Cardinality of the base of a ring of sets

Question

Concretely, my question is:

What is the size of the minimal base of a ring of sets? In my understanding the base is the set of elements that can be used to "deduce the rest". E.g., if I have a ring of sets: $\emptyset$, {a}, {b} and {a,b}, then the base is only {a} and {b}. The latter since {a,b} is the union of those elements and $\emptyset$ is the intersection. Is there any general result to compute the elements in the base of a ring of sets?

I have not been able to solve this question for a while and I am starting to wonder if I am looking in the wrong place or I am not aware of an existing solution. Any help will be appreciated :)

Answer 1

I assume you are referring in particular to the power set ring of all subsets of $X$.

For $X$ finite we have minimum generating size $\kappa = n$. We can't do any better than the singleton base.

For $X$ infinite the minimum generating size is $\kappa = 2^{|X|}$, the cardinality of the power set of $X$. Since the power set is a base of itself we have $\kappa <= 2^{|X|}$. Note that rings are closed under finite operations only. For each $n \in \mathbb{N}$ we have $A_n$, the set of all 'expressions' with set operations of $n$ base subsets. If have a base of minimal cardinality $\kappa$ then you can convince yourself that $A_n$ has cardinality $\kappa^n = \kappa$. (this is hand waving but I'm certain it's true)

For instance if $X = \mathbb{N}$ then $A_3$ might have expressions like $( \{ 0\} \cup \{1,2 \} ) \setminus \{ 1254654 \}$.

Then we have a map $A_{\omega} = \bigcup A_n \to P(X)$ by 'evaluating' these formal combinations of base subsets and set operations, e.g. the expression above evaluates to $\{0, 1, 2\}$. This map is surjective since the base generates $P(X)$, so $2^{|X|} \leq |A_{\omega}| = \aleph_0 \kappa = \kappa$. With both inequalities we have $\kappa = 2^{|X|}$.

Answer 2

Assuming that by a ring of sets you mean a family $L$ of subsets of $X$ which is closed under intersection and union, then in general, the minimum can be as great as $|L|$.

Indeed, if $L = \{ \varnothing, \{ x_1\}, \{x_1,x_2\}, \cdots, \{x_1,x_2, \cdots, x_n\} \}$, then you need them all in the base, since you don't have one as intersection or union of others.

More generally, since a ring of sets is always a distributive lattice, in the finite case you just need the $\vee$-irreducible elements, which in the case above, is all of them (some will say that the minimum element of a lattice is not $\vee$-irreducible, but that is just a convenient convention that doesn't matter here).
In the infinite case, a possible base could be the set of prime ideals (or filters), and I think no other could be of less cardinality. Again, if you have a chain, you still need them all in the base.