Talk:Quasigroup

Where do these beasts show up?

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I'm intending to add some examples of non-associative quasigroups later. The nonzero octonions form a quasigroup under multiplication, and the unit octonions make the 7-sphere into a quasigroup (a Moufang loop, I think). Steiner triple systems are essentially a type of commutative quasigroup (define a * a = a, and a * b = c if (a,b,c) is a triple). Moufang loops apparently arose in some geometric context (projective planes?), but I don't know any details about this. There are also various other types of quasigroups (IP loops, Bol loops, etc.) that have been studied, but I don't know where they arise. --Zundark, 2001-09-04

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How do Quasigroups relate to cwatsets?

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I don't know. What makes you think there is a relationship? --Zundark

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"Lastly, e = e * b = b, so e is a two-sided identity element."

Why is e = e * b? Is this a typo? -- Smjg 15:10, 15 Jan 2004 (UTC)

We've just proved that b is a right identity element, so e * b = e. --Zundark 09:07, 16 Jan 2004 (UTC)

Moufang Loops

Has anyone else [ref.1] observed that, when Moufang Loops are used as Cayley multiplication tables for vectors, the following are true?

1). Every vector has a multiplicative left inverse Ai, with Ai.A={1,0,..} (and a similar right inverse). This unifies multiplication and division.

2). Groups and octonions (and so perhaps all Moufang Loops) have "Frobenius conservation" [ref.2] Det[A]Det[B]= ±Det[AB], where Det is calculated from the inverse table after mapping with the vector. The real (but not the complex) factors of the determinant are "conserved symmetries".

3). Many real algebra multiplication tables {R, C, H, O, Clifford, Davenport, etc} are equivalence relations on Moufang loops (multiplying half-length vectors) and retain these properties.

4). Multiple conserved symmetries provide partial-fraction denominators for the inverse, so quotients factorise and sub-algebras are created when constraints are put on these factors. This does not apply to R, C, H, O, and some Clifford algebras, which only conserve a single size.

Moufang loop algebras appear to be relevant to mathematical physics because they alone have conserved properties, but this remark is probably still excluded as "original research". May I suggest that the existing Moufang Loop material deserves to be a separate entry?

195.92.168.170 08:23, 15 October 2005 (UTC)

[1] http://library.wolfram.com/infocenter/MathSource/4894/ [2] G.Frobenius, Uber die Primfactoren der Gruppendeterminante, Sitzungsber.Preuss. Akad. Wiss. Berlin Phys. Math,KL. 1896, (985-1021). (not seen, but quoted from van der Waarden, History ofAlgebra).

Recent edits

Nice work with this page, Fropuff. As someone who works in this area, I have been thinking about revising this page myself, but was somewhat daunted by how much needed to be done. There is now a good framework from which to begin. I also agree with the decision to spin off Moufang loops. I don't think any other variety of quasigroup or loop deserves a page of its own, but Moufang loops certainly do. Mkinyon 15:13, 26 January 2006 (UTC)

Thanks, I'm glad you like it. The Moufang loop material definitely seemed like it needed its own page. I hope you take the opportunity to expand this page. This isn't my area of expertise, so I only made the most obvious changes. -- Fropuff 20:08, 26 January 2006 (UTC)

One question: there is some inline TeX in the Inverse properties subsection. I haven't touched it, because I'm trying to figure out why it's there. Oversight? Isn't inline TeX a no-no around here, at least until MathML is fully integrated? (And I wonder what the status of that project is?) Mkinyon 23:21, 27 January 2006 (UTC)

It's there because I got lazy. There's no hard rules regarding TeX/HTML; just a set of guidelines. Inline TeX is okay as long as it doesn't generate PNGs (although whether or not that happens depends on your settings). If you want to change it to HTML feel free. You can check out m:blahtex for the status of the MathML project. I suspect integration is still a long way off. -- Fropuff 23:33, 27 January 2006 (UTC)

Yes, I forgot about PNG generation being browser/setting specific. My IE generates PNG for one of the inline equations and leaves the rest alone. Weird. I'm just going to leave it as is, unless I decide on a more substantial edit of the whole subsection. Thanks again. Mkinyon 18:37, 30 January 2006 (UTC)

loop rings?

Has work been published on loop rings, analogous to group rings?Rich 22:43, 25 September 2006 (UTC)

Yes, there has been quite a bit of work on them. A starting place, if you're interested, is Edgar Goodaire's paper A Brief History of Loop Rings. Michael Kinyon 00:06, 26 September 2006 (UTC)

ThanksRich 06:46, 26 September 2006 (UTC)

Inverse properties

Should the word 'element' be inserted here?

Every loop element has a unique left and right inverse given by Scot.parker 13:03, 7 August 2007 (UTC)

Terminology

I removed from the definitions the remark about left and right quotients: "(sometimes called parastrophe)". It doesn't fit the meaning. I believe parastrophe means the process of generating a new quasigroup by permuting the variables. We say the new quasigroup is "parastrophic" or "conjugate" to the original. I put in a couple of later subsections to deal with this.

There is much variation in terminology; it might be desirable to add any (relatively common) missing variants. Zaslav (talk) 03:09, 4 June 2010 (UTC)

Zassenhaus example

The equation defining the Zassenhaus example is wrong. It should read:

(x1, x2, x3, x4) * (y1, y2, y3, y4) = (x1 + y1, x2 + y2, x3 + y3, x4 + y4 + (x3 − y3)(x1y2 − x2y1))

— Preceding unsigned comment added by 204.123.28.70 (talk) 20:43, 21 August 2012 (UTC)

Article text changed accordingly. Could need a verification, though. 81.170.129.141 (talk) 09:31, 12 March 2013 (UTC)

"Trivial quasigroup"

I have on occasion seen the term "trivial quasigroup" show up, generally in the form of "nontrivial quasigroup", used to limit the quasigroups under consideration to those with two or more elements. In my mind, there are two candidates for for the name "trivial quasigroup": the empty quasigroup and the quasigroup with one element. Should this article not discuss or mention these concepts? This is echoed in semigroups, to wit the empty semigroup and the semigroup with one element. — Quondum 05:07, 14 October 2013 (UTC)

This article should indeed mention the concepts, but on the basis of good sources. Can you suggest any? I suspect the meaning used out there is not consistent; for example I see one author writing of "the empty and trivial quasigroups" but surely nobody uses "non-trivial quasigroup" to allow 0 elements as well as 2 or more. McKay (talk) 03:02, 16 October 2013 (UTC)

I am not a fan of the terms "trivial semigroup" and "trivial quasigroup", since they are in some dense not the most trivial. These terms and the terms "empty quasigroup" and "nonempty quasigroup" (these terms being unambiguous) seem to occur, though the author Jonathan D. H. Smith seems to be associated with a substantial proportion of these uses; not working in the field I cannot identify "good sources". For the time being, I've implicitly introduced the concepts (unnamed) by mentioning order 0 and 1 in the table that I added. — Quondum 06:11, 16 October 2013 (UTC)

Good. But I think there is no loop of order 0 since it is supposed to have an identity element. McKay (talk) 07:16, 16 October 2013 (UTC)

Based on limited but nonempty knowledge of the area, I believe no one in the area would allow a quasigroup with no elements. It makes no sense in terms of what quasigroups should do. Thus, "trivial" should mean one element. If J.D.H. Smith has a use for an empty quasigroup, that's fine but probably not general usage. Zaslav (talk) 02:54, 18 October 2013 (UTC)

Our definition is satisfied by the empty set with the empty relation, so we have to do something. Although JDH Smith is the most common author who writes "empty quasigroup" or "non-empty/nonempty quasigroup" he/she isn't the only one. McKay (talk) 06:30, 18 October 2013 (UTC)

Puzzling permutations

The current "Loops" section contains

$(x_{1},x_{2},x_{3})\cdot (y_{1},y_{2},y_{3})=(y_{3}/x_{2},y_{1}\backslash x_{3},x_{1}y_{2})=(x_{2}//y_{3},x_{3}\backslash \backslash y_{1},x_{1}y_{2}),$

where "//" and "\\" are the conjugate division operations; the latter formula more explicitly shows that the construction is exploiting an orbit of S³.

First, I'm deleting "the latter formula more explicitly shows that the construction is exploiting an orbit of S³" because (a) I don't know what it means, (b) so I don't see why it's important enough, and (c) what is S³? For the symmetric group of degree 3 this is nonstandard notation. Anyway, this whole clause needs to be explained if it is worth keeping.

Second, "conjugate division" could and should be defined here rather than leaving the reader to attempt the definition after following the link. I will insert that. Zaslav (talk) 21:17, 8 September 2018 (UTC)

Right-TA-quasigroups?

Total antisymmetry

A quasigroup (Q, ∗) is called totally anti-symmetric if for all c, x, y ∈ Q, both of the following implications hold:

1. (c ∗ x) ∗ y = (c ∗ y) ∗ x implies that x = y
2. x ∗ y = y ∗ x implies that x = y.

Well, it looks like right-side property only. Why? Do these two implications imply another one as follows?

1. x ∗ (y ∗ c) = y ∗ (x ∗ c) implies that x = y.

If they do, then it should be mentioned in the article. If they don't, then it should be discussed in the article.

Merge proposal

I propose merging Latin square property into this article. That article has been a stub since 2005, which indicates to me that it is not going to grow into a full article. It easily fits in the quasigroup article which currently doesn't even hint at the existence of infinite quasigroups (no more than one or two sentences should suffice). The 10 × 10 Latin square on that page would be a nice addition as well. Thanks for considering this. --Bill Cherowitzo (talk) 19:13, 31 May 2020 (UTC)

Associative quasigroups

The image shown in the section "Loops" suggests strongly that an associative quasigroup is the same as an inverse semigroup, but I don't think this is true; the "inverse" property of an inverse semigroup is weaker than that of a quasigroup. In fact, it seems that a nonempty associative quasigroup is automatically a group; see quasigroup on the nLab. -- Michael Shulman (talk) 04:01, 5 February 2022 (UTC)

Yes, nonempty associative quasigroups are groups, and the image falsely suggests that all inverse semigroups are quasigroups. I've now removed the image. --Zundark (talk) 09:14, 5 February 2022 (UTC)