Groupoids have entered the OEIS !

[Benoît Jubin]

Yes, I know that magmas are also called groupoids (for instance... in
the OEIS!), and that's why I wrote everywhere "groupoids (categories
all of whose morphisms are invertible)", to avoid confusion.

For solvable cancellative magmas, I've only seen the term quasigroup.
A loop being a unital quasigroup.

By the way, what do you think about adding similar sequences for
"strongly connected categories" (that is, there exists a morphism
between every couple - as opposed to pair - of objects) ?

Sincerely,
Benoit



On Fri, May 16, 2008 at 7:45 PM,  <franktaw at netscape.net> wrote:
> Umm, you should know that the word "groupoid" has multiple meanings.  And
> (one of) the other one(s) is already in the OEIS.  See A001424 and A001329,
> or take a look at http://mathworld.wolfram.com/Groupoid.html.
>
> In addition to the meanings there, I think I have seen it used for a set
> with a
> binary operation which is solvable and cancellable (equivalent conditions
> for
> finite examples), but not necessarily associative.  In other words, the
> operation
> table is a Latin square.  These are also called quasigroups, or sometimes
> loops.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Benoît Jubin <benoit.jubin at gmail.com>
>
> Dear SeqFans,
>
> I have just contributed six sequences counting finite groupoids
> (categories all of whose morphisms are invertible):
> - A140185 (connected groupoids with n morphisms)
> - A140186 (connected groupoids with n more morphisms than objects)
> - A140187 (connected groupoids with n times as many morphisms as objects)
> - A140188 (groupoids with n morphisms and k objects)
> - A140189 (groupoids with n morphisms)
> - A140190 (groupoids with m morphisms and (m-n) objects for any m>=2n)
> The difficulty of counting groupoids is almost entirely in the
> counting of groups, so that these sequences are not hard to compute
> from A000001.
>
> Maybe this is worth a new entry in the OEIS index ? (and by the way,
> another entry for categories could be added):
> categories: A125696, A125697, A125701
> categories, connected: A125698, A125699, A125700, A125702
> categories, strongly connected: [sequences to be added]
> groupoids (categories with inverses): A140188, A140189, A140190
> groupoids (categories with inverses), connected: A140185, A140186, A140187
>
> For the curious, here are two surveys on groupoids:
> - Ronald Brown, From groups to groupoids: a brief survey
> http://www.bangor.ac.uk/~mas010/groupoidsurvey.pdf
> - Alan Weinstein, Groupoids: unifying internal and external symmetry
> http://www.ams.org/notices/199607/weinstein.pdf
>
> Sincerely,
> Benoit Jubin
> [my oeis page: math.berkeley.edu/~jubin/oeis.html ]
>