We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 303 \(\Rightarrow\) 50 |
Some propositions equivalent to the Sikorski extension theorem for Boolean algebras, Bell, J.L. 1988, Fund. Math. |
| 50 \(\Rightarrow\) 14 |
A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar. |
| 14 \(\Rightarrow\) 229 |
Variants of Rado's selection lemma and their applications, Rav, Y. 1977, Math. Nachr. |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 303: | If \(B\) is a Boolean algebra, \(S\subseteq B\) and \(S\) is closed under \(\land\), then there is a \(\subseteq\)-maximal proper ideal \(I\) of \(B\) such that \(I\cap S= \{0\}\). |
| 50: | Sikorski's Extension Theorem: Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). Sikorski [1964], p. 141. |
| 14: | BPI: Every Boolean algebra has a prime ideal. |
| 229: | If \((G,\circ,\le)\) is a partially ordered group, then \(\le\) can be extended to a linear order on \(G\) if and only if for every finite set \(\{a_{1},\ldots, a_{n}\}\subseteq G\), with \(a_{i}\neq\) the identity for \(i = 1\) to \(n\), the signs \(\epsilon_{1}, \ldots,\epsilon_{n}\) (\(\epsilon_{i} = \pm 1\)) can be chosen so that \(P\cap S(a^{\epsilon_{1}}_{1},\ldots,a^{\epsilon_{n}}_{n})=\emptyset\) (where \(S(b_{1},\ldots,b_{n})\) is the normal sub-semi-group of \(G\) generated by \(b_{1},\ldots, b_{n}\) and \(P = \{g\in G: e\le g\}\) where \(e\) is the identity of \(G\).) |
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