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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?

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  • $\begingroup$ Regarding your first comment yes. $\sigma$-algebras are very useful in measure theory. Namely, measures are defined over $\sigma$-algebras. $\endgroup$ – Dog_69 Aug 18 '18 at 18:39
  • $\begingroup$ (Non countable) algebras are useful constructs in many senses. Generally they form the stage for anything involving (finite) unions and complementation. $\endgroup$ – copper.hat Aug 18 '18 at 18:44
  • $\begingroup$ Boolean Algebras by Roman Sikorski discusses algebras in which "countable" is replaced by various cardinal numbers (not necessarily the same for intersections and unions). See also Is there a $\sigma$-algebra on $\mathbb{R}$ strictly between the Borel and Lebesgue algebras? and papers such as this. $\endgroup$ – Dave L. Renfro Aug 18 '18 at 20:37
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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.

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