# Free frame generated by a poset

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) ?

• 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. – Malice Vidrine Aug 20 '18 at 18:09