Birkhoff's Representation Theorem For Finite Distributive Lattices

Matthew Doty 📧

December 6, 2022

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.

Abstract

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.

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Related publications

  • 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.

Session Birkhoff_Finite_Distributive_Lattices

Cite

× 
@article{Birkhoff_Finite_Distributive_Lattices-AFP,
  author  = {Matthew Doty},
  title   = {Birkhoff's Representation Theorem For Finite Distributive Lattices},
  journal = {Archive of Formal Proofs},
  month   = {December},
  year    = {2022},
  note    = {\url{https://isa-afp.org/entries/Birkhoff_Finite_Distributive_Lattices.html},
             Formal proof development},
  ISSN    = {2150-914x},
}
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