Talk:Latin square property
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Can anyone give an example of such an operation (or two or three)? Something simple would be nice. I see that this requires logical operations, but does it have to be the same operation (that confuses me); but, if it was a different operator such as XOR, and OR, or OR, and AND for x & y respectively, that I would understand. So, any help in clearing this up? Or is this in someway a reference to vector conjugates? Cyberchip (talk) 03:31, 14 December 2014 (UTC)
- XOR has this property; OR and AND do not. I added a larger example as an illustration to the article. —David Eppstein (talk) 04:20, 14 December 2014 (UTC)
- To be more mundane, every Suduko or KenKen puzzle solution has this property. The "operation" is defined by the table and need not have any "nice" interpretation. Bill Cherowitzo (talk) 04:58, 14 December 2014 (UTC)
The description for the example Latin Square seems incorrect? edit
The description currently reads: "...the multiplication table for a quasigroup whose 10 elements are the digits 0–9."
I cannot see how the description would generate a Latin Square (or equivalently, form a quasigroup). In particular, 0*0 = 0*1 = ... 0*9 = 0, so the first row would certainly not contain every element; indeed, it would contain only one.
I offered a different Latin Square in an edit, the Cayley Table generated by the set {1,2,3,4} under multiplication modulo 5, but it was reverted.
Many thanks, Monty — Preceding unsigned comment added by Montyevans (talk • contribs) 20:11, 25 May 2020 (UTC)
- When it says "the multiplication table for a quasigroup", that does not mean "the multiplication table for ordinary multiplication modulo 10". Why would you think it would? —David Eppstein (talk) 20:42, 25 May 2020 (UTC)
- Just to amplify on David's remark ... A Latin square defines the multiplication table of a quasigroup, they are equivalent things. The original image was a Latin square (just scan the rows and columns visually), and therefore defined a multiplication table for a quasigroup. I thought that it was more important to have an example of a Latin square that was not the "multiplication" table of a group to drive this point home. That is the reason that I reverted, there was nothing wrong with your example.--Bill Cherowitzo (talk) 21:45, 25 May 2020 (UTC)
- Ah, many thanks both for the clarification. I understand that the Latin Square defines the value of a*b for any (a,b) in the quasigroup, but I'm not aware that it is common (and indeed, it seems rather misleading) to refer to the operator * generically as 'multiplication'. This seems particularly confusing when the elements in the set are digits, and yet a*b is unrelated to "a multiplied by b" in the usual sense (modulo N, or otherwise). Indeed, as Bill mentions in the first section of the Talk page, ""the "operation" is defined by the table, and need not have any "nice" interpretation"" - to refer to it as "multiplication", then, seems unclear. The point about providing a Latin Square that doesn't derive directly from a multiplication table of a group is well taken: it might be worth changing the caption to "the operator table for a quasigroup...", or even using letters a...j in place of the digits 0...9, to emphasise that the table decidedly does not represent multiplication. Montyevans (talk) 23:08, 25 May 2020 (UTC)
- It is standard to call it multiplication. In many circumstances in abstract algebra, it would not be satisfactory to just refer to "the operation", because there is more than one operation involved. In fact, this is true with quasigroups, because they can also be given a division operation (or actually two different division operations) derived from their multiplication operation. —David Eppstein (talk) 23:54, 25 May 2020 (UTC)
- [Edit Conflict] You make a good point. As an afterthought I realized that I should have mentioned that we often refer to the quasigroup operation as multiplication (hence the scare quotes in my previous comment). This is a matter of being lazy, but it is very common and to do otherwise makes the language stilted and heavy-handed. As for using letters instead of digits, it certainly can be done, but I find that it is much harder to verify that a square of letters is a Latin square visually, at least for me. In fact, if you work with Latin squares long enough, you come around to the viewpoint that numbers are merely symbols and they don't have any numerical properties unless you impose them.--Bill Cherowitzo (talk) 00:04, 26 May 2020 (UTC)
- Thanks again, both of you, for the further clarification. I'm content with the viewpoint that "numbers are merely symbols and they don't have any numerical properties unless you impose them" (and, as such, multiplication is a perfectly reasonable term to use for the operator in question), but my sense is that people with only the beginnings of exposure to the field (such as myself) may not find that a particularly obvious idea, so I edited the image caption to clarify things. Montyevans (talk) 00:27, 26 May 2020 (UTC)
- Your latest edit is fine, although I am thinking that it makes for an overly large caption. It might be better to bring some of this discussion into the article proper. In any event, the article could use some expansion and I'll try to do so next week.--Bill Cherowitzo (talk) 18:06, 26 May 2020 (UTC)
- Thanks again, both of you, for the further clarification. I'm content with the viewpoint that "numbers are merely symbols and they don't have any numerical properties unless you impose them" (and, as such, multiplication is a perfectly reasonable term to use for the operator in question), but my sense is that people with only the beginnings of exposure to the field (such as myself) may not find that a particularly obvious idea, so I edited the image caption to clarify things. Montyevans (talk) 00:27, 26 May 2020 (UTC)
- [Edit Conflict] You make a good point. As an afterthought I realized that I should have mentioned that we often refer to the quasigroup operation as multiplication (hence the scare quotes in my previous comment). This is a matter of being lazy, but it is very common and to do otherwise makes the language stilted and heavy-handed. As for using letters instead of digits, it certainly can be done, but I find that it is much harder to verify that a square of letters is a Latin square visually, at least for me. In fact, if you work with Latin squares long enough, you come around to the viewpoint that numbers are merely symbols and they don't have any numerical properties unless you impose them.--Bill Cherowitzo (talk) 00:04, 26 May 2020 (UTC)
- It is standard to call it multiplication. In many circumstances in abstract algebra, it would not be satisfactory to just refer to "the operation", because there is more than one operation involved. In fact, this is true with quasigroups, because they can also be given a division operation (or actually two different division operations) derived from their multiplication operation. —David Eppstein (talk) 23:54, 25 May 2020 (UTC)
- Ah, many thanks both for the clarification. I understand that the Latin Square defines the value of a*b for any (a,b) in the quasigroup, but I'm not aware that it is common (and indeed, it seems rather misleading) to refer to the operator * generically as 'multiplication'. This seems particularly confusing when the elements in the set are digits, and yet a*b is unrelated to "a multiplied by b" in the usual sense (modulo N, or otherwise). Indeed, as Bill mentions in the first section of the Talk page, ""the "operation" is defined by the table, and need not have any "nice" interpretation"" - to refer to it as "multiplication", then, seems unclear. The point about providing a Latin Square that doesn't derive directly from a multiplication table of a group is well taken: it might be worth changing the caption to "the operator table for a quasigroup...", or even using letters a...j in place of the digits 0...9, to emphasise that the table decidedly does not represent multiplication. Montyevans (talk) 23:08, 25 May 2020 (UTC)
- Just to amplify on David's remark ... A Latin square defines the multiplication table of a quasigroup, they are equivalent things. The original image was a Latin square (just scan the rows and columns visually), and therefore defined a multiplication table for a quasigroup. I thought that it was more important to have an example of a Latin square that was not the "multiplication" table of a group to drive this point home. That is the reason that I reverted, there was nothing wrong with your example.--Bill Cherowitzo (talk) 21:45, 25 May 2020 (UTC)
Merger proposal edit
I've suggested a merge of this article into Quasigroup. This article has been a stub since 2005, so I don't think much can be done to improve it. The quasigroup article would benefit from a short mention of infinite quasigroups and the order 10 Latin square here would make a fine addition to that page.--Bill Cherowitzo (talk) 19:49, 3 June 2020 (UTC)