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Summary A partial order in which every pair is comparable.
Record metadata
Canonical name Total order
Concept ID total_order
Content version 2.3.0
Alternate terminology No alternate terminology is currently authored for this record. Axioms / constraints partial-order axioms totality: x ≤ y x \le y x ≤ y or y ≤ x y \le x y ≤ x Incoming relations (arrows to this concept) Each relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Relation ID e_poset_total
Relation type Impose axiom impose-axiom
Direction source → target
Endpoint roles source: Builds toward; target: Built from
Authored annotation + totality Authored explanation Require each pair to be comparable.
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.
Outgoing relations (arrows from this concept) Each relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Relation ID total_to_j_ordered_field
Relation type Combine compatible structures combine-compatible
Direction source → target
Endpoint roles source: Builds toward; target: Built from
Authored annotation supply total order Authored explanation Use the total order on the common carrier.
How to interpret this relation type Feed several structures into a construction junction and impose compatibility between them.
This is an authored directed relation from the source endpoint to the target endpoint.
Relation ID total_to_j_ordered_group
Relation type Combine compatible structures combine-compatible
Direction source → target
Endpoint roles source: Builds toward; target: Built from
Authored annotation supply total order Authored explanation Use the total order on the common carrier.
How to interpret this relation type Feed several structures into a construction junction and impose compatibility between them.
This is an authored directed relation from the source endpoint to the target endpoint.
Relation ID total_to_j_ordered_ring
Relation type Combine compatible structures combine-compatible
Direction source → target
Endpoint roles source: Builds toward; target: Built from
Authored annotation supply total order Authored explanation Use the total order on the common carrier.
How to interpret this relation type Feed several structures into a construction junction and impose compatibility between them.
This is an authored directed relation from the source endpoint to the target endpoint.
Relation ID total_order_to_well_order
Relation type Impose axiom impose-axiom
Direction source → target
Endpoint roles source: Builds toward; target: Built from
Authored annotation require least elements Authored explanation Require every nonempty subset to have a least element.
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.