Talk:Quasigroup

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Mathematics rating: B Class Mid Priority  Field: Algebra

Where do these beasts show up?

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

How do Quasigroups relate to cwatsets?

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

"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)

Contents

[edit] 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).

[edit] 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)

[edit] 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)

[edit] 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)