Is there anything else than a $\sigma$-algebra or is $\sigma$-algebra the only meaningful algebra on sets?

Question

Is there anything else than a $\sigma$-algebra or is $\sigma$-algebra the only meaningful algebra on sets?

It seems that the $\sigma$-algebra has been invented in order to serve some particular higher level constructs, but I wonder if there are some other useful algebras on sets?

Answer 1

By Stone's representation theorem for Boolean algebras, the theory of algebras on sets coincides essentially with the theory of Boolean algebras in general. As such, general algebras of sets are extremely useful.