The lattice of convex subsemilattices of a semilattice

The convex subsemilattices of a semilattice E form a lattice Co(E) in the natural way. The purpose of this paper is to study how the properties of this lattice relate to the semilattice itself. For instance, lower semimodularity of the lattice is equivalent, along with various properties, to the semilattice being a tree. When E has more than two elements the lattice does, however, fail many common lattice-theoretic tests. It turns out that it is more fruitful to describe those semilattices E for which every “atomically generated” filter of Co(E) satisfies certain lattice-theoretic properties.