Select any concept, construction junction, or annotated relation by pointer, touch, search, or keyboard.

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

Canonical static concept record

Algebra for a finitary signature ( $\Omega$ -algebra)

Open this concept in the interactive graph Read the Markdown equivalent

Summary

A nonempty set interpreting each operation symbol of a fixed finitary algebraic signature.

Record metadata

Canonical name: Algebra for a finitary signature ( $\Omega$ -algebra)
Concept ID: universal_algebra
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Universal algebra
Domains: Universal algebra, Algebra, Logic
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

nonempty set $A$
finitary algebraic signature $\Omega$

Data

an operation $A^{n}\to A$ for each $n$ -ary symbol

Notes

“Universal algebra” is the subject; an individual object is an algebra for a signature, often called an $\Omega$ -algebra. This atlas adopts the common nonempty-carrier convention.

Concept sources

Wikipedia — Universal algebra — Universal algebra · expository · source ID wp-universal-algebra
nLab — Universal algebra — Universal algebra · reference · source ID nlab-universal-algebra
Burris & Sankappanavar — A Course in Universal Algebra — A Course in Universal Algebra · textbook · source ID burris-universal-algebra

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

First-order structure Algebra for a finitary signature ( $\Omega$ -algebra)

Permalink to relation

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

Relation ID: e_fostruct_universal
Relation type: Representation / equivalence representation-equivalence
Direction: source → target
Endpoint roles: source: Representation / equivalence with; target: Representation / equivalence with
Authored annotation: specialize to an algebraic signature

Authored explanation

A first-order structure whose signature has only function symbols (including constants as nullary operations) is exactly an $\Omega$ -algebra.

How to interpret this relation type

Reinterpret an object, pass to an equivalent presentation, or relate canonically corresponding structures; the carrier may change.

Relation sources

Wikipedia — Universal algebra — Universal algebra · expository · source ID wp-universal-algebra
nLab — Universal algebra — Universal algebra · reference · source ID nlab-universal-algebra
Burris & Sankappanavar — A Course in Universal Algebra — A Course in Universal Algebra · textbook · source ID burris-universal-algebra

signature + carrier operations(construction junction)Algebra for a finitary signature ( $\Omega$ -algebra)

Permalink to relation

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

Relation ID: j_universal_to_algebra
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: interpret every operation symbol

Authored explanation

Choose operations of the required arities on the carrier.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Wikipedia — Universal algebra — Universal algebra · expository · source ID wp-universal-algebra
nLab — Universal algebra — Universal algebra · reference · source ID nlab-universal-algebra
Burris & Sankappanavar — A Course in Universal Algebra — A Course in Universal Algebra · textbook · source ID burris-universal-algebra

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.