[Thanks to Alar Leibak, aleibak@ioc.ee, for helping to construct this page]

## 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*.*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** | 1 2 3 4 ----------- 1 | 1 2 3 4 2 | 4 1 2 3 3 | 3 4 1 2 4 | 2 3 4 1

- Any group
(in a group
## Structure

## Representation

## Decision problems

**Identity problem**:**Word problem**:## Spectra and growth

**Finite spectrum****:****Free spectrum****:****Growth series****:****History/Importance****References****V. D. Belousov,***Fundamentals of the theory of quasigroups and loops*(*in Russian*)**Subsystems**