Here is the definition for Boolean lattice, or Boolean algebra.
I've seen several results about how one can represent a Boolean lattice from a certain structure, or to verify a Boolean lattice can be constructed given proper operations. Here are some examples:
(Stone's theorem) Every Boolean lattice is isomorphic to the lattice of open-and-closed subsets of some compact Hausdorff space.
We can form a Boolean lattice structure in a topological space, for example,
The set of all regular open sets forms a Boolean lattice with proper-defined operations,
(the full definition can be found here) and
The set of topologies on a set $X$ form a complete, complemented lattice.
When we define the fundamental group on a topological space $X$ we associate a functor between the category of Top* and Grp in order to classify spaces. Zariski topology, defined on the set of all prime ideals in a commutation ring with identity, is also intensively studied. However, how come that I haven't seen any further discussion about the lattice structure in algebra and geometry? What are some applications of Stone's theorem, and what are some classical results about Boolean lattice?