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 \(I\cap S= \{0\}\).

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.

BPI: Every Boolean algebra has a prime ideal.

Order Extension Principle: Every partial ordering can be extended to a linear ordering.  Tarski [1924], p 78.

Ordering Principle: Every set can be linearly ordered.

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

  \(C(\infty ,2)\):  Every family of pairs has a choice function.

If \({\cal L}\)  is  a  lattice  isomorphic  to the  lattice  of subalgebras of some unary universal algebra (a unary universal algebra is one with only unary or nullary operations) and \(\alpha \) is an automorphism of \({\cal L}\) of order 2 (that is, \(\alpha ^{2}\)  is  the  identity) then there is a unary algebra \(\frak A\)  and an isomorphism \(\rho \) from \({\cal L}\) onto the lattice of subalgebras of \(\frak A^{2}\) with \[\rho(\alpha(x))=(\rho(x))^{-1} (= \{(s,t) : (t,s)\in\rho(x)\})\] for all \(x\in  {\cal L}\).

For every cardinal \(m\), there is a set \(A\) such that \(2^{|A|^2}\ge m\) and there is a choice function on the collection of 2-element subsets of \(A\).