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	IRI	Subclass Of	Disjoint With	Comment
Operation	`https://uor.foundation/op/Operation`	`http://www.w3.org/2002/07/owl#Thing`		An operation on the ring Z/(2^n)Z. The root class for all UOR kernel operations.
UnaryOp	`https://uor.foundation/op/UnaryOp`	`https://uor.foundation/op/Operation`		A unary operation on the ring: takes one datum and produces one datum.
BinaryOp	`https://uor.foundation/op/BinaryOp`	`https://uor.foundation/op/Operation`		A binary operation on the ring: takes two datums and produces one datum.
Involution	`https://uor.foundation/op/Involution`	`https://uor.foundation/op/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	`https://uor.foundation/op/Identity`	`http://www.w3.org/2002/07/owl#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	`https://uor.foundation/op/Group`	`http://www.w3.org/2002/07/owl#Thing`		A group: a set with an associative binary operation, an identity element, and inverses for every element.
DihedralGroup	`https://uor.foundation/op/DihedralGroup`	`https://uor.foundation/op/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	`https://uor.foundation/op/Operation`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The number of arguments this operation takes. 1 for unary operations, 2 for binary operations.
geometricCharacter	Datatype	true	`https://uor.foundation/op/Operation`	`http://www.w3.org/2001/XMLSchema#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	`https://uor.foundation/op/BinaryOp`	`http://www.w3.org/2001/XMLSchema#boolean`	Whether this binary operation satisfies op(x,y) = op(y,x) for all x, y in R_n.
associative	Datatype	true	`https://uor.foundation/op/BinaryOp`	`http://www.w3.org/2001/XMLSchema#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	`https://uor.foundation/op/BinaryOp`	`http://www.w3.org/2001/XMLSchema#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	`https://uor.foundation/op/Operation`	`https://uor.foundation/op/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	`https://uor.foundation/op/Operation`	`http://www.w3.org/1999/02/22-rdf-syntax-ns#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	`https://uor.foundation/op/Identity`	`https://uor.foundation/op/Operation`	The left-hand side operation of an algebraic identity.
rhs	Object	true	`https://uor.foundation/op/Identity`	`https://uor.foundation/op/Operation`	The right-hand side operation of an algebraic identity.
forAll	Datatype	true	`https://uor.foundation/op/Identity`	`http://www.w3.org/2001/XMLSchema#string`	The quantifier scope: the variable(s) over which this algebraic identity holds (e.g., 'x ∈ R_n').
generatedBy	Object	false	`https://uor.foundation/op/Group`	`https://uor.foundation/op/Operation`	An operation that generates this group. The dihedral group D_{2^n} is generated by op:neg and op:bnot.
order	Datatype	true	`https://uor.foundation/op/Group`	`http://www.w3.org/2001/XMLSchema#positiveInteger`	The number of elements in the group. For D_{2^n}, the order is 2^(n+1).
presentation	Annotation	true	`https://uor.foundation/op/Group`	`http://www.w3.org/2001/XMLSchema#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	Properties	Comment
neg	`https://uor.foundation/op/Involution`	`arity`: 1 `geometricCharacter`: ring_reflection	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).
bnot	`https://uor.foundation/op/Involution`	`arity`: 1 `geometricCharacter`: hypercube_reflection	Hypercube reflection: bnot(x) = (2^n - 1) ⊕ x (bitwise complement). The second generator of D_{2^n}. bnot(bnot(x)) = x.
succ	`https://uor.foundation/op/UnaryOp`	`arity`: 1 `geometricCharacter`: rotation `composedOf`: [`https://uor.foundation/op/neg`, `https://uor.foundation/op/bnot`] `inverse`: `https://uor.foundation/op/pred`	Successor: succ(x) = neg(bnot(x)) = (x + 1) mod 2^n. The critical identity: succ is the composition neg ∘ bnot.
pred	`https://uor.foundation/op/UnaryOp`	`arity`: 1 `geometricCharacter`: rotation_inverse `composedOf`: [`https://uor.foundation/op/bnot`, `https://uor.foundation/op/neg`] `inverse`: `https://uor.foundation/op/succ`	Predecessor: pred(x) = bnot(neg(x)) = (x - 1) mod 2^n. The inverse of succ. pred is the composition bnot ∘ neg.
add	`https://uor.foundation/op/BinaryOp`	`arity`: 2 `geometricCharacter`: translation `commutative`: true `associative`: true `identity`: 0 `inverse`: `https://uor.foundation/op/sub`	Ring addition: add(x, y) = (x + y) mod 2^n. Commutative, associative; identity element is 0.
sub	`https://uor.foundation/op/BinaryOp`	`arity`: 2 `geometricCharacter`: translation `commutative`: false `associative`: false	Ring subtraction: sub(x, y) = (x - y) mod 2^n. Not commutative, not associative.
mul	`https://uor.foundation/op/BinaryOp`	`arity`: 2 `geometricCharacter`: scaling `commutative`: true `associative`: true `identity`: 1	Ring multiplication: mul(x, y) = (x × y) mod 2^n. Commutative, associative; identity element is 1.
xor	`https://uor.foundation/op/BinaryOp`	`arity`: 2 `geometricCharacter`: hypercube_translation `commutative`: true `associative`: true `identity`: 0	Bitwise exclusive or: xor(x, y) = x ⊕ y. Commutative, associative; identity element is 0.
and	`https://uor.foundation/op/BinaryOp`	`arity`: 2 `geometricCharacter`: hypercube_projection `commutative`: true `associative`: true	Bitwise and: and(x, y) = x ∧ y. Commutative, associative.
or	`https://uor.foundation/op/BinaryOp`	`arity`: 2 `geometricCharacter`: hypercube_join `commutative`: true `associative`: true	Bitwise or: or(x, y) = x ∨ y. Commutative, associative.
Critical Identity	`https://uor.foundation/op/Identity`	`lhs`: `https://uor.foundation/op/succ` `rhs`: [`https://uor.foundation/op/neg`, `https://uor.foundation/op/bnot`] `forAll`: x ∈ R_n	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.
D_{2^n}	`https://uor.foundation/op/DihedralGroup`	`generatedBy`: `https://uor.foundation/op/neg` `generatedBy`: `https://uor.foundation/op/bnot` `presentation`: ⟨r, s \| r^{2^n} = s² = e, srs = r⁻¹⟩	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.