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

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