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?

  • $\begingroup$ Look at ordered groups and rings. $\endgroup$ – William Elliot Dec 16 '18 at 3:59
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    $\begingroup$ In set theory and logic courses you'd learn a lot about Boolean algebras, e.g. It depends on your subfield of maths. In computer science/electroncis they're also taught (not the abstract structure so much, more application focused, design of circuits etc.) The abstract structure goes into set theory very quickly. They don't really have aplications in geometry. $\endgroup$ – Henno Brandsma Dec 16 '18 at 7:02
  • $\begingroup$ Search for lattice ordered groups. $\endgroup$ – William Elliot Dec 16 '18 at 9:58
  • $\begingroup$ You may interested in point free topology: en.m.wikipedia.org/wiki/Pointless_topology $\endgroup$ – Robert Thingum Dec 16 '18 at 18:09
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    $\begingroup$ Use of Stoen duality in Boolean algebras is quite pervasive and very common. The functoriality isn't used as much as for Zariski or fundamental groups, but look in the Handbook of Boolean Algebras for many applications. E.g. see part 2 amd part1 $\endgroup$ – Henno Brandsma Dec 17 '18 at 6:30

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