Talk:Cyclic order

Infinite sets cyclically ordered

I'm surprised to find that this article only deals with finite sets. Infinite sets can be cyclically (sometimes called circularly ordered) also. Intuitively, it corresponds to placing the set along a circle; rigorously, one does something like the following (I haven't all the details straight in my head):

Suppose your set is S. A cyclic order on S is a subset $A\subset S^{3}$ satisfying all of the following:

• If x,y,z are distinct elements of A then precisely one of (x,y,z) and (z,y,x) is in A.
• (x,y,z) is in A iff (y,z,x) is in A iff (z,x,y) is in A.
• [Some condition on how triples match up with one another. Something along the lines of:] If (x,y,w) and (y,z,w) are in A then so is (x,y,z).

This set can then be ordered with an analogue of the order topology for ordered sets, for those who care about such things (like me). —msh210 15:54, 14 Apr 2005 (UTC)

What applications does this idea have? Charles Matthews 06:15, 17 Apr 2005 (UTC)

I only know one: trying to order cataclysms (to use Calegari-Dunfield's terminology) of an essential lamination. There are probably many more. But does it matter what applications it has? The construction certainly exists and rightfully belongs in an encyclopedia. —msh210 13:52, 17 Apr 2005 (UTC)

A good application makes it much easier to motivate and clarify an abstract definition. We don't just write for mathematicians; and I don't think any non-mathematician would think it a strange question. Charles Matthews 15:31, 17 Apr 2005 (UTC)

June 2011 expansion

So, I've expanded the article. I've been pecking at this expansion in my user space for a while, and I've gotten tired of it, so I'm just adding what I've got; this is kind of a half-assed effort. Let me be the first to admit that the prose is terrible and should be rewritten. :-) Also, I'm sure that the "Symmetries and model theory" section in particular betrays my lack of understanding of the material. Please help! Melchoir (talk) 22:21, 18 June 2011 (UTC)

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Intro not really accurate

The very first sentence of the article is

"In mathematics, a cyclic order is a way to arrange a set of objects in a circle."

This is appropriate only to provide intuition, but it is not strictly true.

For instance, consider the cyclic order O on the Cartesian product S¹×[0,1) of a circle and a half-open interval, defined by the set of triples of the form [(x,r), (y,s), (z,t)] ∊ (S¹×[0,1))³ whenever (x,r), (y,s), (z,t) are distinct elements of S¹×[0,1) with

• [x,y,z] counterclockwise in S¹; or

• x = y ≠ z in S¹ and r < s in [0,1); or

• x = y = z in S¹ and r < s < t in [0,1).

Here [0,1) could be replaced by any linear ordering of any cardinality. So it is not necessarily possible to arrange the "set of objects in a circle". 2601:200:C000:1A0:89C5:5EA1:E6F5:EF37 (talk) 00:35, 2 August 2021 (UTC)

I don't see how your order would satisfy the cyclicity requirement of Cyclic_order#The_ternary_relation: if a ∈ S¹ is arbitrary, then [(a,0.4), (a,0.5), (a,0.6)] holds by your case 3. However, [(a,0.5), (a,0.6), (a,0.4)] does not hold, whereas it should, by cyclicity. - Jochen Burghardt (talk) 08:21, 2 August 2021 (UTC)

Yes, my definition of the cyclic order O was not intended to explicitly write down all cases, but only to provide generators of the cyclic order on S¹×[0,1). In other words, the cyclic order on S¹×[0,1) is defined by the cases that I wrote and application of the axioms for a cyclic ordering. 2601:200:C000:1A0:49B9:2B8:C68E:6D67 (talk) 21:20, 6 August 2021 (UTC)

Then, how can you be sure that no contradiction can be derived this way, i.e., that the deductive closure of the axioms and your definition doesn't contain a contradiction? - Jochen Burghardt (talk) 17:42, 7 August 2021 (UTC)

I can prove this, but since I was not writing a paper for publication I did not include the proof. (Note: I added the condition "≠ z" to the second case above, which I had neglected to mention originally.) 2601:200:C000:1A0:3960:8417:FC1F:8365 (talk) 17:00, 8 August 2021 (UTC)

Do you have a suggestion how to change the introductory sentence? - Jochen Burghardt (talk) 08:47, 12 August 2021 (UTC)

Weird link and no definition of "compatible" and a confusing sentence

In the section Topology this is the first sentence:

1. "The open intervals form a base for a natural topology, the cyclic order topology."

(with the link as shown).

But clicking on that link one finds no instance of the word "cyclic" at that article.

The next sentence is this:

2. "The open sets in this topology are exactly those sets which are open in every compatible linear order."

But nowhere on this page is "compatible linear order" defined (or linked to).

The last sentence in that first paragraph in the Topology section is this:

3. "To illustrate the difference, in the set [0, 1), the subset [0, 1/2) is a neighborhood of 0 in the linear order but not in the cyclic order."

I find this sentence to be very confusing.

I do not even know what things this last sentence is trying to "illustrate the difference" between.

I hope someone knowledgeable in this area can fix these things.

(By the way, the article is in many ways terrific, so much better than the last time I checked it.)