Toward a Non-Commutative Residuated Lattice from Quaternion Multiplication

The quaternionic G-lattice introduced in the companion paper uses component-wise min as its monoidal operation, making the residuated lattice a product of four independent chains. The quaternionic structure of the framework manifests in the SU (2) group action and the product on S³, but not in the logic's conjunction. This paper investigates whether the Hamilton product of quaternions can serve as a genuinely non-commutative conjunction—a monoidal operation that couples the four epistemic axes through the quaternionic cross terms. We establish positive results: the Hamilton product on V = 0, 1 × -1, 1³ satisfies bilateral identity, bilateral annihilation, and associativity; it defines left and right implications via quaternionic division, both satisfying modus ponens; and under every commutative restriction (Boolean, fuzzy, modal), the two implications coincide and the product reduces to the standard commutative t-norm. Non-commutativity is proven to be emergent: it arises only in the full four-dimensional space, not in any proper restriction. We also characterize fundamental obstructions. The Hamilton product is not closed on V and is not monotone with respect to the component-wise order. We prove that no total order on H is compatible with Hamilton multiplication, that the component-wise lattice order and the Hamilton monoid cannot coexist on V, and—via four independent impossibility results—that no non-trivial lattice order on B⁴ = q ∈ H: |q| ≤ 1 is compatible with the Hamilton monoid. The four results are: (i) no proper bi-invariant cone exists in H (SU (2) transitivity), (ii) no non-trivial bi-invariant partial order exists on Q₈ (exhaustive closure), (iii) no real-dominant subcone is closed under multiplication (analytical bound), and (iv) the norm order collapses the structure to a one-dimensional product BL-algebra. These results culminate in a quaternionic trilemma: for any monoidal operation on a subset of V = 0, 1 × -1, 1³, at most two of associativity, non-commutativity, compatible lattice order can hold simultaneously. The trilemma is a theorem for V-domains (Theorem, trilemma-V): De Moivre's theorem shows that every non-real element's iterates eventually exit V, so the only sub-monoids of V under Hamilton product are commutative. For the full ball B⁴, where powers remain bounded but the angular argument fails, Theorem trilemma-B4 proves that the canonical non-commutative sub-monoid (equal-norm generators) admits no compatible lattice order. The equal-norm hypothesis is sharp: for generators of unequal norm, compatible monotone lattice orders exist (Remark, sharpness). An α-parameterized family interpolates between the diagonal product (α = 0) and the full Hamilton product (α = 1), preserving all specializations; associativity holds only at the endpoints.