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

Name	Subclass Of	Disjoint With	Comment
`Partition`	`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`	`Thing`		A single component of a partition: a set of datum values belonging to one of the four categories.
`IrreducibleSet`	`Component`	`ReducibleSet`, `UnitSet`, `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`	`Component`	`IrreducibleSet`, `UnitSet`, `ExteriorSet`	The set of reducible non-unit elements: elements that can be expressed as a product of two or more non-unit elements.
`UnitSet`	`Component`	`IrreducibleSet`, `ReducibleSet`, `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`	`Component`	`IrreducibleSet`, `ReducibleSet`, `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.

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