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:

Class	Description
IrreducibleSet	Elements with no non-trivial factorization
ReducibleSet	Elements that factor into smaller pieces
UnitSet	Invertible elements (units of the ring)
ExteriorSet	Elements outside the kernel

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

Ontology Properties

Property	Domain	Range	Description
irreducibles	Partition	IrreducibleSet	Link to irreducible set
reducibles	Partition	ReducibleSet	Link to reducible set
units	Partition	UnitSet	Link to unit set
exterior	Partition	ExteriorSet	Link to exterior set
cardinality	Component	xsd:nonNegativeInteger	Element count
density	Component	xsd:string	Density as fraction
member	Component	partition:Component	Member element
sourceType	Partition	type:TypeDefinition	Source type
quantum	Partition	xsd:nonNegativeInteger	Ring quantum level

Example: R_4

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

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

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