The following statement is well known in distributive lattices (D, 0, 1, ., +):
(1) Let F be a filter disjoint from an ideal I. Then there exists a prime filter F’ extending F and disjoint from I.
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 I. Then there exists a prime filter F’ extending F and disjoint from I and for every x not in F’ there exists y in F’ such that x.y is in I.
I am interested if (1) and (2) are equivalent (or not ) in ZF. Any references?
Thanks for the suggestions, but do you know the answer of my question?
I don't immediately know the answer to your question myself (my two excellent set theory courses, taught by Kees Doets, took place about 30 years ago), but I have the impression that the answer will not be found in a textbook on 'Naive set theory'.
Possibly you may find the literature references in the Wikipedia article on the Boolean Prime Ideal theorem useful: http://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem
There may also be some useful information in the 1997 article by Herrlich and Steprans: http://www.tandfonline.com/doi/abs/10.1080/16073606.1997.9632237?journalCode=tqma20#.VEo1BufdgeM (which is unfortunately behind a pay-wall).
Hopefully a true up-to-date expert on set theory will react to your question as well.
Please visit : https://www.researchgate.net/profile/Vilas_Kharat/publication/220534031_Semiprime_Ideals_and_Separation_Theorems_for_Posets/links/53d8f4250cf2e38c6331c7db?ev=pub_int_doc_dl&origin=publication_list&inViewer=true
Article Semiprime Ideals and Separation Theorems for Posets
Thanks to Rineke Verbrugge and Vilas Kharat for the information. Vilas, I was not able to download your paper. Is it contains an answer to my question?
Dimiter Vakarelov
No, they are not equivalent in ZF. Consider for example the case where I and F are trivial: I={0} and F={1}. Then (1) just says "there exists a prime filter" while (2) says "there exists a maximal filter". The existence of maximal filters in distributive lattices is equivalent to the axiom of choice AC while the existence of prime filters is equivalent to what is called BPI (boolean prime ideal theorem) which is known to be strictly weaker than AC, although not provable in ZF.
Dear Professor Ramiro De la Vega,
thanks for your short and clear answer. Do you know some references for (2) in order to use it without proof in a paper? Because I want to mention that (2) is equivalent to the axiom of choice, may I mention that I obtained this information from you?
Dear Profs Vakarelov, Prof. De la Vega,
Herrlich's 2002 paper contains exactly the needed references for Prof. De la Vega's (2): http://onlinelibrary.wiley.com/doi/10.1002/malq.200310033/pdf
Thanks to Prof. Rineke Verbrugge for the reference to Herrlich's paper.
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