I may be asking a trivial question, but I am a bit confused about it. I have tried to search for the concept of a minimal generator of an algebra or a sigma-algebra on a set, but have found this concept nowhere. Suppose that I define a minimal generator of an algebra or a sigma algebra A, as a generator of A, none of whose proper subsets generate A. My question is: Does every algebra or sigma algebra on a set have a minimal generator? Also, if the answer is in the affirmitive, then is the proof constructive, or existential?
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$\begingroup$ This sounds like asking for a minimal generating set for the abelian group of the rational numbers under addition: any two elements a and b belong to a cyclic subgroup. $\endgroup$– The Masked AvengerMay 29, 2014 at 15:58
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1$\begingroup$ The Borel sets on a Polish space has a minimal generating set: Without loss of generality, we can simply take the Cantor cube and the collection of all half cubes $\sigma$-freely generates the $\sigma$-algebra of Borel sets. $\endgroup$– Joseph Van NameMay 29, 2014 at 16:36
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Minimal generators of $\sigma$-algebras are treated in
Bhaskara-Rao, K. P. S., & Rao, B. V. (1981). Borel spaces. PWN.
Among other things, it is shown there that every countably generated $\sigma$-algebra has a minimal generator. The problem is posed whether every $\sigma$-algebra has a minimal generator. This question received a negative answer in
Aniszczyk, B., & Frankiewicz, R. (1984). On minimal generators of σ-fields. Fundamenta Mathematicae, 124(2), 131-134.
answered Jun 9, 2014 at 0:27