Partition

Definition

A partition in the UOR framework is a decomposition of the ring R_n into disjoint components. The class Partition represents this decomposition.

Four Components

Every partition of R_n has exactly four component sets:

These four sets are mutually owl:disjointWith and their cardinalities sum to 2^n.

Ontology Properties

Example: R_4

For R_4 = Z/16Z (n=4, 16 elements):

<https://uor.foundation/instance/partition-R4>
    a                   partition:Partition ;
    schema:ringWittLength "4"^^xsd:nonNegativeInteger ;
    partition:irreducibles  <...irred-set-R4> ;
    partition:reducibles    <...red-set-R4> ;
    partition:units         <...unit-set-R4> ;
    partition:exterior      <...ext-set-R4> .

How Elements Are Classified

An element x ∈ R_n is classified by its factorization behavior:

• Irreducible: x has no non-trivial factorization — its only factorizations involve units. These are the "primes" of the ring.
• Reducible: x factors into two or more non-units. These elements decompose further under the dihedral factorization.
• Unit: x is invertible in R_n (i.e., gcd(x, 2^n) = 1). Units are exactly the odd elements.
• Exterior: x lies outside the multiplicative kernel — typically the zero element and nilpotents.

Worked Example: R_4

For R_4 = Z/16Z with 16 elements (0–15):

The 8 units are exactly the odd numbers (invertible mod 16). The sole irreducible is 2, the generator of the maximal ideal. All even non-zero non-powers-of-two elements are reducible (e.g., 6 = 2 × 3). Zero is exterior.

Role in Resolution

The DihedralFactorizationResolver produces a Partition as its output. The partition is then used by:

• Observable to measure properties
• Certificate to certify correctness
• Transform to apply transformations

See Factorization for the dihedral decomposition that produces the partition.