# Talk:Schröder-Bernstein property/Draft

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 Definition:  A mathematical term used to describe properties that have the same structure as the Schröder-Bernstein theorem of set theory. [d] [e]
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## Three remarks

Nice, Boris. I have only minor comments:

1. I think the "noncommutative counterpart" needs some more explanation.
Now done. Boris Tsirelson 05:47, 7 October 2010 (UTC)
1. What do you think of discussing the relation to antisymmetry of order?
Now done. Boris Tsirelson 05:47, 7 October 2010 (UTC)

--Peter Schmitt 23:55, 25 September 2010 (UTC)

1. "noncommutative counterpart" — on one hand, I am waiting for a book from our library about this point; but on the other hand, it is anyway beyond the form "If X is similar to a part of Y and also Y is similar to a part of X then X and Y are similar (to each other)" and only inspired by it.
2. "relation to antisymmetry of order" — you are right, this should be noted.
3. "structure by (sub)headings" — I also feel the need, but the article is so short, I do not see which headings; do you? Boris Tsirelson 06:23, 26 September 2010 (UTC)
As for headings: "Examples" is obvious, I think.
The introduction could say something like: "(Remark) In contrast to most mathematical terms this only gives the general form, the pattern, of a statement instead of a definite and precise definition."
Then the explanation following it could have the heading "The general pattern" (or similar). For tactical reasons :-) I do not make changes myself.
--Peter Schmitt 12:41, 28 September 2010 (UTC)
Being unsure I did the change in my sandbox; please look there. Is it close to what you mean? Boris Tsirelson 13:48, 28 September 2010 (UTC)
Yes, that is what I meant. What do you think? In the introduction, I would exchange the displayed definition with the second sentence -- but this is only a matter of taste. And -- perhaps -- it is not necessary to repeat this in the first section? --Peter Schmitt 23:52, 28 September 2010 (UTC)

## General issues

Peter, I feel that our interaction model "author - approver" is restrictive. A good unapproved article is better than a bad approved one. Thus I propose to leave the approval to a remote future and do the wiki: you edit as you like, and me, and so on.

The problem is that you have an idea how it should be, but I fail to guess it. I do not feel that the version in my sandbox is better. Yes, it has some headings, and this is good. However, it is not for free. In the old version an example appears early and illustrates the general idea. In the new version they are separated, and so, "the general pattern" section is quite dry. Of course, the reader can jump to examples and return. But still, I am not convinced that the advantage overweight the drawback. Trying to implement your last remarks I make more mess.

A more general remark. I feel you have some rather firm idea of a good article. Could you please publish your idea? (Say, on your home page, or mine, or talk, or even my sandbox, no matter where.) It would help. It seems you implicitly assume that others have the same idea but are lazy to follow it; this is not so.

And another general remark. Your phrase "In my opinion, good and high-quality articles cannot and should not be "whipped up". They require much thought and careful planning." is a bit frightening. It would be OK in a prestigious publishing house with high salaries for authors. But here? We know what the set of active math authors is today. I am inclined to say "Yes - but not now." Boris Tsirelson 09:08, 4 October 2010 (UTC)

Boris, our interaction model is indeed awkward (and I hope that future rules will allow more flexible ways), but -- if I remember it correctly -- it was you who said that we should try to do more approvals (while I did not much care about approval). Thus I did not make direct edits.
In this case, my suggestion to insert some headings was indeed only a suggestion (with a question mark). I do not insist on a particular style or a fixed structure. However, I thought of your remark that you do not like to write a lead and thought this might be similar. I cannot be sure, but I think that I would have discussed this first even if I else had edited the page freely. I do not intend to force any specific headings.
As to your first general remark. Probably I have some firm ideas of a good article, but I think (or hope) that they are flexible and can allow for different styles of exposition. I certainly have some personal style characteristics (who has not?) but I hope that I am open enough to accept other personal styles. By the way, the issue of personal styles is a reason to avoid major edits of another author's article -- usually mixing styles produces suboptimal results.
As to the other remark. The phrase was not meant to be frightening, but I have to admit that I was angry about Tom's demands and his use of the word "whip up". Essentially, he implied that, if I (or the experts = mathematicians) would not provide an article fast then we were not able to and thus he would show us how to (fast). (Or, at least, that was how I understood him.) I wanted to advise him to take his time. (Do you write quickly? I usually can't.)
--Peter Schmitt 01:10, 6 October 2010 (UTC)
After having written this it occurred to me that there is indeed a rule that probably should be followed with almost all (main page) articles. It is not "my" rule but follows from the encyclopedic nature of the project: There should be an introduction that provides the essential information, and that allows the user to judge whether he needs or wants to read the detailed exposition. This introduction is probably best separated from the page by some section title. --Peter Schmitt 13:02, 6 October 2010 (UTC)
Yes, I see.
About this last rule: of course, generally I agree. My only problem is that I feel unable to implement it in this case (without making the article worse in some other aspects). Thus I wonder, whether you see such a solution, or you just hope that it exists (and I can find it)? Boris Tsirelson 13:14, 6 October 2010 (UTC)

## Introduction

I may have found a solution for the introduction. What do you think?

A mathematical property that matches the following pattern
If, for some mathematical objects X and Y, both X is "similar" to a part of Y and Y is "similar" to a part of X then X and Y are "similar" (to each other).
is often called a Schröder–Bernstein (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) property
in analogy to the theorem of the same name (from set theory).

== Schröder-Bernstein properties ==


followed by the current page. (the first sentence may need to be adapted.)

--Peter Schmitt 23:36, 13 October 2010 (UTC)

OK, I did. A bit tricky, but anyway I do not see a better solution.
I did not put "similar" into quotation marks, since otherwise, according to the section, I should do so also for "object" and "part", which would look clumsy. Boris Tsirelson 09:59, 14 October 2010 (UTC)

## Copyedits

The references are better moved to the bibliography, I think.

If I am right, the following changes are stylistic improvements:

• "is a preorder (that is, reflexive and transitive relation)" --> "is a preorder (that is, a reflexive and transitive relation)".
• "A problem of deciding, whether" --> "The problem of deciding whether" (no comma, I think).

--Peter Schmitt 21:26, 14 October 2010 (UTC)

Operator algebra, (von Neumann algebra too?), Banach space should be probably be linked. --Peter Schmitt 23:48, 14 October 2010 (UTC)

The stylistic improvements are done.
Banach space is (and was) linked; operator algebra and von Neumann algebra are not present in CZ.
"references - to the bibliography", really? Why? As far as I understand, our policy is, to put inline refs into the main article, while recommended reading to the bibliography CZ:Citation style#Placing the citation list. Do you really want to change it?
Boris Tsirelson 08:11, 15 October 2010 (UTC)
Sorry about the Banach space link. On my display the blue of a link is sometimes difficult to distinguish from black. But CZ allows "red" links (it even encourages it) to indicate "missing articles".
As for the bibliography: I thought the "Notes" are the inline references and considered the "References" as bibliography ... However, I consider this only as a stylistic question and a matter of taste. I don't think there is strict rule ... not important.
--Peter Schmitt 20:48, 15 October 2010 (UTC)

"It has a noncommutative counterpart": counterpart to what -- to the previously mentioned measurable spaces? --Peter Schmitt 23:37, 21 October 2010 (UTC)

Yes. Is it unclear language? Or, unclear mathematics? Boris Tsirelson 07:07, 22 October 2010 (UTC)
Anyway, it is removed. Now, if (and only if) you like the following text I'll insert it to the end of the "noncommutative" paragraph:
Taking into account that commutative von Neumann algebras are closely related to measurable spaces, one may say that the Schröder–Bernstein theorem for operator algebras is in some sense a noncommutative counterpart of the Schröder–Bernstein theorem for measurable spaces.
Boris Tsirelson 11:56, 22 October 2010 (UTC)
Something relevant: WP:Commutative von Neumann algebras. Boris Tsirelson 12:08, 22 October 2010 (UTC)
Yes, I think that this (or a similar) comment would help. I don't think that this is an obvious connection for all who may read it. (It was not for me.) --Peter Schmitt 00:20, 23 October 2010 (UTC)
I did. Boris Tsirelson 16:24, 23 October 2010 (UTC)

## Toward approval

Nominated (by Peter). Boris Tsirelson 07:08, 22 October 2010 (UTC)

## Approved Version 1.0

Good work gentlemen! D. Matt Innis 03:09, 28 October 2010 (UTC)

## Imported to Wikipedia

Imported [1] to WP on December 2010 by Rich_Farmbrough. Boris Tsirelson 12:06, 20 September 2011 (UTC)