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) ?
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