Boolean prime ideal theorem
Theorem 1 (Boolean prime ideal theorem).
Every Boolean algebra contains a prime ideal.
Let be a Boolean algebra. If is trivial (the two-element algebra), then is the prime ideal we want. Otherwise, pick , where , and let be the trivial ideal. By Birkhoff’s prime ideal theorem for distributive lattices, , considered as a distributive lattice, has a prime ideal (containing obviously) such that . Then is also a prime ideal of considered as a Boolean algebra. ∎
Every Boolean algebra has a prime ideal.
Every ideal in a Boolean algebra can be enlarged to a prime ideal.
Given a set in a Boolean algebra , and an ideal disjoint from , then there is a prime ideal containing and disjoint from .
An ideal and a filter in a Boolean algebra, disjoint from one another, can be enlarged to an ideal and a filter that are complement (as sets) of one another.
Because the Boolean prime ideal theorem has been extensively studied, it is often abbreviated in the literature as BPI. Since the prime ideal theorem for distributive lattices uses the axiom of choice, ZF+AC implies BPI. However, there are models of ZF+BPI where AC fails.
It can be shown (see John Bell’s online article http://plato.stanford.edu/entries/axiom-choice/here) that BPI is equivalent, under ZF, to some of the well known theorems in mathematics:
the completeness theorem for first order logic.
- 1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
- 2 T. J. Jech, The Axiom of Choice, North-Holland Pub. Co., Amsterdam, (1973).
- 3 R. Sikorski, Boolean Algebras, 2nd Edition, Springer-Verlag, New York (1964).
|Title||Boolean prime ideal theorem|
|Date of creation||2013-03-22 18:45:52|
|Last modified on||2013-03-22 18:45:52|
|Last modified by||CWoo (3771)|