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I have added to alternative algebra the characterization in terms of skew-symmetry of the associator.
Also added statement and reference for Artin’s theorem.
Also added statement and reference for Zorn’s theorem.
John Baez in “The Octonions” says that there is a modern proof of this theorem in Schafer’s book. But where?
Added a Related Concepts section to alternative algebra and Cayley-Dickson construction.
Re #3
Hence there are real alternative division algebras of dimensions 1, 2, 4, 8. It has recently been proved (see reference [12] of the appended bibliography of recent papers) that finite-dimensional real division algebras can have only these dimensions. (p. 27)
suggests it isn’t there. [12] is
R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. vol. 64 (1958), pp. 87–89.
Presumably this means Shafer didn’t know of Zorn 1930.
Oh no, Bott and Milnor aren’t speaking about alternative algebras. So does Shafer know that those four are the only real alternative division algebras?
David, this must be about the different statement that dropping the requirement of a norm, then the real division algebras are no longer constrained to be R,C,H,O, but still need to have these dimensions.
It’s about dropping the requirement of being alternative. He continues
It is not true, however, that the only finite-dimensional real division algebras are the four listed above; they are the only alternative ones. For other examples of finite-dimensional real division algebras (necessarily of these specified dimensions of course) see reference [23]
It’s either. Among the real division algebras, the algebras R,C,H,O are singled out as being precisely the normed ones (Hurwitz) or the alternative ones (Zorn).
In “The Octonions” John Baez claims that there is a proof of Zorn’s statement in Schafer’s book. But apparently it’s not the case.
That link given is to the 1961 edition. The 1995 book is a reprint of a 1966 version, and is longer. It at least mentions Zorn. Maybe there’s a proof there. On the other hand, there’s the same statement “they are the only alternative ones” (p. 48). But perhaps the means to show this are available in the book.
Thanks! I’ll check.
I generalized this page to cover flexible algebras.
I also mentioned alternation (rather than just skew-symmetry) in the theorem about the associator. It seems to me that this theorem is really a combination of two facts: that an algebra is alternative iff its associator is alternating (which is true even in characteristic $2$), and that (if $2$ is cancellable) its associator is alternating iff it is skew-symmetric. The first of these is really about the subject of this page, although it's a fairly trivial observation; the second of these is really about more general alternating maps (and is stated but not proved at alternating multifunction).
I don’t mind if you take it apart into two statements.
(And it’s okay to state fairly trivial observations. It’s good to try to keep in mind the spectrum of potential readers.)
I've now done this, with new material at both alternating function and alternative algebra, although the proofs remain combined at the latter.
Incidentally, the trivial observation isn't quite as trivial as I thought; it still includes the result that an alternative algebra is flexible.
I have created commutative operation and associative operation to un-gray two of the links at alternative algebra. Slightly expanded the text at magma accordingly.
Also made a 1-line entry nonassociative ring, with cross-links, to ungray that link.
You still demand equational law. Either this should redirect to algebraic theory or else also algebraic theory should point to equational law.
I think that algebraic theory should point to equational law. What we cover at the former is more general than the latter. Actually, there's already a spot where that link would fit, so I'll put it in now.
There's also a theorem in Johnstone about precisely which algebraic theories can be described by equational laws. I kind of thought that we had it in the article, but we don't seem to.
I guess I’m confused by the last comment. I looked in the Elephant for a little while but didn’t find anything on this topic. But more to the point: I thought every algebraic theory could be described by operations and equational laws!
Let me argue this in the case where we treat “monad on $Set$” as virtually a synonym for “algebraic theory”. (Even if one wants to quibble with that, it’s a convenient starting place to make my point.) Let $(T, \mu, \eta)$ be a monad. For each cardinality $n$ the set of $n$-ary operation symbols (for the signature of the algebraic theory) may be taken to be $T n$.
The functoriality of $T$ means that for a set $X$, elements of $T X$ can be represented by tuples $(t, x) \in T n \times X^n$ in the sense that there is a canonical map $T n \times X^n \to T X$ mated to the functorial structure map $X^n = \hom(n, X) \to \hom(T n, T X)$, and every element of $T X$ can be so represented by some pair $(t, x) \in T n \times X^n$ for some $n$ (essentially by the Yoneda lemma). For example, an algebra over the endofunctor $T$, say $(X, \alpha: T X \to X)$, yields a structure [in the sense of logic or model theory] of the signature of $T$, where we curry the composite map $T n \times X^n \to T X \stackrel{\alpha}{\to} X$ to get $T n \to \hom(X^n, X)$ which interprets $n$-ary function symbols $f \in T n$ as $n$-ary operations on $X$.
Turning to the monad structure, I would say that any $(x, y) \in T T n \times T T n$ that lies in the kernel pair of $\mu n: T T n \to T n$ can be considered an “equational law”. Take for example the theory of monoids and the associativity law. In the usual syntax we might write this as $m(m(x_1, x_2), x_3) = m(x_1, m(x_2, x_3))$, but here I want to view the two sides of the equation formally as elements in $T T 3$, represented by elements in $T 2 \times (T 3)^2$ (i.e., put $X = T 3$ and apply the canonical map $T 2 \times X^2 \to T X$). The left side is represented by $(m; s, t)$ where $s \in T 3$ is “$m(x_1, x_2)$” or more formally the image of $m \in T 2$ under the map $T 2 \stackrel{T i}{\to} T 3$ where $i: \{1, 2\} \to \{1, 2, 3\}$ is the inclusion map, and $t$ is “$x_3$” or more formally the image of the singleton under the map $\ast \stackrel{\eta \ast}{\to} T \ast \stackrel{T j}{\to} T 3$ where $j: \ast \to \{1, 2, 3\}$ names the element $3$. A similar interpretation of the right side $m(x_1, m(x_2, x_3))$ as an element of $T T 3$, represented again by an element in $T 2 \times (T 3)^2$, can of course be given. The two elements in $T T 3$ are of course distinct, but they map to the same $3$-ary operation in $T 3$ under $\mu 3$.
In fact one could say that the monad equations directly encapsulate the equational laws of the algebraic theory. What I mean is this: the monad associativity $\mu \circ \mu T = \mu \circ T \mu$ delivers sufficiently many of equational laws $(x, y)$, parametrized by elements $z \in T T T n$ by putting $x = (\mu T)_n(z)$ and $y = (T \mu)_n(z)$. (Clearly such pairs belong to the kernel pair of $m_n$.) By “sufficiently many”, I mean that any equational law in arity $n$ belongs to the equivalence relation generated by the image of $\langle \mu T_n, T \mu_n \rangle: T T T n \to T T n \times T T n$. This is because $\mu_n$ is the coequalizer of $\mu T_n, T \mu_n$, and the kernel pair of the coequalizer of a parallel pair $(f, g)$ is the equivalence relation generated by the image of $\langle f, g \rangle$, by effective regularity of the topos $Set$.
I didn't read through your entire argument carefully, Todd, but I was sure that it would be correct when I read
Let me argue this in the case where we treat “monad on $Set$” as virtually a synonym for “algebraic theory”.
As I recall, Johnstone did say that every monadic theory can be described by equational laws.
But there are also large algebraic theories that are not monadic. Some of these can still be described by equational laws, and some cannot. That is, finitary-algebraic ⊂ small-algebraic ⊂ monadic^{1} ⊂ equational ⊂ large-algebraic.
Or at least, that's how I remember it. I should look in Johnstone to check. (And by ‘Johnstone’ I mean Stone Spaces, not the Elephant!)
Equivalently, locally-small-algebraic ↩
Well, looking at some of these links, I have noticed two things:
At algebraic theory, the term ‘large algebraic’ seems to mean precisely that it can be described by equational laws, so why am I thinking that there's something even more general that cannot always be so described, and what could that possibly be?
We may not have an article about equational laws themselves, but we do have equationally presentable category, which is probably what I was thinking of.
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