Implications

We have the following indirect implication of form equivalence classes:

303 \(\Rightarrow\) 387
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
303 \(\Rightarrow\) 50 Some propositions equivalent to the Sikorski extension theorem for Boolean algebras, Bell, J.L. 1988, Fund. Math.
50 \(\Rightarrow\) 14 A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
14 \(\Rightarrow\) 49 A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
49 \(\Rightarrow\) 30 clear
30 \(\Rightarrow\) 387 "Dense orderings, partitions, and weak forms of choice", Gonzalez, C. 1995a, Fund. Math.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
303:

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

50:

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.

14:

BPI: Every Boolean algebra has a prime ideal.

49:

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

30:

Ordering Principle: Every set can be linearly ordered.

387:

DPO:  Every infinite set has a non-trivial, dense partial order.  (A partial ordering \(<\) on a set \(X\) is dense if \((\forall x, y\in X)(x \lt y \to (\exists z \in X)(x \lt z \lt y))\) and is non-trivial if \((\exists x,y\in X)(x \lt y)\)).

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