[MrQuincle]

Just like a Laplace transform maps differential equations into algebraic equations and convolution into multiplication - or Fourier for that matter - it's not so hard to imagine that there are encryption maps (that are hard to invert) but where something like a sum operation becomes a feasible operation in the encrypted domain. A max operation can similarly have an equivalent operation in the encrypted space.

I guess your concern is that the output is "one of the encrypted input" values and hence identified, although not decrypted. Subsequently, all the input values would be fed into the "max" module and their complete order can be determined by the one running the homeomorphic server.

In that case we will need to have an output where all inputs are returned. Perhaps a map with indices and values (all encrypted) as input and as output would be sufficient.