What are universal abstract $\sigma$-algebras on $\sigma$-frames?

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?

(I do not have an idea, yet)

This construction is not immediate. It is described in details in

• Madden, James J. "κ-frames." Journal of Pure and Applied Algebra 70.1-2 (1991): 107-127.

It's described below Corollary 5.2. If I understand it correctly, it goes as follows:

1. To a $\kappa$-frame $L$ assign the free $\kappa$-frame $L'$ obtained from $L$ by adjoining a complement to every element of it (described in the paragraph before Proposition 5.1).

2. Construct a sequence of $\kappa$-frames $(L_\alpha)_{\alpha\leq\kappa}$ as follows:

$$L_0 = L, \quad L_{\alpha+1} = (L_\alpha)', \quad \text{and}\quad L_\lambda = \text{colim}_{\alpha < \lambda}\ L_\alpha$$ (Where $\lambda$ is a limit ordinal.)

Then, $L_\kappa$ is a Boolean $\kappa$-frame. In order to obtain the result you asked for, run the above procedure for a $\sigma$-frame $L$ in case when $\kappa = \omega_1$.