Talk:Planar ternary ring

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I am puzzled about this sentence: "However, two nonisomorphic planar ternary rings can lead to the construction of nonisomorphic projective planes." Isn't that exactly what one would want? Is the second "nonisomorphic" supposed to be "isomorphic"? --Michael Kinyon 16:38, 1 August 2006 (UTC)

Yes, that is correct, I must have been a little bit distracted. Thanks! Evilbu 18:40, 1 August 2006 (UTC)

In the literature, the term "ternary field" is used almost as often as "planar ternary ring". (I personally prefer it.) I have created a page for the former that redirects to here, and have modified the first sentence of this article accordingly. --Michael Kinyon 17:44, 1 August 2006 (UTC)

I looked at the referenced paper Hall [1943]. The axioms in the article are NOT the same as in that paper. So are they correct? and if so where did they come from? The reference to Hall [1943] is not the source of these exact axioms. Michael Beeson


 * They are correct. I'll add a reference (Albert and Sandler) that gives them in this form (up to an alphabetic isomorphism). Bill Cherowitzo 04:28, 21 September 2011 (UTC)  — Preceding unsigned comment added by Wcherowi (talk • contribs)

from plane to ring
Can one go from a projective plane to a ternary ring in some circumstances? Tkuvho (talk) 05:13, 26 November 2010 (UTC)
 * Yes, always. Bill Cherowitzo (talk) 04:31, 21 September 2011 (UTC)

Terminology
I added a paragraph to the intro to point out that terminology varies widely and is often not as stated in the first paragraph. I can provide references if necessary. A cursory search of relevant literature will probably turn up great variety. Zaslav (talk) 07:11, 25 May 2014 (UTC)


 * That paragraph is really very vague and full of weasel words. Please come up with some references and make some concrete statements.Bill Cherowitzo (talk) 21:22, 25 May 2014 (UTC)


 * I said I can provide references. All you have to do is say they should be provided.  I will do that soon.  (Lack of time may postpone action a few days.)  Zaslav (talk) 18:55, 26 May 2014 (UTC)
 * P.S. I see you've done some nice work in finite geometry.  Zaslav (talk) 19:17, 26 May 2014 (UTC)


 * Sorry, I didn't mean to sound harsh. I have never seen "planar ternary ring" refer to anything else. "Ternary field" was used by Pickert and Michael Kinyon put that reference in (Michael is an expert in non-associative algebra and is familiar with the terminology in that area, so if he says its widely used I'll believe him). Stevenson uses "ternary ring" as a synonym for a PTR, but I have never seen the term used in the generality given in the article. By the way, I changed the page in the Stevenson reference back to 274 as page 270 in my edition is a blank page (I think I have a first printing, but it is the book in the references, perhaps this was just a typo. I don't know how I managed to put that reference in the wrong part of the sentence.) I really am interested in seeing what references you come up with. Bill Cherowitzo (talk) 00:11, 27 May 2014 (UTC)


 * All is forgiven! I'm travelling and it's hard to edit WP, but I've been seeing lots of old and new books and articles lately and the terminology is all over the place.  I have a little list for use when time allows.  Re page 274, I think that's what we looked up in our Polygonal Press reprint and decided 270 was the right page, but I'm away from my books so can't say for sure.  Can editions differ?  Yes . . . to be resolved later.  Zaslav (talk) 03:17, 31 May 2014 (UTC)

Addition
A footnote states that the sum x+y is defined either as t(1,x,y) or t(x,1,y) by different authors and then says "The difference comes from the alternative ways these authors coordinatize the plane." Having thought about it, I believe that the difference is independent of the coordinatization. In other words, the coordinatization can be carried out by some system, and addition can then be defined by either method, arbitrarily. Is that correct? If so, the quoted sentence should be removed. Zaslav (talk) 20:22, 26 May 2014 (UTC)


 * Well, yessss, but ... there is nothing that would say you would get isomorphic additions. I know of three published ways to define the ternary operation after the plane is coordinatized. Hall uses T(x,a,b) = y to mean point (x,y) I line [a,b], while Hughes and Piper would have it as point (a,b) I line [x,y], and Pickert would say point (a,y) I line [x,b]. In order to have addition be a reflection of the coordinates assigned to the lines with slope 1, if you follow Hall the sum should be T(x,1,b), while H&P and Pickert would have it as T(1,a,b). As to what the actual values will be, that will depend on how you have assigned the coordinates to the points on these lines. There is a discussion of this in Hughes & Piper p. 125. Bill Cherowitzo (talk) 00:31, 27 May 2014 (UTC)


 * I didn't know of the many ways to interpret t(x,a,b)=y. That is a mess.  I see some of the authors are actually coordinatizing the dual plane (according to the other authors).  However, I would prefer not to say the difference "comes from" coordinatization,  because it isn't determined by the coordinatization; it is related but is a separate choice, even though there may be a good reason for the choice.  Zaslav (talk) 17:53, 31 May 2014 (UTC)

Added a subsection showing the relationship to Cartesian coordinates
I think this was too implicit before. A lot of readers probably didn't recognise this. --Svennik (talk) 15:38, 28 February 2024 (UTC)

I should add I haven't read dedicated literature on this topic. Somebody who has done so should change the notation to conform to whatever the convention might be. --Svennik (talk) 13:37, 1 March 2024 (UTC)