Quasigroups

Definition

A set with a binary operation (not necessarily associative) which satisfies unique solvability of equations. That is, for all a,b there exist unique x, y satisfying

              x a = b     and    a y = b 
        

Using three binary operations, written as (nonassociative) multiplication, /, and \ (where x = b/a and y = a\b are the unique solutions to the equations above), the variety of quasigroups is defined by the identities

  
             (y/x)x = y, x(x\y) = y,
             (xy)/y = x, x\(xy) = y
            
        

Examples

• Any loop (including any group )
• Here's a quasigroup that isn't a loop (thanks to Alar Leibak (aleibak@ioc.ee) for contributing this):

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