I(x)(xy) = y and (yx)J(x) = y
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)) = yHere 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