This post seems a bit incomplete in terms of terminology. So T is a torus? P is the toric variety? What does it mean for a set to be Milnor. Are you talking about Milnor fibrations? I also don't see what induces the norm in your case.
If you want to explain tropological stuff it makes more sense to start from tropological geometry. Because tropological algebra has a clear intuitive meaning.
RE: Associativity in Tropical Set Theory