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 $\kappa = \omega_1$.

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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