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Canonical static concept record

Difference pairs of natural numbers

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Summary

Pairs $(a,b)\in\mathbb N^2$ representing formal differences $a-b$ .

Record metadata

Canonical name: Difference pairs of natural numbers
Concept ID: integer_difference_pairs
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Number theory
Domains: Number theory, Set theory, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Quotient presentation

Carrier(s)

set $\mathbb N\times\mathbb N$

Data

equivalence relation $(a,b)\sim(c,d)$ iff $a+d=b+c$

Canonically induces

quotient abelian group

Notes

This is the Grothendieck group completion of the additive commutative monoid $\mathbb N$ .

Concept sources

Wikipedia — Integer: Construction — Integer · encyclopedic reference · source ID wp-construction-integers
nLab — integer — integer · mathematical reference · source ID nlab-integer

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Natural numbers $\mathbb N$ Difference pairs of natural numbers

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: natural_numbers_to_integer_pairs
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form formal differences

Authored explanation

Use $\mathbb N\times\mathbb N$ to represent differences before quotienting.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Integer: Construction — Integer · encyclopedic reference · source ID wp-construction-integers
nLab — integer — integer · mathematical reference · source ID nlab-integer

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.