Canonical static concept record

First-order signature

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Summary

A vocabulary of function, relation, and constant symbols together with their arities.

Record metadata

Canonical name: First-order signature
Concept ID: first_order_signature
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Logic
Domains: Logic, Universal algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

sets of symbols

Data

arity assignments

Concept sources

Wikipedia — Signature (logic) — Signature (logic) · expository · source ID wp-signature-logic
Open Logic Project — Models and Theories — Open Logic Project: Models and Theories · open textbook · source ID openlogic-model-theory

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Set First-order signature

Permalink to relation

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

Relation ID: set_to_signature
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: choose symbols and arities

Authored explanation

Choose sets of function, relation, and constant symbols with arity assignments.

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 — Signature (logic) — Signature (logic) · expository · source ID wp-signature-logic
Open Logic Project — Models and Theories — Open Logic Project: Models and Theories · open textbook · source ID openlogic-model-theory

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

First-order signature Algebraic signature

Permalink to relation

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

Relation ID: e_signature_algebraic
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: operation symbols only

Authored explanation

Restrict a first-order signature to function symbols and constants.

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 — Signature (logic) — Signature (logic) · expository · source ID wp-signature-logic
nLab — Algebraic theory — Algebraic theory · reference · source ID nlab-algebraic-theory
Burris & Sankappanavar — A Course in Universal Algebra — A Course in Universal Algebra · textbook · source ID burris-universal-algebra

First-order signature First-order theory

Permalink to relation

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

Relation ID: e_signature_theory
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: choose a set of sentences

Authored explanation

Choose a set $T$ of sentences formed in the fixed signature $L$ .

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 — Model theory — Model theory · expository · source ID wp-model-theory
Open Logic Project — Models and Theories — Open Logic Project: Models and Theories · open textbook · source ID openlogic-model-theory

First-order signature signature + carrier(construction junction)

Permalink to relation

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

Relation ID: signature_to_j_structure
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: supply signature

Authored explanation

Use the specified vocabulary and arities.

How to interpret this relation type

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

Relation sources

Wikipedia — Structure (mathematical logic) — Structure (mathematical logic) · expository · source ID wp-structure-logic
Open Logic Project — Models and Theories — Open Logic Project: Models and Theories · open textbook · source ID openlogic-model-theory