Ring

Definition

A ring in the UOR framework is the algebraic substrate Z/(2^n)Z — integers modulo 2^n. The ontology class Ring represents this structure.

Mathematical Basis

For quantum level n, the ring R_n = Z/(2^n)Z has 2^n elements: {0, 1, ..., 2^n - 1}.

The ring is equipped with:

Standard addition and multiplication (mod 2^n)
Two canonical involutions (self-inverse operations):
- neg: additive negation, neg(x) = 2^n - x mod 2^n
- bnot: bitwise complement, bnot(x) = 2^n - 1 - x

These two involutions generate the dihedral group D_{2^n}, captured by DihedralGroup.

Ontology Representation

schema:Ring has the following properties:

Property	Type	Description
ringQuantum	xsd:nonNegativeInteger	The quantum level n
modulus	xsd:nonNegativeInteger	2^n
generator	op:Operation	The primary generator (op:neg)
negation	op:Operation	Ring negation (op:neg)
complement	op:Operation	Bitwise complement (op:bnot)

Example: The Byte Ring

For n=8, R_8 = Z/256Z is the byte ring — the algebraic structure underlying every byte of digital information:

<https://uor.foundation/instance/ring-R8>
    a                   schema:Ring ;
    schema:ringQuantum  "8"^^xsd:nonNegativeInteger ;
    schema:modulus      "256"^^xsd:nonNegativeInteger ;
    schema:generator    op:neg ;
    schema:negation     op:neg ;
    schema:complement   op:bnot .

The Critical Identity

The ring's two involutions satisfy the critical identity (see CriticalIdentityProof):

neg(bnot(x)) = succ(x)  for all x ∈ R_n

This means the successor operation succ is the composition of negation and complement: composedOf [op:neg, op:bnot].