Loops

Definition

A quasigroup with an (identity) element e satisfying, for all x,

              xe = x     and    ex = x 
        

Examples

• Any group
• Here's a loop that isn't a group (thanks to Alar Leibak, aleibak@ioc.ee):

             * | 1 2 3 4 5
             -------------
             1 | 1 2 3 4 5
             2 | 2 5 1 3 4
             3 | 3 4 5 1 2
             4 | 4 1 2 5 3 
             5 | 5 3 4 2 1
          

  This is not associative because, for example, 3*(2*2)=3*5=2, but (3*2)*2=4*2=1.

Structure

Representation

Decision problems

Identity problem:Solvable (complete set)

Word problem:Solvable [Evans1951]

Spectra and growth

Finite spectrum:

Free spectrum:

Growth series:

History/Importance

References

Subsystems