# 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
```

## Decision problems

Identity problem:Solvable (complete set)
Word problem:Solvable [Evans1951]

Finite spectrum:
Free spectrum:
Growth series:

## Subsystems

A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu