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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?

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    $\begingroup$ My guess is that you get the universal abstract $\sigma-$algebra by adding formal complements of the elements in your $\sigma-$frame in a similar way to how you construct the Grothendieck group for a commutative monoid or localize a commutative ring. Note that the explicit construction isn't that important, you get the existence of the adjoint by the adjoint functor theorem (the forgetful functor preserves limits). $\endgroup$ – Georg Lehner Dec 18 '15 at 23:18
  • $\begingroup$ I tried to answer your question in the corresponding MSE thread. $\endgroup$ – Tomáš Jakl Jun 18 '18 at 13:16
  • $\begingroup$ @TomasJakl Thanks. It'll take some time for me to look trough it. I have literally not dealt with this topic since then. $\endgroup$ – Stefan Perko Jun 18 '18 at 13:39

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