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.

Restricted Choice for Families of Well Ordered Sets:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that the family of non-empty well orderable subsets of \(Y\) has a choice function.

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\), \(A\) has an infinite subset \(B\) such that for every \(n\in\omega\), \(n>0\), the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco [1998b]

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\) and every \(n\in\omega\), there is an infinite subset \(B\) of \(A\) such the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco} [1998b]

(For \(n\in\omega\), \(n\ge 2\).)  For every infinite set \(X\), there is an infinite \(Y \subseteq X\) such that the set of all \(n\)-element subsets of \(Y\) has a choice function.