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Does there exist a partial order, nontrivial for forcing, that is countably closed, but whose separative quotient is not countably closed? Supposing the answer is yes, then is there a partial order, nontrivial for forcing, that is countably closed, but is not forcing equivalent to any countably closed separative partial order?

For those of you unfamiliar with the separative quotient of a partial order, it is defined as follows. Two elements of a partial order are compatible iff there is some element below both of them. We form the separative quotient of a partial order by taking equivalence classes: x is equivalent to y when x and y are compatible with the exact same things. We then define a new partial order for the separative quotient -- $x \leq y$ iff everything compatible with x is compatible with y.

A partial order is said to be separative if whenever $x \nleq y$, there is $z \leq x$ such that z is incompatible with y. The separative quotient of any partial order is separative.

Some of the ways, order-theoretically speaking, that two partial orders can be forcing equivalent are

(1) They are isomorphic, or more generally, (2) A dense subset of one of them is isomorphic to a dense subset of the other.

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Welcome to MO, Norman! I added the 'forcing' and 'order-theory' tags to your question. Also, since it seems, in principle, that this question might be answerable by someone outside forcing---it is at heart a combinatorial question about partial orders---I would recommend that you might explain a bit more about what you mean by 'separative quotient' and indeed 'separative', in order to assist this possibility. –  Joel David Hamkins Jan 12 '10 at 13:40
    
Thanks. I thought about adding the forcing tag, but it only had one other occurrence so far, and the site suggests not creating new tags. –  Norman Lewis Perlmutter Jan 13 '10 at 5:09
    
You need 250 reputation to create tags. But it is encouraged to use all the appropriate tags that do exist, and I had created the forcing tag for precisely this sort of question. I believe that as MO grows enormous, the tags will play an increasingly critical role for people to find the questions that might interest them. –  Joel David Hamkins Jan 13 '10 at 13:57
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2 Answers

I don't have a definite answer for either question, but here are some facts that may be useful (mostly for the second one) though you may be aware of them already.

If $P$ is countably closed then the complete Boolean algebra $RO(P)$ is strategically closed. So, by a result of Boban Velickovic (Playful Boolean Algebras, TAMS 296, 1986), if $RO(P)$ has a dense subset of size $2^{\aleph_0}$ (e.g. when $|P| \leq 2^{\aleph_0}$), $RO(P)$ must have a countably closed dense subset. Another result of Matt Foreman (Games Played on Boolean Algebras, JSL 48, 1983) shows that if $RO(P)$ is $(\kappa,\infty)$-distributive and has a dense set of size $\kappa$, then $RO(P)$ has a countably closed dense subset. This puts some constraints on potential counterexamples.

Also Jech and Shelah (On countably closed complete Boolean algebras, JSL 61, 1996) contains a potentially inspirational example, and Bernard König (Dense subtrees in complete Boolean algebras, MLQ 52, 2006) shows in more detail what to avoid.

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up vote 3 down vote accepted

Stevo Todorcevic answered this question for me at the MAMLS conference in honor of Richard Laver last weekend in Boulder, CO. Apparently, the answer is that examples of forcings that are closed, whose separative quotients are not closed, come up frequently, with one particular example being forcings involving semi-selective coideals studied by Ilija and Farah.

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I think you mean Ilijas Farah. –  François G. Dorais Feb 10 '10 at 21:27
    
But can we have the details? –  Joel David Hamkins Feb 10 '10 at 23:17
    
Yes, it must have been Ilijas Farah. I have a handwritten name in my notes, so it was hard to transcribe. I didn't get any details, though, and was unable to find the work online after a quick search. –  Norman Lewis Perlmutter Feb 11 '10 at 17:01
    
Have you looked at #43 on Ilijas's home page math.yorku.ca/~ifarah/preprints.html –  François G. Dorais Feb 11 '10 at 23:07
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