Algebraic Laws

Definition

The UOR Foundation ontology formalizes 7 core algebras (with additional identity families from Amendments 23–53) that govern computation over the ring R_n = Z/(2^n)Z. Each algebra is encoded as a set of named Identity individuals in the op/ namespace, with lhs, rhs, and forAll properties specifying the algebraic equation and its quantifier domain.

The Seven Algebras

1. Ring Algebra (R_A, R_M)

The additive group (R_A1–R_A6) and multiplicative monoid (R_M1–R_M5) axioms for the commutative ring Z/(2^n)Z. These are the foundation: associativity, commutativity, identity elements, inverses, and distributivity.

2. Boolean Algebra (B_)

Thirteen identities (B_1–B_13) encoding XOR, AND, OR, and BNOT as a Boolean algebra on the n-bit representation. Includes De Morgan's laws (B_11, B_12) and the involution property (B_13).

3. Cross-Structure Laws (X_)

Seven identities (X_1–X_7) connecting ring operations to Boolean operations. The key identity X_5: neg(x) = add(bnot(x), 1) (two's complement) bridges the ring and Boolean worlds.

4. Dihedral Group (D_)

Four identities (D_1, D_3–D_5) describing the dihedral group D_{2^n} generated by negation and bitwise complement. D_5 gives the full presentation.

5. Unit Group (U_)

Five identities (U_1–U_5) characterizing the group of invertible elements R_n× ≅ Z/2 × Z/2^{n-2} (for n ≥ 3), including order formulas and the step function.

6. Affine Group (AG_)

Four identities (AG_1–AG_4) describing the affine group Aff(R_n) = R_n× ⋉ R_n, which extends the dihedral group by non-trivial unit multiplications.

7. Carry Algebra (CA_)

Six identities (CA_1–CA_6) encoding the carry propagation rules that govern how ring addition differs from Boolean XOR. The Witt identification (WC_1–WC_12) proves these are the 2-typical Witt addition polynomials over F_2. The Ostrowski– Archimedean bridge (OA_1–OA_5) grounds the Landauer temperature β* = ln 2 via the product formula at p=2.

Cross-Algebra Maps

Six inter-algebra maps (phi_1–phi_6) formalize the relationships between algebras. See the Resolution concept page for the full pipeline φ₄ = φ₃ ∘ φ₂ ∘ φ₁.

Thermodynamic Interpretation

The identities TH_1–TH_10 reinterpret the resolution process as a thermodynamic system: site entropy, Landauer bounds, and phase transitions provide physical intuition for computational cost.