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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).
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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
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First, I'm deleting "the latter formula more explicitly shows that the construction is exploiting an orbit of S" because (a) I don't know what it means, (b) so I don't see why it's important enough, and (c) what is S? For the symmetric group of degree 3 this is nonstandard notation. Anyway, this
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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
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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
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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
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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
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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?)
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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.
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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
203:) 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. --
452:)". 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.
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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.
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2). Groups and octonions (and so perhaps all
Moufang Loops) have "Frobenius conservation" DetDet= ±Det, 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".
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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?
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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
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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. —
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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.
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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. --
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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
306:
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).
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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.
909:{\displaystyle (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}),}
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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.
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Yes, nonempty associative quasigroups are groups, and the image falsely suggests that all inverse semigroups are quasigroups. I've now removed the image. --
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1). Every vector has a multiplicative left inverse Ai, with Ai.A={1,0,..} (and a similar right inverse). This unifies multiplication and division.
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Has anyone else observed that, when Moufang Loops are used as Cayley multiplication tables for vectors, the following are true?
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If they do, then it should be mentioned in the article. If they don't, then it should be discussed in the article.
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Yes, there has been quite a bit of work on them. A starting place, if you're interested, is Edgar Goodaire's paper
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the quasigroup with one element. Should this article not discuss or mention these concepts? This is echoed in
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There is much variation in terminology; it might be desirable to add any (relatively common) missing variants.
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Well, it looks like right-side property only. Why? Do these two implications imply another one as follows?
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think any other variety of quasigroup or loop deserves a page of its own, but Moufang loops certainly do.
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The 10 × 10 Latin square on that page would be a nice addition as well. Thanks for considering this. --
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478:(x1, x2, x3, x4) * (y1, y2, y3, y4) = (x1 + y1, x2 + y2, x3 + y3, x4 + y4 + (x3 − y3)(x1y2 − x2y1))
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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Good. But I think there is no loop of order 0 since it is supposed to have an identity element.
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I removed from the definitions the remark about left and right quotients: "(sometimes called
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for the status of the MathML project. I suspect integration is still a long way off. --
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I'm intending to add some examples of non-associative quasigroups later. The nonzero
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The equation defining the Zassenhaus example is wrong. It should read:
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I don't know. What makes you think there is a relationship? --
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405:A Brief History of Loop Rings
109:and see a list of open tasks.
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919:where "//" and "\\" are the
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