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From: Saviola on 26 Jul 2010 05:33
Here it is. Sorry for any trouble:

(AC): Given a non-empty family A={A_i}_(i belongs to I) of non-empty sets, there exists a choice function for A.

(BPI): Given a proper ideal J of a boolean lattice B, there exists a prime ideal I of B such that I contains J.

(DPI): Given a distributive lattice L, an ideal J of L and a filter G of L such that J and G are disjoint, there exists a prime ideal I of L such that I contains J and L\I contains G.

(DMI): Every distributive lattice with 1, which has more than one element, contains a maximal ideal.

Prove that (BPI) ===> (DPI) and (DMI) ==> (AC).

Supposedly (BPI) ===> (DPI) can be proved by constructing an embedding of a given distributive lattice into a Boolean lattice, to which (BPI) is applied.

Similarly, (DMI) ==> (AC) can be proved by applying (DMI) to a suitable lattice of sets.
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