\(SPI^*\): Uniform weak ultrafilter principle: For each family \(F\) of infinite sets \(\exists f\) such that \(\forall x\in F\), \(f(x)\) is a non-principal ultrafilter on \(x\).

\(C(\infty,< \aleph_{0})\):  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 \(x\in  {\cal L}\).