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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

Totally ordered group

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Summary

A group equipped with a translation-invariant total order.

Record metadata

Canonical name: Totally ordered group
Concept ID: ordered_group
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Order theory
Domains: Order theory, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

set $G$

Data

group structure
total order

Axioms / constraints

$a\le b$ implies $xay\le xby$ for all $x,y$ (equivalently, left and right translations preserve order)

Notes

“Ordered group” can mean a partially ordered group in some literature. This node specifically means a totally ordered group.

Concept sources

Wikipedia — Ordered group — Ordered group · expository · source ID wp-ordered-group
Cambridge University Press — A Primer on Ordered Sets and Lattices — A Primer on Ordered Sets and Lattices · textbook chapter · source ID gierz-order

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

group + total order(construction junction)Totally ordered group

Permalink to relation

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

Relation ID: j_ordered_group_to_output
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: require translation invariance

Authored explanation

Require the order to be invariant under the group action by translation.

How to interpret this relation type

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

Relation sources

Wikipedia — Ordered group — Ordered group · expository · source ID wp-ordered-group
Cambridge University Press — A Primer on Ordered Sets and Lattices — A Primer on Ordered Sets and Lattices · textbook chapter · source ID gierz-order

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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