[FOM] Question on Distributive Lattices

[A. Mani]

(due to Dimiter Vakarelov)

Are the following two statements equivalent in ZF?

The following is well known in distributive lattices (L, ., +, 0, 1):

(1) Let F be a filter disjoint from an ideal H. Then there exists a
prime filter K extending F and disjoint from H.

Usually the proof of (1) follows by an application of Zorn Lemma. But
then the proof yields a stronger version:

(2) Let F be a filter disjoint from an ideal H. Then there exists a
prime filter K extending F and disjoint from H and
(\foall x\notin K) (\exists y\in K) x.y \in H

for every x not in K there exists y in K such that x.y is in H.
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I think it should be possible to deduce the equivalence, but might
have been  proved already.



Best

A. Mani



A. Mani
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