Groupoids
Thanks to Alar Leibak, aleibak@ioc.ee, for helping to construct this page
Definition
- A groupoid is a pair (S,*), where S is a set and
* is any binary operation on S.
There is no restriction on the binary operation -- it does not need
to satisfy any special properties.
- Note that in topology and category theory, "groupoid" is
used differently.
Examples
- Any set S with the map f
f : SxS --> S
- Any semigroup, monoid, group, quasigroup, loop etc.
Structure
-
Representation
Decision problems
- Identity problem: There is hardly a problem.
- Word problem: Solvable uniformly for all finite presentations
Spectra and growth
- Finite spectrum: all positive integers
- Free spectrum: the free groupoid on even one generator is countably infinite
- Growth series: (1 - sqrt(1 - 4rz))/2 for the free groupoid on r generators
History/Importance
-
References
-
Subsystems
- Semigroups, quasigroups, medial groupoids
A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu