{"id":7460,"date":"2023-10-20T14:20:34","date_gmt":"2023-10-20T12:20:34","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=7460"},"modified":"2023-10-20T14:20:34","modified_gmt":"2023-10-20T12:20:34","slug":"compact-semilattices-without-small-semilattices-i-interval-homomorphisms-products-and-the-hoare-hyperspace","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=7460","title":{"rendered":"Compact semilattices without small semilattices I: interval homomorphisms, products, and the Hoare hyperspace"},"content":{"rendered":"\n

We have already talked about compact semilattices here<\/a> (and we will give back the definition below). A puzzling property was that one needs them to have small semilattices in a number of applications. For example, the fundamental theorem of compact semilattices<\/a> says that there is a tight connection between compact semilattices with small semilattices and bc-domains (more on this below). Most compact semilattices seem to have small semilattices, so it was a natural question whether there existed compact semilattices without small semilattices at all. This question was answered in the positive by Jimmie Lawson in 1970 [1], and we have described his counterexample here<\/a>, although under a slightly different angle.<\/p>\n\n\n\n

The purpose of today’s post is to show that there are a lot of other compact semilattices without small semilattices. Although we had eluded this (pretty fundamental) fact in the previous posts on compact semilattices, we will state another characterization of the ‘small semilattice’ property: a compact semilattice has small semilattices if and only if it has enough of something called interval homomorphisms (namely, enough continuous semilattice homomorphisms to ([0, 1], min) to separate the points, see later), or equivalently it embeds, both as a topological space and as a semilattice, into a product of copies of [0, 1], with pointwise min as addition.<\/p>\n\n\n\n

This gives a practical way of showing that a given compact semilattice does not have small semilattices: it suffices to show that it does not have enough interval homomorphisms. This was how Lawson’s counterexample was originally proved to not have small semilattices: it does not have enough interval homomorphisms, in the worst possible way, as all its interval homomorphisms are constant, and none can therefore separate any pair of distinct points.<\/p>\n\n\n\n

We will see, by a simple argument, that any product of a compact semilattice without small semilattices (e.g., Lawson’s) with any compact semilattice must also fail to have small semilattices. This will give us quite a lot of compact semilattices without small semilattices.<\/p>\n\n\n\n

Then we will see that the Hoare hyperspace of any compact semilattice (L<\/em>, +) (with its upward topology) is again a compact semilattice, and that if (L<\/em>, +) does not have small semilattices, then this Hoare hyperspace will not have small semilattices either, giving us even more examples of compact semilattices without small semilattices. This part will require us to use a flurry of results. Most of them have been presented on this blog, but not all of them; apologies for that!<\/p>\n\n\n\n

Compact semilattices, bc-domains<\/h2>\n\n\n\n

A semilattice<\/em> is a pair (L<\/em>, +) where L<\/em> is a set and + is an associative, commutative, and idempotent operator +. It is topological if + is (jointly) continuous. In this previous post<\/a>, we talked about the fundamental theorem of compact semilattices: there is an isomorphism of categories between:<\/p>\n\n\n\n