{"id":4939,"date":"2022-04-19T19:25:51","date_gmt":"2022-04-19T17:25:51","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4939"},"modified":"2023-07-11T14:49:23","modified_gmt":"2023-07-11T12:49:23","slug":"topological-semilattices-with-small-semilattices","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4939","title":{"rendered":"Topological semilattices with small semilattices"},"content":{"rendered":"\n

My purpose today is to explain a counterexample due to Jimmie Lawson [3], or rather a slight variant of it, pertaining to the theory of topological semilattices (which I will not assume to be Hausdorff, contrarily to [3]) and to a property that crops up naturally, namely having small semilattices<\/em>. Before I can do this, I will have to spend some time explaining what topological semilattices are, and how small semilattices arise naturally.<\/p>\n\n\n\n

Semilattices<\/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<\/em> operator +. By idempotence, I mean that x<\/em>+x<\/em>=x<\/em> for every x<\/em> in L<\/em>. Examples abound. Notably, there are sup-semilattices<\/em>, which are posets in which every pair of elements x<\/em>, y<\/em> has a supremum x<\/em> \u22c1 y<\/em>, in which case we can define + as \u22c1; dually, there are inf-semilattices<\/em>, in which every pair of elements x<\/em>, y<\/em> has an infimum x<\/em> \u22c0 y<\/em>, in which case we define + as \u22c0.<\/p>\n\n\n\n

Every semilattice (L<\/em>, +) has an associated ordering \u2291, defined by x<\/em>\u2291y<\/em> if and only if x<\/em>=x<\/em>+y<\/em>. Notice that, if we start from an inf-semilattice, we recover the original ordering this way. Conversely, this definition of \u2291 turns L<\/em> into a poset, in which x<\/em>+y<\/em> is the infimum of x<\/em> and y<\/em>, for any two points x<\/em> and y<\/em>. Therefore semilattices are merely another presentation of inf-semilattices. (Or also of sup-semilattices, if you decide to take the opposite of \u2291 as your preferred ordering!)<\/p>\n\n\n\n

Semilattices occur in a variety of other situations. Instead of making a comprehensive list, let me concentrate on one particular case where they arise naturally: the question of characterizing the algebras of the finitary Smyth hyperspace monad, as studied by Andrea Schalk [1]. That as well is something I need to introduce.<\/p>\n\n\n\n

The finitary Smyth hyperspace Q<\/strong>fin<\/sub>(X<\/em>)<\/h2>\n\n\n\n

For every topological space X<\/em>, let the finitary Smyth hyperspace<\/em> Q<\/strong>fin<\/sub>(X<\/em>) be the set of non-empty finitary compact subsets of X<\/em>, by which I mean the subsets of X<\/em> of the form \u2191{x<\/em>1<\/sub>, …, xn<\/sub><\/em>}, where n<\/em>\u22651, and \u2191 denotes upward closure with respect to the specialization preordering of X<\/em>.<\/p>\n\n\n\n

It is easy to see that (Q<\/strong>fin<\/sub>(X<\/em>), \u222a) is a semilattice. Its associated ordering \u2291 is defined by: Q<\/em> is below Q’<\/em> if and only if Q<\/em> = Q<\/em> \u222a Q’<\/em>, if and only if Q’<\/em> is included in Q<\/em>. In other words, its associated ordering \u2291 is the familiar reverse inclusing<\/em> ordering that we are accustomed to when handling Smyth powerdomains<\/a> and their variants.<\/p>\n\n\n\n

What other properties does it have?<\/p>\n\n\n\n

We equip Q<\/strong>fin<\/sub>(X<\/em>) with the upper Vietoris topology<\/em>, whose basic open subsets are the sets \u2610U<\/em> \u225d {Q<\/em> \u2208 Q<\/strong>fin<\/sub>(X<\/em>) | Q<\/em> \u2286 U<\/em>}, where U<\/em> ranges over the open subsets of X<\/em>. This is the subspace topology induced by the topology with the same name on the Smyth hyperspace<\/em> Q<\/strong>(X<\/em>); Q<\/strong>(X<\/em>) is the set of all non-empty compact saturated subsets of X<\/em>, whose basic open subsets I usually also write as \u2610U<\/em>, and are instead defined as {Q<\/em> \u2208 Q<\/strong>(X<\/em>) | Q<\/em> \u2286 U<\/em>}. I am only considering Q<\/strong>fin<\/sub>(X<\/em>) for now instead of Q<\/strong>(X<\/em>) for simplicity.<\/p>\n\n\n\n

With that topology, (Q<\/strong>fin<\/sub>(X<\/em>), \u222a) is even a topological<\/em> semilattice. A topological semilattice<\/em> is a semilattice (L<\/em>, +) where L<\/em> is given a topology such that + is jointly<\/em> continuous. In the case of Q<\/strong>fin<\/sub>(X<\/em>), we check that binary union is jointly continuous by verifying that \u222a\u20131<\/sup>(\u2610U<\/em>) is simply \u2610U<\/em> \u00d7 \u2610U<\/em>.<\/p>\n\n\n\n

Rather easily, Q<\/strong>fin<\/sub>(X<\/em>) is T0<\/sub>, too: its specialization preordering is reverse inclusion.<\/p>\n\n\n\n

Finally, for all elements Q<\/em> and Q’<\/em> of Q<\/strong>fin<\/sub>(X<\/em>), we always have Q<\/em> \u2286 Q<\/em> \u222a Q’<\/em>, namely Q<\/em> \u222a Q’<\/em> is below Q<\/em> in the specialization ordering. The topological semilattices (L<\/em>, +) in which x<\/em>+y<\/em> \u2264 x<\/em> for all points x<\/em> and y<\/em>, where \u2264 is the specialization preordering of L<\/em> (not the canonical ordering \u2291 defined from +) are called deflationary<\/em>. You may compare this with the notion of inflationary<\/em> semilattices, which we have looked at here<\/a>.<\/p>\n\n\n\n

Algebras of the Q<\/strong>fin<\/sub> monad<\/h2>\n\n\n\n

This raises the question whether (Q<\/strong>fin<\/sub>(X<\/em>), \u222a) would be the free<\/em> T0<\/sub> deflationary topological semilattice over the space X<\/em>. For that question to make sense, we need to define a category of T0<\/sub> deflationary topological semilattices, with continuous, addition-preserving maps as morphisms, and we ask whether the forgetful functor from that category to the category Top<\/strong>0<\/sub> of T0<\/sub> topological spaces has a left adjoint.<\/p>\n\n\n\n

Let us approach the problem in a different way. If (Q<\/strong>fin<\/sub>(X<\/em>), \u222a) were the free T0<\/sub> deflationary topological semilattice over X<\/em>, then the required adjunction would produce a monad<\/a> (T<\/em>, \u03b7, \u03bc) on Top<\/strong>0<\/sub> such that TX<\/em>=Q<\/strong>fin<\/sub>(X<\/em>) for every space X<\/em>, at least up to natural isomorphism. We already know of such a monad: the unit \u03b7X<\/sub><\/em> : X<\/em> \u2192 Q<\/strong>fin<\/sub>(X<\/em>) maps every point x<\/em> of X<\/em> to \u2191x<\/em>, and the multiplication \u03bcX<\/sub><\/em> : Q<\/strong>fin<\/sub>(Q<\/strong>fin<\/sub>(X<\/em>)) \u2192 Q<\/strong>fin<\/sub>(X<\/em>) maps every element Q<\/em><\/strong> of Q<\/strong>fin<\/sub>(Q<\/strong>fin<\/sub>(X<\/em>)) to \u222aQ<\/em><\/strong>, the union of all the elements of Q<\/em><\/strong>; explicitly, \u03bcX<\/sub><\/em>(\u2191{\u2191E<\/em>1<\/sub>, …, \u2191En<\/sub><\/em>}) = \u2191(E<\/em>1<\/sub> \u222a … \u222a En<\/sub><\/em>). (I will let you check that this is indeed a monad by yourself.)<\/p>\n\n\n\n

Then we characterize the algebras<\/a> of that monad. The category Top<\/strong>0<\/sub>Q<\/sup><\/strong>fin<\/sup><\/sub> of Q<\/strong>fin<\/sub>-algebras, also known as the Eilenberg-Moore category of the monad, is automatically equipped with an adjunction, which turns Q<\/strong>fin<\/sub>(X<\/em>) into the free Q<\/strong>fin<\/sub>-algebra over the T0<\/sub> space X<\/em>. Our question then becomes: are the Q<\/strong>fin<\/sub>-algebras on Top<\/strong>0<\/sub> exactly the T0<\/sub> deflationary topological semilattices?<\/p>\n\n\n\n

This was studied, among other things, by Andrea Schalk [1]. One can check that every Q<\/strong>fin<\/sub>-algebra has a canonical structure of deflationary topological semilattice, but not all deflationary topological semilattices give rise to a Q<\/strong>fin<\/sub>-algebra. The reasons are the following, and I will explain them below.<\/p>\n\n\n\n