representing a distributive lattice by ring of sets
In this entry, we present the proof of a fundamental fact that every distributive lattice is lattice isomorphic to a ring of sets, originally proved by Birkhoff and Stone in the 1930’s. The proof uses the prime ideal theorem of Birkhoff (http://planetmath.org/BirkhoffPrimeIdealTheorem). First, a simple results from the prime ideal theorem:
Let be a distributive lattice and with . Then there is a prime ideal containing one or the other.
Definition. Let be a distributive lattice, and the set of all prime ideals of . Define , the powerset of , by
is an injection.
If , then by the lemma there is a prime ideal containing one but not another, say and . Then and , so that . ∎
is a lattice homomorphism.
There are two things to show:
preserves : If , then , so that and , since is a sublattice. So and as a result. On the other hand, if , then and . Since is prime, , so that . Therefore, .
preserves : If , then , which implies that or , since is a sublattice of . So . On the other hand, if , then , since is a lattice ideal. Hence .
Therefore, is a lattice homomorphism. ∎
Every distributive lattice is isomorphic to a ring of sets.
Let be as above. Since is an embedding, is lattice isomorphic to , which is a ring of sets. ∎
Remark. Using the result above, one can show that if is a Boolean algebra, then is isomorphic to a field of sets. See link below for more detail.
|Title||representing a distributive lattice by ring of sets|
|Date of creation||2013-03-22 19:08:24|
|Last modified on||2013-03-22 19:08:24|
|Last modified by||CWoo (3771)|