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