Finite families of finite sets closed under union and intersection

Question

At first I thought that any finite family of finite sets closed under union and intersection must be a power set. But then I came up with the following counterexamples:

$F_1=\{\emptyset, \{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\},\{1,2,3,4\} \}$.

$F_2=\{\emptyset,\{1\},\{2\},\{3,4\},\{1,2\},\{1,3,4\},\{2,3,4\},\{1,2,3,4\}\}$.

So my question is what types of finite families of finite sets are closed under union and intersection? Is there any literature on this subject? I could only find this: https://en.wikipedia.org/wiki/Ring_of_sets

Answer 1

These are the open sets of a finite topological space. As you can read on Wikipedia, they can be described by a surprisingly small amount of information: the $\Theta(n^2)$ bits of information needed to define the specialization preorder on the ground set $\{1,\dots,n\}.$ This is in contrast to the $2^{\Theta(n)}$ bits needed to describe a general set system, or monotone set system for example.

They are also order-isomorphic to finite distributive lattices, which give a natural setting for finite-space statistical physics as in the FKG inequality, where it is convenient to be able to study things like $\{0,1,2,3\}^d$ with the product order.

Answer 2

Any Boolean algebra is an example of this, however they are also closed under complement. They should however constitute a wide range of examples for you. Some more nice examples (the second one is not a boolean algebra but at least satisfy your conditions) $$\{\emptyset, \{1,2\},\{3,4\},\{1,2,3,4\}\}$$ $$\{\{1,2\},\{1,2,3,4,5\},\{1,2,3,4,5,6,7,8\}\}$$