# IP-quasigroups

## Definition

A quasigroup with two additional unary operations (I and J), such that the following identities hold:
```	I(x)(xy) = y   and  (yx)J(x) = y
```

## Examples

• Any group (in a group I = J)
• Any IP-loop
• Let G be a group and f an automorphism of order two [f(f(x) = x]. Define the binary operation by
```	x*y = f(x)y.
```
Then G(*) is an IP-quasigroup, where
```	I(x)=x' and  J=f(x')
```
(x' is the inverse element for x in group G). The identities are verified as follows:
```	I(x)*(x*y) = I(x)*(f(x)y) = x'*(f(x)y) = f(x')f(x)y = y
(y*x)*J(x) = (f(y)x)*J(x) = (f(y)x)*f(x') = f(f(y)x)f(x') =
= f(f(y))f(x)f(x') = f(f(y)) = y
```
Here is a finite example.
```	          * | 1 2 3 4
-----------
1 | 1 2 3 4
2 | 4 1 2 3
3 | 3 4 1 2
4 | 2 3 4 1
```

## Decision problems

Identity problem:
Word problem:

Finite spectrum:
Free spectrum:
Growth series:

## References

V. D. Belousov, Fundamentals of the theory of quasigroups and loops (in Russian)

## Subsystems

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