It should presumably satisfy:
$\int e(x,y,z) e(x,u,v) dx = \delta(y-u)\delta(z-v) - \delta(y-v)\delta(z-u)$
$\int e(x,y,z) e(x,y,v) dx dy = 2 \delta(z-v)$
$\int e(x,y,z) e(x,y,z) dx dy dz = 6$
where $\delta(x)$ is the Dirac delta function.
If it exits, is there an integral formula for it? Or a series? And would it necessarily be non-differentiable? If so, what are its derivatives?
Edit: I just realised that for two variables $e(x,y)=sgn(x-y)$ and its derivative is therefor the dirac delta function. The others would be made up of products of the sgn function. Is that correct?