Groups of groups
August 22nd, 2008 by WaltYou know, mathematical terminology cannot be parodied. Mathematicians have invented groups, semigroups, quasigroups, pseudogroups, and two mostly-unrelated concepts both known as groupoids. They have invented both formal groups and quantum groups, neither of which are kinds of groups. And while the study of groups is a branch of algebra, most groups are not, in fact, algebraic.
August 23rd, 2008 at 5:14 am
What’s the other kind of groupoid? The only one I’ve heard is a (small) category where all the morphism are isomorphisms. This is the kind of groupoid the fundamental groupoid of a topological space is.
August 23rd, 2008 at 6:04 am
A set with an arbitrary binary operation. These are sometimes known as magmas. No one really studies these for themselves, but it’s a catch-all term for various alternatives to associativity. There are medial groupoids, entropic groupoids, etc.
August 23rd, 2008 at 7:18 am
Quantum groups are groups. They just fail to be groups in a certain way that’s more salient than the way that they are groups, so it’s OK to say they’re not groups and everyone knows what you mean.
August 23rd, 2008 at 2:35 pm
Wikipedia has a pretty nice chart outlining the most famous concepts intermediate between groups and magmas (formerly known as groupoids).
August 23rd, 2008 at 6:22 pm
“locally ringed spaces” is one that always gets to me…
August 23rd, 2008 at 10:04 pm
Let’s not forget left and right semimedial magmas, along with their fertile cousins the trimedial magmas, which can generate a medial submagma out of any three elements, and finally the most exotic branch of the family tree, the left and right unars, which the Wikipedia article cited by John Baez links to Unar (Urdu: انڑ ), “one of the most purest [sic] and oldest Sindhi tribe [sic] in Sindh, Pakistan.”
August 24th, 2008 at 9:45 pm
Structures that lie above a magma in the mathematical hierarchy have been catalogued so exhaustively that it’s only fair to glance in the other direction, and inquire about the possibility of mathematical objects even more formless than a magma.
Since a magma only conforms to the sole condition of closure under a binary operation, the most obvious way to weaken conditions would allow partial closure, so that the result of a few pairings leaks out of the underlying set.
I can already see a whole theory of non-leaky cycles developing on subsets where repeated application of the “operation” never oozes out of the original domain, but before research on our newly invented mathematical object can begin in earnest, it needs a name, and in analogy with the magma that engendered it, I propose that we call it a smegma.
August 25th, 2008 at 11:09 am
You jest, Jacob, but it’s been done. Pseudogroups have a partially-defined multiplication. The natural generalization is to allow any number of partially-defined operations of any arity, which has been studied as part of universal algebra.
August 25th, 2008 at 2:35 pm
Just when I was poised to enter the history of mathematics (Jacob Freeze, the inventor of mathematical smegma…), you want to fold my baby into pseudogroups?
Wikipedia, which must be a respectable source if an outstanding mathematician like John Baez cites it, says a pseudogroup “is not an axiomatic algebraic idea; rather it defines a set of closure conditions on sets of homeomorphisms defined on open sets U of a given Euclidean space E or more generally of a fixed topological space S.”
Is this another mess like the two sorts of groupoids, and I just haven’t seen the other pseudogroups floating around in the literature?
The only instances that I know of where pseudogroups appear in the context of universal algebra apparently take over the usual topological concept of pseudogroups, although I can never quite imagine how any sort of group emerges on the other side of the universal-algebra grinder that breaks down the binary operation of groups into a weird mix of nullary, unary, and binary operations.
There’s obviously something behind your all-too-brief comment that I don’t understand, and maybe you can provide a link to some manifestation of pseudogroups that doesn’t involve higher-order topological operations like homeomorphisms, or at least presents them in a way that makes the reduction of the relevant set of operations on homeomorphisms to something simpler than a magma more transparent.
At a level far below homeomorphisms in the hierarchy of topological concepts, even some components of the Wikipedia definition of a pseudogroup as simple as inverses may not exist for any element of a smegma, where the range of leaky operations is absolutely unrestricted. For example, it’s perfectly possible to define a smegma that maps every real number onto Pamela Anderson, and once you arrive at Pamela Anderson, there’s no way back.
August 25th, 2008 at 10:09 pm
If you throw away all of the topological information, a pseudogroup is just a group with the closure axiom dropped. Pseudogroups are interesting mainly for the interplay of topological and algebraic aspects. If you drop both of these, and just consider a set with a single partially-defined binary operation, there isn’t much you can say. At that point, nothing much is gained by restricting yourself to a single binary operation, so you might as well study a set with any number of nullary, unary, binary, etc. operations.
August 28th, 2008 at 2:51 am
@lavelle: locally ringed space means the following.
You have a topological space, and you want to build on it something like, say, continuous or holomorphic functions. So you insiste that they are a sheaf (i.e., you can define a continuous function locally and than glue) and that they are rings (sum and product of continuous functions is continuous). This gives you a ringed space.
But then… you want them to be functions in the sense that you want to be able to say if at a given point they are nonzero. Moreover, if they are nonzero at a point, they should be nonzero and invertible in a neighborhood of that point. Surprisingly enough, this is equivalent to the locally ringed space assumption. At least that’s what I think - I never saw it written anywhere.
September 3rd, 2008 at 10:41 pm
Despite what some might have you believe, the magma sense of “groupoid” is not dead. It is still the preferred term for sets with a single binary operation in much of central and eastern Europe. For instance, no one in the “Prague school” (Charles University’s algebra department and their descendants) would ever use “magma”. Also note that MSC 20N02 is still called “Sets with a single binary operation (groupoids)”, and many papers are published with that usage. To those folks, the other type are Brandt groupoids.
I say all this as someone who personally prefers the term “magma”, by the way. Perhaps the next time Mathematical Reviews carries out one of their classification revisions, someone should lobby for MSC 20N02 to be called “Magmas”.
September 26th, 2008 at 6:37 am
*shudders* I don’t like groups very much. I sometimes don’t like algebra very much if I’m being honest. And if I carry on in this honest mode, I don’t really know what I like! No wonder I can’t choose what flippin’ modules to choose!
November 7th, 2008 at 11:13 am
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