UOR Partitions

IRI: https://uor.foundation/partition/
Prefix: partition:
Space: bridge
Comment: Irreducibility partitions produced by type resolution. A partition divides the ring into four disjoint components: Irreducible, Reducible, Units, and Exterior.

Imports

https://uor.foundation/schema/
https://uor.foundation/type/

Classes

Name	IRI	Subclass Of	Disjoint With	Comment
Partition	`https://uor.foundation/partition/Partition`	`http://www.w3.org/2002/07/owl#Thing`		A four-component partition of R_n produced by resolving a type declaration. The four components — Irreducible, Reducible, Units, Exterior — are mutually disjoint and exhaustive over the carrier.
Component	`https://uor.foundation/partition/Component`	`http://www.w3.org/2002/07/owl#Thing`		A single component of a partition: a set of datum values belonging to one of the four categories.
IrreducibleSet	`https://uor.foundation/partition/IrreducibleSet`	`https://uor.foundation/partition/Component`	`https://uor.foundation/partition/ReducibleSet`, `https://uor.foundation/partition/UnitSet`, `https://uor.foundation/partition/ExteriorSet`	The set of irreducible elements under the active type: elements whose only factorizations involve units or themselves. Analogous to prime elements in a ring.
ReducibleSet	`https://uor.foundation/partition/ReducibleSet`	`https://uor.foundation/partition/Component`	`https://uor.foundation/partition/IrreducibleSet`, `https://uor.foundation/partition/UnitSet`, `https://uor.foundation/partition/ExteriorSet`	The set of reducible non-unit elements: elements that can be expressed as a product of two or more non-unit elements.
UnitSet	`https://uor.foundation/partition/UnitSet`	`https://uor.foundation/partition/Component`	`https://uor.foundation/partition/IrreducibleSet`, `https://uor.foundation/partition/ReducibleSet`, `https://uor.foundation/partition/ExteriorSet`	The set of invertible elements (units) in the carrier: elements with a multiplicative inverse. In Z/(2^n)Z, the units are the odd integers.
ExteriorSet	`https://uor.foundation/partition/ExteriorSet`	`https://uor.foundation/partition/Component`	`https://uor.foundation/partition/IrreducibleSet`, `https://uor.foundation/partition/ReducibleSet`, `https://uor.foundation/partition/UnitSet`	Elements of R_n that fall outside the active carrier — i.e., outside the type's domain. These are ring elements that do not participate in the current type resolution.

Properties

Name	Kind	Functional	Domain	Range	Comment
irreducibles	Object	true	`https://uor.foundation/partition/Partition`	`https://uor.foundation/partition/IrreducibleSet`	The irreducible component of this partition.
reducibles	Object	true	`https://uor.foundation/partition/Partition`	`https://uor.foundation/partition/ReducibleSet`	The reducible component of this partition.
units	Object	true	`https://uor.foundation/partition/Partition`	`https://uor.foundation/partition/UnitSet`	The units component of this partition.
exterior	Object	true	`https://uor.foundation/partition/Partition`	`https://uor.foundation/partition/ExteriorSet`	The exterior component of this partition.
member	Object	false	`https://uor.foundation/partition/Component`	`https://uor.foundation/schema/Datum`	A datum value belonging to this partition component.
cardinality	Datatype	true	`https://uor.foundation/partition/Component`	`http://www.w3.org/2001/XMLSchema#nonNegativeInteger`	The number of elements in this partition component. The cardinalities of the four components must sum to 2^n.
density	Datatype	true	`https://uor.foundation/partition/Partition`	`http://www.w3.org/2001/XMLSchema#decimal`	The irreducible density of this partition: \|Irr\| / \|A\|, where A is the active carrier.
sourceType	Object	true	`https://uor.foundation/partition/Partition`	`https://uor.foundation/type/TypeDefinition`	The type declaration that was resolved to produce this partition.
quantum	Datatype	true	`https://uor.foundation/partition/Partition`	`http://www.w3.org/2001/XMLSchema#positiveInteger`	The quantum level n at which this partition was computed. The ring has 2^n elements at this level.