If you mean:
> $$x \preceq_p y \iff p \cdot x \leq p \cdot y$$

then \$$f(x) = p \cdot x \$$ and we see that \$$f(x) \otimes_Y f(x') = p \cdot x + p \cdot x' = p \cdot (x+x') = f( x \otimes_Y x')\$$ since multiplication distributes over addition.