Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction (Q2570178)

Ultraboolean semilattices are defined as directed unions of finite boolean algebras (among semilattices with \(0\)). Every finite distributive semilattice with \(0\) is a retract of a finite boolean algebra. The author proves that every distributive semilattice with \(0\) is a retract of an ultraboolean semilattice. This result is interesting and difficult and the author uses methods of category theory to prove it.