Sikorski’s extension theorem
Theorem 1 (Sikorski’s ).
We prove this using Zorn’s lemma. Let be the set of all pairs such that is a subalgebra of containing , and is an algebra homomorphism extending . Note that is not empty because . Also, if we define by requiring that and that extending , then becomes a poset. Notice that for every chain in ,
If , pick . Let be the join of all elements of the form where and , and the meet of all elements of the form where and . and exist because is complete. Since preserves order, it is evident that . Pick an element such that .
Let . Every element in has the form , with . Define by setting , where . We now want to show that is a Boolean algebra homomorphism extending . There are three steps to showing this:
is a function. Suppose . Then, by the last remark of this entry (http://planetmath.org/BooleanSubalgebra), , so that , which in turn implies that . Hence is well-defined.
is a Boolean homomorphism. All we need to show is that respects and . Let and . Then , where and . So
so respects . In addition, respects , as , so that
extends . If , write . Then
Let be an ideal of a Boolean algebra . Let , the Boolean subalgebra generated by . The function given by iff is a Boolean homomorphism. First, notice that iff iff iff . Next, if at least one of is in , , so that . If neither are in , then , so , or . This means that .
As the proof of the theorem shows, ZF+AC (the axiom of choice) implies Sikorski’s extension theorem (SET). It is still an open question whether the ZF+SET implies AC.
Next, comparing with the Boolean prime ideal theorem (BPI), the proof of the corollary above shows that ZF+SET implies BPI. However, it was proven by John Bell in 1983 that SET is independent from ZF+BPI: there is a model satisfying all axioms of ZF, as well as BPI (considered as an axiom, not as a consequence of AC), such that SET fails.
- 1 R. Sikorski, Boolean Algebras, 2nd Edition, Springer-Verlag, New York (1964).
- 2 J. L. Bell, http://plato.stanford.edu/entries/axiom-choice/The Axiom of Choice, Stanford Encyclopedia of Philosophy (2008).
|Title||Sikorski’s extension theorem|
|Date of creation||2013-03-22 18:01:31|
|Last modified on||2013-03-22 18:01:31|
|Last modified by||CWoo (3771)|
|Synonym||Sikorski extension theorem|