BPI: Every Boolean algebra has a prime ideal.

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 \(G\).)