Consider a set $S$ with a preorder $\preceq$ (a *preorder* is a reflexive and transitive relation). A lower set $A$ of $S$ is defined as a subset of $S$ such that for all $x \in S$ and $y \in A$, if $x \preceq y$ then $x \in A$.


I believe the set of all lower sets of $S$ form a complete, distributive lattice. Is there a reference which states this?



Reference request: lower sets of a preorder form a lattice
It's even better than completely distributive: it's a complete ring of sets, or abstractly, a "Stone lattice". Moreover, this construction describes a dual categorical equivalence between posets and Stone lattices. See [Section 6.3](https://books.google.com/books?id=IBZeDwAAQBAJ&pg=PA201) of my book *Lipschitz Algebras* (second edition).


