4
$\begingroup$

Suppose I have a poset $P$, is there a "best" frame for $P$; that is a frame $L$ with a monotone map $P\to L$ that is universal ?

What if I add some nice conditions on $P$: the $P$'s I'm interested in have suprema for arbitrary chains, they have finite infima, they satisfy the distributive law of frames whenever it makes sense ?

In other words, does the inclusion functor $\mathbf{Frm}\to \mathbf{Pos}$ have a left adjoint ? If not, is there an interesting subcategory of $\mathbf{Pos}$ that contains $\mathbf{Frm}$ such that the inclusion functor has a left adjoint ? A big subcategory (not necessarily full) ?

If the answer's no, is there a canonical/natural/interesting way of associating a frame to a poset (if necessary, a poset that has nice properties such as the ones I described) ?

$\endgroup$
  • 1
    $\begingroup$ I know there's a left adjoint to the forgetful functor from frames to semilattices; so it seems you can reduce the question to whether there are free semilattices on an arbitrary poset. $\endgroup$ – Malice Vidrine Aug 20 '18 at 18:09
4
$\begingroup$

Take the free semilattice on P and then take the free frame on the free semilattice, which is just the set of downclosed subsets of the semilattice. The free semilattice construction is a bit involved, but is essentially the set of finite subsets subject to an equivalence relation determined by the order on P. I can’t find it easily on the web and it’s certainly not a result that is orginal to my PhD but it is Lemma 2.9.2 in http://www.christophertownsend.org/Documents/townsendphd.pdf

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.