Let $X$ be an ordered set. A *down-set* (also called a *lower set* or an *order ideal*) of $X$ is a subset $D$ of $X$ such that for every $x, y \in D$, if $x \in D$ and $y \leq\_X x$, then $y \in D$. The *down-set lattice* $\mathcal{O}(X)$ of $X$ is the set of all down-sets of $X$ ordered by inclusion.


Let $P$ and $Q$ be ordered sets, and let $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ be their respective down-set lattices. Are $P$ and $Q$ order-isomorphic if and only if $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ are order-isomorphic? If $P$ and $Q$ are order-isomorphic, then it's easy to show that $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ are order-isomorphic. And if $P$ and $Q$ are totally ordered, then $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ are order-isomorphic to the ideal completions of $P$ and $Q$ respectively, and it's again easy to show that if $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ are order-isomorphic, then $P$ and $Q$ are order-isomorphic. But what if $P$ and $Q$ are only partially ordered?



Order-isomorphic down-set lattices
If $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ are order-isomorphic, then $P$ and $Q$ are isomorphic as well. Furthermore, not only can we recover the poset $P$ from the lattice $\mathcal{O}(P)$, but the mapping $P\mapsto\mathcal{O}(P)$ gives us a duality between the category of all posets and certain kinds of lattices called superalgebraic lattices.


If $L$ is a complete lattice, then an element $x\in L$ is said to be supercompact if $x\leq\bigvee R$ implies that $x\leq r$ for some $r\in R$. A complete lattice $L$ is said to be superalgebraic if every element in $L$ is the least upper bound of supercompact elements.


In the lattice $\mathcal{O}(P)$ it is easily seen that the supercompact elements are precisely elements of the form $\downarrow x=\{y\in P|y\leq x\}$. Therefore the original poset $P$ is order isomorphic to the set of all supercompact elements in $\mathcal{O}(P)$. Furthermore, it is easily seen that $\mathcal{O}(P)$ is a superalgebraic lattice since every lower set is the union of sets of the form $\downarrow x$.


For more information on superalgebraic lattices and this duality see for example the paper Complete Congruences on Topologies and Down-set Lattices by Erne, Gehrke, and Pultr.


