Transportable Monoids and Multiplicative Lattice Structure

This paper develops a unified transport framework for multiplicative structures, showing that when multiplication admits an injective, order-reflecting transport into a totally ordered additive domain, lattice structure necessarily emerges. The central result, the Universal Multiplicative Collapse Theorem, proves that such transport forces divisibility to form a distributive lattice with meet and join induced canonically by minimum and maximum in the target order. A complementary Structural Characterization Theorem classifies exactly which commutative cancellative monoids admit this transport, showing equivalence with distributive lattice divisibility and embeddability into products of totally ordered groups. Classical phenomena - including prime valuations on ℕ⁺, logarithms on ℝ₊, and rotational exponentials generated by quadratic elements - are unified as instances of the same transport principle. The framework provides a structural explanation for identities involving gcd/lcm, valuation additivity, logarithmic order transport, and unit-circle parameterization, demonstrating that these results arise from transportability rather than ad hoc domain-specific arguments. All results are proved using explicit algebraic and order-theoretic assumptions, and the presentation is self-contained.