antisymmetric relation
Let an agent explain antisymmetric relations, relay definitions, examples and connections to order theory from mathematics sources, describe the named relation types and flag the lower bound alias, and distinguish antisymmetry from asymmetry, symmetry, non-symmetry and total orders.
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 antisymmetric relations, relay definitions, examples and connections to order theory from mathematics sources, describe the named relation types and flag the lower bound alias, and distinguish antisymmetry from asymmetry, symmetry, non-symmetry and total orders.
A binary relation R on a set such that whenever a R b and b R a, then a equals b, meaning no two distinct elements are related in both directions, as with less than or equal to and divisibility on the natural numbers; asymmetric relations, where a R b excludes b R a, are exactly the antisymmetric and irreflexive ones, and antisymmetry together with reflexivity and transitivity defines partial orders such as lexicographical and pointwise orders, while the registry alias lower bound is an element-level concept in ordered sets rather than a relation type.
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
No distinct mutual pairs
Basis of partial orders
Compatible with reflexivity
Differs from asymmetry
What it looks like
Not a visible object; seen in diagrams without two-way arrows between distinct points.
Physical character
definition: aRb and bRa imply a = b formula
partial order: reflexive, antisymmetric, transitive list
registry parent: endorelation note
How it is recognised
Relation with no distinct two-way pairs
Asymmetric relation, asymmetric property, partial order, lexicographical order, pointwise order, lower bound
Asymmetric relations are also irreflexive; symmetric relations go both ways; non-symmetric only means not symmetric; total orders compare all pairs
Related models
is a kind of - in registry terms
underlies -
includes - as irreflexive case
is contrasted with -
In practice
Families and kinds
partial orders
total orders
lexicographical orders
pointwise orders
asymmetric relations as a subclass
Standards and regulation
No regulation
Failure modes and hazards
Confusing antisymmetric and asymmetric
Treating non-symmetric as antisymmetric
Registry alias for an element concept
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 an antisymmetric relation is.
Science.
Definition
Definition.
Definition
Definition.
- What is an antisymmetric relation, and how does it differ from asymmetric, symmetric and non-symmetric relations? definition
- Is the question about maths, databases or logic? boundary
Named
Named concepts.
Named
Named.
- What are asymmetric relations, partial orders, lexicographical and pointwise orders, and lower bounds? definition
- Which entry fits the specific concept? action
Order theory Order theory.
Science.
Partial orders
Partial orders.
Partial orders
Partial orders.
- How does antisymmetry help define partial orders? provenance
- Which references are standard? provenance
Bounds
Bounds.
Bounds
Bounds.
- What are lower and upper bounds in ordered sets? provenance
- Which sources are cited? provenance
Examples Examples.
Application.
Numbers
Number relations.
Numbers
Numbers.
- Why is divisibility on natural numbers antisymmetric but not on integers? provenance
- Which entry fits divisibility? action
Orders
Lexicographic order.
Orders
Orders.
- How does lexicographic ordering sort words? provenance
- Which entry fits lexicographic order? action
Context Related properties.
Context.
Properties
Relation properties.
Properties
Properties.
- How do reflexive, symmetric, antisymmetric and transitive properties combine? provenance
- Which entry fits binary relation? action
Graphs
Graphs.
Graphs
Graphs.
- How are antisymmetric relations drawn as directed graphs and Hasse diagrams? provenance
- Which entry fits Hasse diagram? action
What the second pass must settle
- Should asymmetric relation be a separate primary entry?
- How should mathematics sources be linked?
- The registry alias lower bound should be moved to order theory element concepts