The universal algebra of a $\sigma$-algebra I am searching for the 'dual' algebraic structure of a $\sigma$-algebra. The notion of duality is like in the case of the Boolean algebra and set algebra.
If $X$ is a set, the complement and intersection on the power set of $X$ is called a set algebra and the series of equations that define a Boolean algebra is the dual of this structure. 
I found this link that seems related to my question:
Is there such a thing as the sigma-completion of a Boolean algebra?
but still it does not solve my problem.
 A: In the paper [1], Sikorski constructs a duality that generalizes Stone duality to certain $\sigma$-complete Boolean algebras and more generally certain $\kappa$-complete Boolean algebras. I shall outline the duality mentioned in Sikorski's paper here.
Suppose that $\lambda$ is a cardinal. Then a Boolean algebra $B$ is said to be $\lambda$-complete if the least upper bound $\bigvee R$ exists whenever $|R|<\lambda$. A filter $Z$ on a $\lambda$-complete Boolean algebra $B$ is said to be a $\lambda$-complete filter if whenever $|R|<\lambda$ and $R\subseteq Z$, then $\bigwedge R\in Z$ as well. We shall call a $\lambda$-complete Boolean algebra strongly $\lambda$-representable if every $\lambda$-complete filter can be extended to a $\lambda$-complete ultrafilter.
A $P_{\lambda}$-space is a completely regular space such that the intersection of less than $\lambda$ many open sets is open, and a topological space $X$ is said to be $\lambda$-compact if every open cover of $X$ has a subcover of cardinality less than $\lambda$. Sikorski gave a correspondence between all $\lambda$-compact $P_{\lambda}$-spaces and all $\lambda$-representable $\lambda$-complete Boolean algebras. The proof of this result is exactly the same as the proof of the duality between Boolean algebras and compact totally disconnected spaces.


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*Sikorski, Roman
Remarks on some topological spaces of high power. 
Fund. Math. 37, (1950). 125–136. 

