Compact semilattices without small semilattices I: interval homomorphisms, products, and the Hoare hyperspace

I have already talked about compact semilattices before, but there is a lot more to say, especially on the subject of having small semilattices or not. Zhenchao Lyu is joining me this month, and we will pursue this next month. We start with Jimmie Lawson’s characterization of those compact semilattices that have small semilattices as being those that have enough interval homomorphisms. We will then exploit that and show that, once we know just one compact semilattice with small semilattices (for example, one given by J. Lawson himself), then we can produce a whole lot of them, using products or the Hoare hyperspace construction. Read the full post.