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Canonical static concept record

Commutative monoid

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Summary

A monoid whose operation is commutative.

Record metadata

Canonical name: Commutative monoid
Concept ID: commutative_monoid
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebra
Domains: Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

set $M$

Data

binary operation
identity

Axioms / constraints

monoid axioms
$x\cdot y = y\cdot x$

Concept sources

Wikipedia — Commutative monoid — Commutative monoid · expository · source ID wp-comm-monoid
nLab — Commutative monoid — Commutative monoid · mathematical reference · source ID nlab-commutative-monoid
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Abelian group Commutative monoid

Permalink to relation

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

Relation ID: abelian_group_to_commutative_monoid
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 additive inverses

Authored explanation

Retain the commutative associative operation and identity.

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 — Abelian group — Abelian group · expository · source ID wp-abelian
Wikipedia — Commutative monoid — Commutative monoid · expository · source ID wp-comm-monoid
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Monoid Commutative monoid

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

Relation ID: e_monoid_comm
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + commutativity

Authored explanation

Require $xy=yx$ .

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Commutative monoid — Commutative monoid · expository · source ID wp-comm-monoid
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.