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A magma is a set $M$ equipped with a binary operation $* : M \times M \to M$. In abstract algebra we typically begin by studying a special type of magma: groups. Groups satisfy certain additional axioms that "symmetries of things" should satisfy. This is made precise in the sense that for any object $A$ in a category $C$, the invertible morphisms $A \to A$ have a group structure again given by composition. An alternate definition of "group," then, is "one-object category with invertible morphisms," and then the additional axioms satisfied by groups follow from the axioms of a category (which, for now, we will trust as meaningful). Groups therefore come equipped with a natural notion of representation: a representation of a group $G$ (in the loose sense) is just a functor out of $G$. Typical choices of target category include $\text{Set}$ and $\text{Hilb}$.

It seems to me, however, that magmas (and their cousins, such as non-associative algebras) don't naturally admit the same interpretation; when you throw away associativity, you lose the connection to composition of functions. One can think about the above examples as follows: there is a category of groups, and to study the group $G$ we like to study the functor $\text{Hom}(G, -)$, and to study this functor we like to plug in either the groups $S_n$ or the groups $GL_n(\mathbb{C})$, etc. on the right, as these are "natural" to look at. But in the category of magmas I don't have a clue what the "natural" examples are.

Question 1: Do magmas and related objects like non-associative algebras have a "natural" notion of "representation"?

It's not entirely clear to me what "natural" should mean. One property I might like such a notion to have is an analogue of Cayley's theorem.

For certain special classes of non-associative object there is sometimes a notion of "natural": for example, among not-necessarily-associative algebras we may single out Lie algebras, and those have a "natural" notion of representation because we want the map from Lie groups to Lie algebras to be functorial. But this is a very special consideration; I don't know what it is possible to say in general.

(If you can think of better tags, feel free to retag.)

Edit: Here is maybe a more focused version of the question.

Question 2: Does there exist a "nice" sequence $M_n$ of finite magmas such that any finite magma $M$ is determined by the sequence $\text{Hom}(M, M_n)$? (In particular, $M_n$ shouldn't be an enumeration of all finite magmas!) One definition of "nice" might be that there exist compatible morphisms $M_n \times M_m \to M_{n+m}$, but it's not clear to me that this is necessarily desirable.

Edit: Here is maybe another more focused version of the question.

Question 3: Can the category of magmas be realized as a category of small categories in a way which generalizes the usual realization of the category of groups as a category of small categories?

Edit: Tom Church brings up a good point in the comments that I didn't address directly. The motivations I gave above for the "natural" notion of representation of a group or a Lie algebra are in some sense external to their equational description and really come from what we would like groups and Lie algebras to do for us. So I guess part of what I'm asking for is whether there is a sensible external motivation for studying arbitrary magmas, and whether that motivation leads us to a good definition of representation.

Edit: I guess I should also make this explicit. There are two completely opposite types of answers that I'd accept as a good answer to this question:

  • One that gives an "external" motivation to the study of arbitrary magmas (similar to how dynamical systems motivate the study of arbitrary unary operations $M \to M$) which suggests a natural notion of representation, as above. This notion might not look anything like the usual notion of either a group action or a linear representation, and it might not answer Question 3.

  • One that is "self-contained" in some sense. Ideally this would consist of an answer to Question 3. I am imagining some variant of the following construction: to each magma $M$ we associate a category whose objects are the non-negative integers where $\text{Hom}(m, n)$ consists of binary trees with $n$ roots (distinguished left-right order) and $m$ "empty" leaves (same), with the remaining leaves of the tree labeled by elements of $M$. Composition is given by sticking roots into empty leaves. I think this is actually a 2-category with 2-morphisms given by collapsing pairs of elements of $M$ with the same parent into their product. An ideal answer would explain why this construction, or some variant of it, or some other construction entirely, is natural from some higher-categorical perspective and then someone would write about it on the nLab!

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I can't think obvious non-associative structures to represent into. The free magma consists of finite binary trees with the operation that attaches two trees at a new root. Maybe thinking about this case might help... –  François G. Dorais Apr 12 '10 at 20:39
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H-spaces are essentially continuous representations of magmas. Sometimes you require associativity up to a homotopy - these homotopies are then all nicely parametrized by the associahedra. Your condition about compatible morphisms from $M_n x M_m \to M_{n+m}$ is starting to look like you want some kind of operad structure. Just throwing some terminology out there for you to google and see if you think it is relevant. If this is the kind of thing you are after I could try to craft an actual answer to your question. –  Steven Gubkin Apr 12 '10 at 21:27
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Harry, I'm afraid I don't see the relevance of those links. –  Qiaochu Yuan Apr 12 '10 at 22:39
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I imagine you came across this, which discusses in broad terms what are representations of non-associative algebras. projecteuclid.org/DPubS/Repository/1.0/… Probably any reasonable notion of (linear) representation of magmas would extend to a non-associative algebra, simply because linear maps can be added and scaled. Of course, if you study completely general magmas, there won't be much you can say... But they suggest various specializations. I didn't give this as an answer since I don't know anything about this topic. –  David Jordan Apr 13 '10 at 1:14
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Also, without picking a fight, I would like to quietly object to: "This is made precise in the sense that for any object A in a category C , the invertible morphisms A->A have a group structure." "Groups therefore come with a natural notion of representation: a functor out of G." "Lie algebras have a 'natural' notion of representation because we want the map from Lie groups to Lie algebras to be functorial." None of this is false, but it seems to be putting the dictionary before the horse. –  Tom Church Apr 13 '10 at 1:16
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6 Answers

up vote 5 down vote accepted

Since magmas in general don't have much structure, we can't reasonably expect a representation to preserve much structure. We can therefore define a left representation of a magma $M$ to be a set $V$ equipped with a map $M \times V \to V$. We do the analogous thing for general nonassociative algebras. Serge Lang liked to describe a notion of left regular representation of an algebra $A$, which is just the linear map $A \to \operatorname{End} (A)$ that takes an element to the linear transformation it induces by left multiplication. As expected, this map is a homomorphism if and only if the algebra is associative.

There are special cases of nonassociative algebras that admit good notions of representation, and in the cases I know, these arise from operads that have "good relationships" with the associative operad. The standard example is the natural map from the Lie operad to the Associative operad that yields the forgetful functor from associative algebras to Lie algebras. This functor admits the universal enveloping algebra functor as left adjoint. There is a formalism of enveloping operads, which generalizes this case. The upshot is that these special cases have a lot more structure than a simple composition law, so we can demand more from a representation (namely, that it respect the operad structure as manifested through the universal enveloping algebra).

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That's a depressing definition of a representation of a magma; you just pick a function V -> V for each element of the magma in a way that is independent of the the binary operation. The kind of answer I have in mind won't necessarily have anything to do with functions from a set to itself... –  Qiaochu Yuan Apr 13 '10 at 2:02
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Picking a bunch of functions is pretty much what you do when you define a magma, but with V=M. –  S. Carnahan Apr 13 '10 at 2:53
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Qiaochu$_{2010}$ was silly. This answer is clearly correct. –  Qiaochu Yuan Feb 10 at 8:35
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Let me concentrate on your first question (frankly speaking, the way you formulate your second question slightly lacks motivation).

The case where there is a reasonable suggestion, assumes that you work with some type of nonassociative algebras over a field, and all identities follow from the multilinear ones. In other words, the category of algebras you are studying is the category of algebras over some operad O. In this case, there is a nice way to describe a module over such an algebra. For an algebra A, a module structure on V is given by a collection of operations defined by all possible operations from O, where you are allowed to plug an element from V into one slot of operations, and plug elements of A into other slots. To write down the module axioms, take the defining identities of O, and form new identities, marking one element there in all possible ways; now treat the unmarked elements as elements of A, and marked elements as belonging to V.

For example, for associative algebras the original identity is (ab)c=a(bc), which leads to the following definition. A module structure is defined by two operations, a,v\mapsto av and a,v\mapsto va satisfying the identities (ab)v=a(bv), (av)b=a(vb), (va)b=v(ab). This means that in the case of associative algebras we defined bimodules. Also, for Lie algebras we get the module structure which, as it is immediate to check, coincides with the usual module/representation structure. In general, this construction provides a reasonable "enveloping algebra" for your nonassociative algebra. Thus, one way to approach your question is to study representations of the enveloping algebra, and sometimes it's the best you can get.

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Taking the cue from Lie algebras you could try considering something like the enveloping algebra. In the Lie case a representation of $g$ is just a usual representation of $U(g)$ so maybe here you can make the same construction.

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The construction of the universal enveloping algebra privileges the bilinear operation AB - BA; my guess is that this operation isn't generic enough to really capture the behavior of an algebra that is very far from being associative, e.g. many interesting non-associative algebras might collapse. –  Qiaochu Yuan Apr 12 '10 at 20:20
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My guess would be then that representations in a vector space in such a case would be meaningless, and you would need to search for some other structure. –  Adam Gal Apr 12 '10 at 20:23
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Supporting Neel's answer: yes Joel Vanderwerf did generalize the wreath product to certain universal algebras as his dissertation. (I believe there was an issue of similarity type, and that the algebras were given with their clones. Or not.) Last I knew, he was in industry helping develop models for traffic patterns. Ralph McKenzie or Japheth Wood might remember more detail; sadly I do not.

Gerhard "Ask Me About System Design" Paseman, 2010.04.12

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For monoids (which are associative) the Krohn-Rhodes theorem gives a powerful decomposition result: every finite monoid is a quotient of a submonoid of an alternating wreath product of finite groups and monoids. Google suggests that Joel Wanderwerf may have generalized this theorem to arbitrary algebras in a 1996 article for the Semigroup Forum journal, but I don't have access to this Springer journal so I can't say for sure.

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There is an instant method to produce "modules" over about anything: given an object $C$ in any category $\mathcal C$ with pullbacks, one defines $$ C-\textrm{mod}:=\textrm{Ab}(\mathcal C/C) $$ (the category of internal abelian groups in the slice over $C$).

Of course as every quickie, it is not always what you want. But at least it is always there :)

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