transitive relation
Let an agent explain transitive relations, relay definitions, examples and related order concepts from mathematics and logic sources, describe the named order types and closures, flag the Wikidata property aliases, and distinguish transitivity from reflexivity, symmetry, quasitransitivity and non-transitive relations such as rock-paper-scissors dominance.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent explain transitive relations, relay definitions, examples and related order concepts from mathematics and logic sources, describe the named order types and closures, flag the Wikidata property aliases, and distinguish transitivity from reflexivity, symmetry, quasitransitivity and non-transitive relations such as rock-paper-scissors dominance.
A binary relation R on a set such that whenever a R b and b R c, then a R c, as with less than, ancestor of or equality, underpinning preorders, which are reflexive and transitive, strict orders, which are irreflexive and transitive, and dense preorders, while the transitive closure of a relation is the smallest transitive relation containing it; the registry aliases follows and followed by are Wikidata sequence properties, which are not transitive in themselves, and appear misfiled.
What it is for: Not applicable; a mathematical property.
It can be explain the property; relay examples; describe named order types; distinguish related properties.
Distinguishing features
Chaining property
Basis of orders
Closure operation
Aliases include data properties
What it looks like
Not a visible object; seen in diagrams as paths implying direct arrows.
Physical character
definition: for all a, b, c: aRb and bRc imply aRc formula
Warshall algorithm: O(n^3) complexity - transitive closure
registry parents: endorelation, quasitransitive relation note
How it is recognised
Relation where a R b and b R c imply a R c
Strict order, preorder, transitive closure, dense preorder, follows, followed by
Reflexive relations relate each element to itself; symmetric ones go both ways; quasitransitive relations are weaker; follows and followed by are sequence properties
Related models
is a kind of - in registry terms
underlies -
has closure -
is contrasted with -
In practice
Families and kinds
preorders
partial orders
strict orders
equivalence relations
dense orders
transitive closures
Standards and regulation
No regulation
Failure modes and hazards
Assuming preferences are always transitive
Registry aliases from Wikidata properties
Confusing transitive and quasitransitive
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Understand What a transitive relation is.
Science.
Definition
Definition.
Definition
Definition.
- What is a transitive relation, and how does it differ from reflexive, symmetric and quasitransitive relations? definition
- Is the question about maths, logic, databases, preferences or grammar s transitive verbs? boundary
Named
Named concepts.
Named
Named.
- What are strict orders, preorders, dense preorders and transitive closures, and why are follows and followed by different? definition
- Which entry fits the specific concept? action
Mathematics Mathematics.
Science.
Orders
Order theory.
Orders
Orders.
- How do transitive relations build order theory? provenance
- Which references are standard? provenance
Closure
Transitive closure.
Closure
Closure.
- How is the transitive closure computed, for example by Warshall s algorithm? provenance
- Which sources are cited? provenance
Applications Applications.
Application.
Graphs
Graphs and databases.
Graphs
Graphs.
- How is transitivity used in graph reachability and database queries? provenance
- Which entry fits reachability? action
Preferences
Preferences.
Preferences
Preferences.
- Why is transitivity of preferences debated in economics, with evidence attributed? provenance
- Which entry fits rational choice theory? action
Context Logic.
Context.
Logic
Logic.
Logic
Logic.
- How is transitivity expressed in syllogisms and description logics? provenance
- Which entry fits syllogism? action
Non-transitive
Non-transitive examples.
Non-transitive
Non-transitive.
- What are examples of non-transitive relations such as nontransitive dice? provenance
- Which entry fits nontransitive dice? action
What the second pass must settle
- Should preorder and strict order be separate primary entries?
- How should mathematics sources be linked?
- The registry aliases follows and followed by should be removed or moved to Wikidata property entries