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Canonical static concept record

Integers $\mathbb Z$

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Summary

The ordered commutative ring obtained from formal differences of natural numbers.

Record metadata

Canonical name: Integers $\mathbb Z$
Concept ID: integers
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Number theory
Domains: Number theory, Algebra, Order theory, Set theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Number system

Carrier(s)

quotient $(\mathbb N\times\mathbb N)/\sim$

Data

addition
multiplication
total order

Axioms / constraints

integral-domain laws

Canonically induces

Euclidean-domain structure

Notes

The map $n\mapsto[(n,0)]$ canonically embeds $\mathbb N$ into $\mathbb Z$ ; it need not be literal subset inclusion in the quotient realization.

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.

Difference pairs of natural numbers Integers $\mathbb Z$

Permalink to relation

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

Relation ID: integer_pairs_to_integers
Relation type: Quotient construction quotient-construction
Direction: source → target
Endpoint roles: source: Quotients to; target: Obtained as quotient of
Authored annotation: identify equal differences

Authored explanation

Quotient by $(a,b)\sim(c,d)$ exactly when $a+d=b+c$ ; induced operations produce $\mathbb Z$ .

How to interpret this relation type

Identify elements by a stated equivalence relation and equip the quotient with the induced structure. The edge detail must state the representatives and equivalence relation.

Relation sources

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

Natural numbers $\mathbb N$ Integers $\mathbb Z$

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

Relation ID: natural_numbers_into_integers
Relation type: Canonical embedding canonical-embedding
Direction: source → target
Endpoint roles: source: Embeds canonically into; target: Contains a canonical copy of
Authored annotation: $n\mapsto[(n,0)]$

Authored explanation

The map $n\mapsto[(n,0)]$ is an injective semiring homomorphism into the nonnegative integers.

How to interpret this relation type

Map one structure injectively into another by the standard structure-preserving inclusion or representation. This records a canonical copy, not necessarily literal set containment.

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.

Integers $\mathbb Z$ Countable infinity $\aleph_0$

Permalink to relation

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

Relation ID: integers_to_aleph_zero
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: countably infinite

Authored explanation

$\mathbb Z$ is countably infinite.

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

nLab — integer — integer · mathematical reference · source ID nlab-integer
Wikipedia — Aleph number — Aleph number · encyclopedic reference · source ID wp-aleph-number

Integers $\mathbb Z$ Integral domain

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

Relation ID: integers_to_integral_domain
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: forget order

Authored explanation

Forgetting the order leaves an integral domain.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

nLab — integer — integer · mathematical reference · source ID nlab-integer

Integers $\mathbb Z$ Ordered ring

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

Relation ID: integers_to_ordered_ring
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: forget the distinguished integer realization

Authored explanation

Forgetting the canonical origin of its elements leaves the usual ordered-ring structure.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

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

Integers $\mathbb Z$ Integer fraction pairs

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

Relation ID: integers_to_rational_pairs
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form nonzero-denominator fractions

Authored explanation

Use integer numerator–denominator pairs with nonzero denominator 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 — Rational number — Rational number · encyclopedic reference · source ID wp-rational-number
nLab — rational number — rational number · mathematical reference · source ID nlab-rational-number

Integers $\mathbb Z$ Rational numbers $\mathbb Q$

Permalink to relation

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

Relation ID: integers_into_rationals
Relation type: Canonical embedding canonical-embedding
Direction: source → target
Endpoint roles: source: Embeds canonically into; target: Contains a canonical copy of
Authored annotation: $z\mapsto z/1$

Authored explanation

The map $z\mapsto[(z,1)]$ embeds $\mathbb Z$ canonically into its field of fractions.

How to interpret this relation type

Map one structure injectively into another by the standard structure-preserving inclusion or representation. This records a canonical copy, not necessarily literal set containment.

Relation sources

Wikipedia — Rational number — Rational number · encyclopedic reference · source ID wp-rational-number
nLab — rational number — rational number · mathematical reference · source ID nlab-rational-number