices, and order ideals

A lattice is a poset $L$ in which every pair $x,y$ has a greatest lower bound, its meet $x\wedge y$, and a least upper bound, its join $x\vee y$. A lattice is distributive when, for all $x,y,z\in L$,

$x\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)$

and

$x\vee(y\wedge z)=(x\vee y)\wedge(x\vee z).$

Let $P$ be a poset. An order ideal, or down-set, is a subset $I\subseteq P$ such that $y\in I$ and $x\le y$ imply $x\in I$. The set of all order ideals of $P$, ordered by inclusion, is denoted $J(P)$. Both $\varnothing$ and $P$ are order ideals.

A lattice isomorphism is a bijection preserving meets and joins. Such a map also preserves and reflects the order, since $x\le y$ is equivalent to $x\wedge y=x$.