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

• 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.
• 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). – Georg Lehner Dec 18 '15 at 23:18