UOR Operations

IRI: https://uor.foundation/op/
Prefix: op:
Space: kernel
Comment: Ring operations, involutions, algebraic identities, and the dihedral symmetry group D_{2^n} generated by neg and bnot.

Imports

https://uor.foundation/schema/

Classes

Name	Subclass Of	Comment
`Operation`	`Thing`	An operation on the ring Z/(2^n)Z. The root class for all UOR kernel operations.
`UnaryOp`	`Operation`	A unary operation on the ring: takes one datum and produces one datum.
`BinaryOp`	`Operation`	A binary operation on the ring: takes two datums and produces one datum.
`Involution`	`UnaryOp`	A unary operation f such that f(f(x)) = x for all x in R_n. The two UOR involutions are neg (ring reflection) and bnot (hypercube reflection).
`Identity`	`Thing`	An algebraic identity: a statement that two expressions are equal for all inputs. The critical identity is neg(bnot(x)) = succ(x) for all x in R_n.
`Group`	`Thing`	A group: a set with an associative binary operation, an identity element, and inverses for every element.
`DihedralGroup`	`Group`	The dihedral group D_{2^n} of order 2^(n+1), generated by the ring reflection (neg) and the hypercube reflection (bnot). This group governs the symmetry of the UOR type space.

Properties

Name	Kind	Functional	Domain	Range	Comment
`arity`	Datatype	true	`Operation`	`nonNegativeInteger`	The number of arguments this operation takes. 1 for unary operations, 2 for binary operations.
`geometricCharacter`	Datatype	true	`Operation`	`string`	A description of the geometric role of this operation in the UOR ring and hypercube geometry. Examples: 'ring_reflection', 'hypercube_reflection', 'rotation', 'translation', 'scaling'.
`commutative`	Datatype	true	`BinaryOp`	`boolean`	Whether this binary operation satisfies op(x,y) = op(y,x) for all x, y in R_n.
`associative`	Datatype	true	`BinaryOp`	`boolean`	Whether this binary operation satisfies op(op(x,y),z) = op(x,op(y,z)) for all x, y, z in R_n.
`identity`	Datatype	true	`BinaryOp`	`integer`	The identity element of this binary operation: the value e such that op(x, e) = op(e, x) = x for all x in R_n.
`inverse`	Object	true	`Operation`	`Operation`	The inverse operation: the operation inv_op such that op(x, inv_op(x)) = e for all x, where e is the identity.
`composedOf`	Object	true	`Operation`	`List`	Ordered list of operations this operation is composed from. Uses rdf:List to preserve application order (first element applied innermost). E.g., succ = neg ∘ bnot is encoded as [op:neg, op:bnot] meaning neg applied to the result of bnot.
`lhs`	Object	true	`Identity`	`Operation`	The left-hand side operation of an algebraic identity.
`rhs`	Object	true	`Identity`	`Operation`	The right-hand side operation of an algebraic identity.
`forAll`	Datatype	true	`Identity`	`string`	The quantifier scope: the variable(s) over which this algebraic identity holds (e.g., 'x ∈ R_n').
`generatedBy`	Object	false	`Group`	`Operation`	An operation that generates this group. The dihedral group D_{2^n} is generated by op:neg and op:bnot.
`order`	Datatype	true	`Group`	`positiveInteger`	The number of elements in the group. For D_{2^n}, the order is 2^(n+1).
`presentation`	Annotation	true	`Group`	`string`	The group presentation (generators and relations). Annotation only — not used for reasoning. Example: ⟨r, s \| r^{2^n} = s² = e, srs = r⁻¹⟩

Named Individuals

Name	Type	Comment
`neg`	`Involution`	Ring reflection: neg(x) = (-x) mod 2^n. One of the two generators of the dihedral group D_{2^n}. neg(neg(x)) = x (involution property).
`arity`: 1 `geometricCharacter`: ring_reflection
`bnot`	`Involution`	Hypercube reflection: bnot(x) = (2^n - 1) ⊕ x (bitwise complement). The second generator of D_{2^n}. bnot(bnot(x)) = x.
`arity`: 1 `geometricCharacter`: hypercube_reflection
`succ`	`UnaryOp`	Successor: succ(x) = neg(bnot(x)) = (x + 1) mod 2^n. The critical identity: succ is the composition neg ∘ bnot.
`arity`: 1 `geometricCharacter`: rotation `composedOf`: [`neg`, `bnot`] `inverse`: `pred`
`pred`	`UnaryOp`	Predecessor: pred(x) = bnot(neg(x)) = (x - 1) mod 2^n. The inverse of succ. pred is the composition bnot ∘ neg.
`arity`: 1 `geometricCharacter`: rotation_inverse `composedOf`: [`bnot`, `neg`] `inverse`: `succ`
`add`	`BinaryOp`	Ring addition: add(x, y) = (x + y) mod 2^n. Commutative, associative; identity element is 0.
`arity`: 2 `geometricCharacter`: translation `commutative`: true `associative`: true `identity`: 0 `inverse`: `sub`
`sub`	`BinaryOp`	Ring subtraction: sub(x, y) = (x - y) mod 2^n. Not commutative, not associative.
`arity`: 2 `geometricCharacter`: translation `commutative`: false `associative`: false
`mul`	`BinaryOp`	Ring multiplication: mul(x, y) = (x × y) mod 2^n. Commutative, associative; identity element is 1.
`arity`: 2 `geometricCharacter`: scaling `commutative`: true `associative`: true `identity`: 1
`xor`	`BinaryOp`	Bitwise exclusive or: xor(x, y) = x ⊕ y. Commutative, associative; identity element is 0.
`arity`: 2 `geometricCharacter`: hypercube_translation `commutative`: true `associative`: true `identity`: 0
`and`	`BinaryOp`	Bitwise and: and(x, y) = x ∧ y. Commutative, associative.
`arity`: 2 `geometricCharacter`: hypercube_projection `commutative`: true `associative`: true
`or`	`BinaryOp`	Bitwise or: or(x, y) = x ∨ y. Commutative, associative.
`arity`: 2 `geometricCharacter`: hypercube_join `commutative`: true `associative`: true
`Critical Identity`	`Identity`	The foundational theorem of the UOR kernel: neg(bnot(x)) = succ(x) for all x in R_n. This identity links the two involutions (neg and bnot) to the successor operation, making succ derivable from neg and bnot.
`lhs`: `succ` `rhs`: [`neg`, `bnot`] `forAll`: x ∈ R_n
`D_{2^n}`	`DihedralGroup`	The dihedral group of order 2^(n+1), generated by neg (ring reflection) and bnot (hypercube reflection). Every element of this group acts as an isometry on the type space 𝒯_n.
`generatedBy`: `neg` `generatedBy`: `bnot` `presentation`: ⟨r, s \| r^{2^n} = s² = e, srs = r⁻¹⟩