## 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* - 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

- 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
## Examples

## 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**