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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Algebra over a field

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Summary

A vector space equipped with a bilinear multiplication, not necessarily associative or unital.

Record metadata

Canonical name: Algebra over a field
Concept ID: algebra_over_field
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)

field $F$
carrier $A$

Data

vector-space operations
bilinear multiplication $A\times A\to A$

Axioms / constraints

bilinearity

Notes

Associativity, a multiplicative identity, commutativity, Lie identities, and similar laws define more specialized kinds of algebra.

Concept sources

Wikipedia — Algebra over a field — Algebra over a field · expository · source ID wp-algebra-field
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

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Vector space Algebra over a field

Permalink to relation

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

Relation ID: vector_space_to_algebra
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + bilinear multiplication

Authored explanation

Choose a bilinear map $A\times A\to A$ ; do not yet require associativity or a unit.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Wikipedia — Algebra over a field — Algebra over a field · expository · source ID wp-algebra-field
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

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Each relation below starts at this concept.