This theory proves a theorem of Birkhoff that asserts that any finite distributive lattice is isomorphic to the set of down-sets of that lattice's join-irreducible elements. The isomorphism preserves order, meets and joins as well as complementation in the case the lattice is a Boolean algebra. A consequence of this representation theorem is that every finite Boolean algebra is isomorphic to a powerset algebra.
- Birkhoff, G. (1937). Rings of sets. Duke Mathematical Journal, 3(3). https://doi.org/10.1215/s0012-7094-37-00334-x
- B. A. Davey and H. A. Priestley, “Chapter 5. Representation: The Finite Case,” in Introduction to Lattices and Order, 2nd ed., Cambridge, UK ; New York, NY: Cambridge University Press, 2002, pp. 112–124.