Description:
This note contains the proof that forms 14 (The Boolean Prime Ideal Theorem), [14 BM],
[14 BN], and [14 BO] are equivalent.
Content:
Forms 14 (The Boolean Prime Ideal Theorem), [14 BM],
[14 BN], and [14 BO] are equivalent.
(Recall: [14 BM]: A system of equations over a finite ring \(R\) has a solution in \(R\) if
and only if every finite sub-system has a solution in \(R\). [14 BN]: A system of equations
over a finite field \(F\) has a solution in \(F\) if and only if every finite sub-system has a solution in \(F\).
[14 BO]: A system of equations over a finite structure \((A,+,\cdot)\) has a solution in \(A\), if and only if
every finite sub-system has a solution in \(A\).)
First, it is clear that [14 BO] implies [14 BM] implies
[14 BN]. We shall show that [14 J] implies
[14 BO] and that [14 BN] implies
Form 14.
1
[14 J] implies [14 BO].
Let \(A\) be a finite set with two operations \(+\) and \(\cdot\) defined on it. Suppose \(\Sigma\) is a set of equations over \(A\) such
that every finite subset of \(\Sigma\) has a solution. Let \(V\) be the set of variables which occur in \(\Sigma\). Since \(A\) is finite
the discrete topology on \(A\) is compact. By [14 J], the product \(P = \prod_{i\in V}A\) is compact.
For each finite subset \(\sigma\) of \(\Sigma\), let \(x_\sigma =\{ f\in P : f\) is a solution for \(\sigma\}\). Let
\(B = \{ x_\sigma : \sigma\) is a finite subset of \(\Sigma\}\). The set \(B\) is a set of closed subsets of \(P\) with the finite
intersection property. Since \(P\) is compact,the intersection of \(B\) is non-empty. Any element of the intersection is a solution of the system \(\Sigma\).
2
[14 BN] implies Form 14.
Let \(B\) be a Boolean algebra and for each \(b\in B\), let \(x_b\) be a variable. For \(b\in B\), let \(b'\) be the complement of \(b\).
For each \(b, c\in B\), consider the following set of equations \(\Sigma\) over the two element field \(\{0,1\}\):
\[x_b + x_b' = 1,\qquad\qquad x_{b\cap c} = x_b \cdot x_c.\]
It is easy to see that each finite subset of \(\Sigma\) has a solution. [14 BN] implies that \(\Sigma\) has a
solution. Therefore, the set \(\{b\in B: x_b = 1\}\) is an ultrafilter.
Howard-Rubin number:
30
Type:
Proofs and statements
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