UOR Division Algebras

IRI: https://uor.foundation/division/
Prefix: division:
Space: kernel
Comment: The four normed division algebras R, C, H, O and the Cayley-Dickson construction.

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This is a kernel-space namespace in the Define stage of the PRISM pipeline. It provides the immutable algebraic substrate — ring structure, schema vocabulary, and operation algebra.

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Class hierarchy

Imports

https://uor.foundation/op/
https://uor.foundation/convergence/

Classes

Name	Subclass Of	Disjoint With	Comment
`NormedDivisionAlgebra`	`Thing`		An algebra over R that is a division ring with multiplicative norm. Exactly four exist (Hurwitz theorem): R, C, H, O.
`CayleyDicksonConstruction`	`Thing`		The doubling construction that builds each division algebra from the previous: R → C → H → O.
`MultiplicationTable`	`Thing`		The explicit product rules for a division algebra’s basis elements.
`AlgebraCommutator`	`Thing`		The commutator [a,b] = ab − ba. Zero for R and C; non-zero for H and O.
`AlgebraAssociator`	`Thing`		The associator [a,b,c] = (ab)c − a(bc). Zero for R, C, H; non-zero for O.

Properties

Name	Kind	Functional	Domain	Range	Comment
`algebraDimension`	Datatype	true	`NormedDivisionAlgebra`	`nonNegativeInteger`	The dimension of this division algebra (1, 2, 4, or 8).
`isCommutative`	Datatype	true	`NormedDivisionAlgebra`	`boolean`	Whether multiplication in this algebra is commutative.
`isAssociative`	Datatype	true	`NormedDivisionAlgebra`	`boolean`	Whether multiplication in this algebra is associative.
`basisElements`	Datatype	true	`NormedDivisionAlgebra`	`string`	The basis elements of this division algebra.
`algebraMultiplicationTable`	Object	true	`NormedDivisionAlgebra`	`MultiplicationTable`	The multiplication table for this algebra.
`cayleyDicksonSource`	Object	true	`CayleyDicksonConstruction`	`NormedDivisionAlgebra`	The source algebra of the Cayley-Dickson doubling.
`cayleyDicksonTarget`	Object	true	`CayleyDicksonConstruction`	`NormedDivisionAlgebra`	The target algebra of the Cayley-Dickson doubling.
`adjoinedElement`	Datatype	true	`CayleyDicksonConstruction`	`string`	The new basis element adjoined by this doubling step.
`conjugationRule`	Datatype	true	`CayleyDicksonConstruction`	`string`	The conjugation and multiplication rule for the adjoined element.

Named Individuals

Name	Type	Comment
`RealAlgebra`	`NormedDivisionAlgebra`	The real numbers R: dimension 1, commutative, associative.
`algebraDimension`: 1 `isCommutative`: true `isAssociative`: true `basisElements`: {1}
`ComplexAlgebra`	`NormedDivisionAlgebra`	The complex numbers C: dimension 2, commutative, associative.
`algebraDimension`: 2 `isCommutative`: true `isAssociative`: true `basisElements`: {1, i}
`QuaternionAlgebra`	`NormedDivisionAlgebra`	The quaternions H: dimension 4, non-commutative, associative.
`algebraDimension`: 4 `isCommutative`: false `isAssociative`: true `basisElements`: {1, i, j, k}
`OctonionAlgebra`	`NormedDivisionAlgebra`	The octonions O: dimension 8, non-commutative, non-associative.
`algebraDimension`: 8 `isCommutative`: false `isAssociative`: false `basisElements`: {1, i, j, k, l, il, jl, kl}
`cayleyDickson_R_to_C`	`CayleyDicksonConstruction`	Cayley-Dickson doubling R → C: adjoin i with i² = −1.
`cayleyDicksonSource`: `RealAlgebra` `cayleyDicksonTarget`: `ComplexAlgebra` `adjoinedElement`: i `conjugationRule`: i² = −1
`cayleyDickson_C_to_H`	`CayleyDicksonConstruction`	Cayley-Dickson doubling C → H: adjoin j with j² = −1, ij = k, ji = −k.
`cayleyDicksonSource`: `ComplexAlgebra` `cayleyDicksonTarget`: `QuaternionAlgebra` `adjoinedElement`: j `conjugationRule`: ij = k, ji = −k
`cayleyDickson_H_to_O`	`CayleyDicksonConstruction`	Cayley-Dickson doubling H → O: adjoin l, non-associative Fano plane products.
`cayleyDicksonSource`: `QuaternionAlgebra` `cayleyDicksonTarget`: `OctonionAlgebra` `adjoinedElement`: l `conjugationRule`: non-associative Fano plane products