Antisymmetric Relation Ontology

FLOW ALGEBRA
ANTISYMMETRIC RELATION ONTOLOGY
Kireth Semantic Library | Entity Graph | Mathematical Entities | Relation Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/relation-entities/antisymmetric-relation-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | 2026-10-13 | Locale: en-GB
Source status: Formal specification based on established mathematics; source page not independently verified.

01. PURPOSE
flow "AntisymmetricRelationOntology" v1.0
  domain: set_theory.relation_theory
  entity_type: AntisymmetricRelation
  intent: classify, validate, reason, transform, explain
  formalism: classical_two_valued_logic

02. SIGNATURE
  sorts: Set, Element, BinaryRelation, Proposition, Proof, Graph
  R: subset(A x A)
  contains(R,a,b): Boolean
  antisymmetric(R,A): Boolean
  inverse(R): BinaryRelation
  intersection(R,S): BinaryRelation
  identity(A): BinaryRelation

03. DEFINITION
  antisymmetric(R,A) iff
    forall a,b in A: (R(a,b) and R(b,a)) implies a=b
  equivalent_condition:
    intersection(R,inverse(R)) subseteq identity(A)
  counterexample:
    exists a,b in A: a != b and R(a,b) and R(b,a)

04. DOMAIN CONSTRAINTS
  require R subseteq A x A
  require all compared endpoints belong to A
  require equality semantics defined for A
  if domain unspecified: result UNKNOWN, not FALSE

05. CORE AXIOMS AND CONSEQUENCES
  AS1: forall a,b: R(a,b) and R(b,a) -> a=b
  AS2: forall a!=b: R(a,b) -> not R(b,a)
  AS3: intersection(R,inverse(R)) subseteq identity(A)
  AS4: any subset S of antisymmetric R is antisymmetric
  AS5: antisymmetry does not imply reflexivity
  AS6: antisymmetry does not imply transitivity
  AS7: antisymmetry does not imply totality

06. DISTINCTIONS
  symmetric: R(a,b) -> R(b,a)
  asymmetric: R(a,b) -> not R(b,a), including a=b
  antisymmetric: mutual edges permitted only when endpoints equal
  irreflexive: not R(a,a) for every a
  asymmetric -> antisymmetric and irreflexive
  antisymmetric does not imply asymmetric
  symmetric and antisymmetric -> R subseteq identity(A)
  symmetric is NOT the negation of antisymmetric

07. EXAMPLES
  E1: <= on real numbers: antisymmetric TRUE
  E2: subseteq on sets: antisymmetric TRUE
  E3: divides on positive integers: antisymmetric TRUE
  E4: divides on all integers: antisymmetric FALSE
      witness: 2 divides -2 and -2 divides 2, but 2 != -2
  E5: equality: antisymmetric TRUE
  E6: strict less-than: antisymmetric TRUE (also asymmetric)
  E7: universal relation on a two-element set: FALSE
  E8: empty relation on any set: TRUE
  E9: symmetric friendship between distinct people: generally FALSE

08. FINITE VALIDATION
  function is_antisymmetric(A,R):
    assert R subseteq A x A
    for each (a,b) in R:
      if a != b and (b,a) in R:
        return FALSE with witness (a,b)
    return TRUE
  complexity: O(|R|) expected with hash-set membership
  alternative: O(|A|^2) with adjacency matrix

09. ADJACENCY MATRIX
  for finite ordered A, M[i,j]=1 iff R(ai,aj)
  antisymmetric iff for every i!=j:
    not (M[i,j]=1 and M[j,i]=1)
  diagonal entries unrestricted

10. DIRECTED GRAPH SEMANTICS
  vertices: elements of A
  directed_edges: ordered pairs in R
  valid: self-loops allowed
  invalid: a two-way edge between distinct vertices
  note: directed cycles of length >=3 may exist

11. RELATIONAL OPERATIONS
  inverse: R antisymmetric iff inverse(R) antisymmetric
  intersection: intersection of antisymmetric relations is antisymmetric
  union: not generally preserves antisymmetry
  composition: not generally preserves antisymmetry
  transitive_closure: not generally preserves antisymmetry
  reflexive_closure: preserves antisymmetry
  restriction_to_subset: preserves antisymmetry

12. ORDER RELATION CONNECTIONS
  partial_order(R): reflexive(R) and antisymmetric(R) and transitive(R)
  total_order(R): partial_order(R) and comparable_all_pairs(R)
  preorder(R): reflexive(R) and transitive(R)
  partial_order(R) -> antisymmetric(R)
  preorder(R) does not necessarily imply antisymmetric(R)
  strict_partial_order(R): irreflexive(R) and transitive(R)
  strict_partial_order(R) -> asymmetric(R) -> antisymmetric(R)

13. INFERENCE RULES
  rule mutual_edges_imply_identity:
    premises: antisymmetric(R,A), R(a,b), R(b,a)
    conclusion: a=b
  rule distinct_forward_excludes_reverse:
    premises: antisymmetric(R,A), a!=b, R(a,b)
    conclusion: not R(b,a)
  rule witness_refutes:
    premises: a!=b, R(a,b), R(b,a)
    conclusion: not antisymmetric(R,A)
  rule partial_order_entails:
    premises: partial_order(R,A)
    conclusion: antisymmetric(R,A)

14. KNOWLEDGE GRAPH ENTITY SCHEMA
  AntisymmetricRelation:
    id: stable_uri
    domain_set: Set.id
    ordered_pairs: optional finite extension
    predicate_definition: optional formal expression
    antisymmetry_status: TRUE | FALSE | UNKNOWN
    proof_reference: optional Proof.id
    counterexample_pair: optional [Element.id,Element.id]
    assumptions: list
    provenance: source_uri
    confidence: evidence-dependent

15. KNOWLEDGE GRAPH EDGES
  AntisymmetricRelation --is_a--> BinaryRelation
  AntisymmetricRelation --defined_over--> Set
  AntisymmetricRelation --constrains--> ReciprocalPairs
  PartialOrder --requires_property--> Antisymmetry
  TotalOrder --requires_property--> Antisymmetry
  AsymmetricRelation --implies_property--> Antisymmetry
  AntisymmetricRelation --contrasts_with--> SymmetricRelation
  AntisymmetricRelation --has_counterexample--> DistinctMutualPair

16. SEMANTIC EQUIVALENCE
  equivalence:
    forall a,b: (R(a,b) and R(b,a)) -> a=b
    forall a!=b: not (R(a,b) and R(b,a))
    R intersect R_inverse subseteq diagonal(A)
  preserve domain and equality interpretation when rewriting
  avoid conflating antisymmetric with asymmetric

17. VALIDATION AND TESTS
  test singleton_universal: TRUE
  test empty_relation: TRUE
  test identity_relation: TRUE
  test one_direction_edge: TRUE
  test two_direction_distinct: FALSE
  test loops_and_one_direction_edge: TRUE
  test directed_three_cycle: TRUE
  test invalid_outside_domain: ERROR
  test unknown_predicate: UNKNOWN unless proof supplied

18. CONFIDENCE
  proof_based: 1.0 relative to accepted assumptions
  exhaustive_finite_check: 1.0 relative to complete input
  sampled_relation: insufficient for universal confirmation
  unknown_or_incomplete_data: UNKNOWN
  never infer global antisymmetry from absence of sampled counterexamples

19. FLOW HASH
  canonicalise:
    - normalise ontology URI and version
    - normalise typed variables and predicates
    - preserve assumptions, domain, equality and quantifier scope
    - sort unordered metadata fields only
  FLOW_HASH: SHA-256(canonical_serialisation)
  note: calculate digest on deployment; no fabricated hash value

20. UPDATE MODEL
  triggers:
    - changed source page
    - modified formal definitions
    - changed relation extension or domain
    - corrected examples or counterexamples
  steps:
    - increment version
    - revalidate proofs and tests
    - recompute FLOW_HASH
    - retain provenance and revision history

21. AI CONSUMPTION
  inputs: relation specification, domain, assumptions, query
  outputs: classification, proof_or_witness, explanation, provenance
  policy:
    - use three-valued status TRUE/FALSE/UNKNOWN for incomplete evidence
    - do not substitute probabilistic confidence for logical validity
    - return distinct reciprocal pair as a negative certificate
    - preserve type constraints and source attribution

22. SUMMARY INVARIANT
  antisymmetric(R,A) <=> forall a,b in A:
    (R(a,b) and R(b,a)) => (a=b)
  essential_warning: antisymmetric != asymmetric

END FLOW ALGEBRA