TITLE: Is the Pierce spectrum useful elsewhere in Mathematics?
QUESTION [5 upvotes]: In Borceaux and Janelidze's Galois Theories, a construction of the Pierce spectrum is given. It is the poset of ideals in a Boolean ring. It's construction is reminiscent of the Zariski spectrum in commutative ring theory, however I've not seen it used elsewhere.
Q. Is the Pierce spectrum important elsewhere in mathematics?

REPLY [6 votes]: For Boolean rings, the Pierce spectrum coincides with the Zariski spectrum
and is one of the functors implementing the
Stone duality between Boolean algebras and compact
totally disconnected Hausdorff spaces and Stonean duality
between complete Boolean algebras and compact extremally disconnected
Hausdorff spaces.
Restricting further to complete Boolean algebras of projections
of commutative von Neumann algebras produces
an equivalence of categories of commutative von Neumann algebras
and compact strictly localizable enhanced measurable spaces,
i.e., a version of the Gelfand duality in the setting of measure theory.