Originally asked on MSE.
In this paper, the authors make the following definitions:
- An (abstract) $\sigma$-algebra is a boolean algebra with countable joins.
- A $\sigma$-frame is a bounded lattice with countable joins, where the distributive law holds ($-\wedge x$ preserves countable joins)
Respective notions of morphisms are the obvious ones; morphisms preserve all the given structure. On page 7 (above Lemma 3) they claim, that there is a left adjoint to the forgetful functor from the category of $\sigma$-algebras to the category of $\sigma$-frames.
How does "this" left-adjoint look like explicitely?