TITLE: Minimal generator of an algebra or a sigma-algebra
QUESTION [7 upvotes]: 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?

REPLY [5 votes]: 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.