Strong Linking Axiom for Boolean Algebras: Each linked system in a Boolean algebra is contained in some maximal linked system. (If \(B\) is a Boolean algebra, \(L\subseteq B\) is linked if \(a\wedge b\neq 0\) for all \(a\) and \(b\in L\).)

Disjoint Transversal axiom (2,2): If \(X\) is a set and \({\cal U}\) and \({\cal V}\) are subsets of \(\{ y\subseteq X : |y|\le 2\}\) and if for each two finite subsets \({\cal U}_{0}\subseteq {\cal U}\) and \({\cal V}_{0} \subseteq {\cal V}\) there are transversals \(Y_{1}\) and \(Y_{2}\) for \({\cal U}_{0}\) and \({\cal V}_{0}\) respectively such that \(Y_{1} \cap Y_{2} = \emptyset\), then there are disjoint transversals \(X_{1}\) and \(X_{2}\) for \({\cal U}\) and \({\cal V}\) respectively. (\(Y\subseteq X\) is a transversal for \({\cal U}\) if for all \(y\in \cal U\), \(y\cap Y \ne\emptyset\).)