Mathematical Relation Ontology

FLOW ALGEBRA
MATHEMATICAL RELATION ONTOLOGY
Kireth Semantic Library | Entity Graph > Mathematical Entities > Foundations
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/foundations-of-mathematical-entities/mathematical-relation-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | 2026-10-12
Status: Subject-based formalisation; source page content not independently verified.

01. PURPOSE
flow MathematicalRelationOntology v1.0:
  intent: represent, validate, compose and infer mathematical relations
  scope: finite/infinite sets, n-ary relations, orders, equivalences, functions, graph relations
  logic: typed first-order logic with explicitly stated assumptions

02. TYPES
  Entity := MathematicalObject
  Set[T] := collection of elements of type T
  Tuple[T1,...,Tn] := ordered n-tuple
  Relation[A,B] := subset(A × B)
  NaryRelation[T1,...,Tn] := subset(T1 × ... × Tn)
  Predicate[A,B] := (A,B) -> Boolean
  Proof := {premises, derivation, conclusion, system}
  Evidence := {source, method, timestamp?, limitations}
  TruthStatus := PROVED | DISPROVED | UNKNOWN | INCONSISTENT

03. CORE ENTITIES
  MathematicalRelation {id, arity, domains, extension?, predicate?, provenance}
  RelationInstance {relation_id, arguments, truth_status, proof?}
  RelationProperty {name, domain_constraints, verification_method}
  RelationComposition {left_relation, right_relation, result_relation}
  RelationGraph {vertices, directed_edges, labels}
  EquivalenceClass {relation, representative, members}
  QuotientSet {base_set, equivalence_relation, classes}
  PartialOrder {carrier, relation}

04. SEMANTIC RELATIONSHIPS
  MathematicalRelation --HAS_DOMAIN--> Set
  MathematicalRelation --HAS_CODOMAIN--> Set
  MathematicalRelation --HAS_ARITY--> NaturalNumber
  RelationInstance --INSTANCE_OF--> MathematicalRelation
  RelationInstance --RELATES--> MathematicalObject
  MathematicalRelation --SATISFIES--> RelationProperty
  MathematicalRelation --COMPOSES_WITH--> MathematicalRelation
  MathematicalRelation --REPRESENTED_BY--> Predicate | Matrix | Graph
  EquivalenceClass --PARTITIONS--> Set
  PartialOrder --INDUCES--> HasseDiagram

05. MEMBERSHIP SEMANTICS
  R ⊆ A × B
  R(a,b) ⇔ (a,b) ∈ R
  dom(R) = {a ∈ A | ∃b ∈ B: R(a,b)}
  ran(R) = {b ∈ B | ∃a ∈ A: R(a,b)}
  inverse(R) = {(b,a) | (a,b) ∈ R}
  identity(A) = {(a,a) | a ∈ A}

06. RELATION OPERATIONS
  union(R,S) = R ∪ S                requires same ambient product
  intersect(R,S) = R ∩ S           requires same ambient product
  difference(R,S) = R \ S          requires same ambient product
  complement(R) = (A × B) \ R      requires declared A and B
  compose(S,R) = {(a,c) | ∃b: R(a,b) ∧ S(b,c)}
  restrict(R,C,D) = R ∩ (C × D)

07. RELATION PROPERTIES (R ⊆ A × A)
  reflexive(R) ⇔ ∀a∈A: R(a,a)
  irreflexive(R) ⇔ ∀a∈A: ¬R(a,a)
  symmetric(R) ⇔ ∀a,b∈A: R(a,b) ⇒ R(b,a)
  antisymmetric(R) ⇔ ∀a,b∈A: (R(a,b) ∧ R(b,a)) ⇒ a=b
  asymmetric(R) ⇔ ∀a,b∈A: R(a,b) ⇒ ¬R(b,a)
  transitive(R) ⇔ ∀a,b,c∈A: (R(a,b) ∧ R(b,c)) ⇒ R(a,c)
  total(R) ⇔ ∀a,b∈A: a≠b ⇒ (R(a,b) ∨ R(b,a))

08. SPECIAL RELATIONS
  equivalence(R) ⇔ reflexive(R) ∧ symmetric(R) ∧ transitive(R)
  preorder(R) ⇔ reflexive(R) ∧ transitive(R)
  partial_order(R) ⇔ preorder(R) ∧ antisymmetric(R)
  total_order(R) ⇔ partial_order(R) ∧ total(R)
  strict_partial_order(R) ⇔ irreflexive(R) ∧ transitive(R)
  tolerance(R) ⇔ reflexive(R) ∧ symmetric(R)
  note: tolerance need not be transitive

09. EQUIVALENCE CLASSES
  requires equivalence(R) on A
  class_R(a) = {x ∈ A | R(x,a)}
  quotient(A,R) = {class_R(a) | a ∈ A}
  rule: R(a,b) ⇒ class_R(a) = class_R(b)
  rule: ¬R(a,b) ⇒ class_R(a) ∩ class_R(b) = ∅
  invariant: quotient classes are nonempty and partition A

10. FUNCTIONS AS RELATIONS
  functional(R) ⇔ ∀a,b1,b2: (R(a,b1) ∧ R(a,b2)) ⇒ b1=b2
  total_on_A(R) ⇔ ∀a∈A: ∃b∈B: R(a,b)
  function_A_to_B(R) ⇔ functional(R) ∧ total_on_A(R)
  injective(R) ⇔ function_A_to_B(R) ∧ ∀a1,a2,b: (R(a1,b) ∧ R(a2,b)) ⇒ a1=a2
  surjective(R) ⇔ function_A_to_B(R) ∧ ∀b∈B: ∃a∈A: R(a,b)
  bijective(R) ⇔ injective(R) ∧ surjective(R)

11. CLOSURE OPERATIONS
  reflexive_closure(R) = R ∪ identity(A)
  symmetric_closure(R) = R ∪ inverse(R)
  transitive_closure(R) = R+ = ⋃_{n≥1} R^n
  reflexive_transitive_closure(R) = R* = identity(A) ∪ R+
  note: composition powers are well-typed on A × A

12. GRAPH REPRESENTATION
  directed_graph(R) = (A, {(a,b) | R(a,b)})
  adjacency_matrix[i,j] = 1 iff R(ai,aj)
  path(a,b) ⇔ R+(a,b)
  strongly_connected(a,b) ⇔ R*(a,b) ∧ R*(b,a)
  note: graph algorithms require explicit finiteness for termination guarantees

13. INFERENCE RULES
  [REFLEXIVITY] reflexive(R), a∈A ⊢ R(a,a)
  [SYMMETRY] symmetric(R), R(a,b) ⊢ R(b,a)
  [TRANSITIVITY] transitive(R), R(a,b), R(b,c) ⊢ R(a,c)
  [ANTISYMMETRY] antisymmetric(R), R(a,b), R(b,a) ⊢ a=b
  [COMPOSITION] R(a,b), S(b,c) ⊢ compose(S,R)(a,c)
  [EQUIVALENCE] equivalence(R), R(a,b) ⊢ class_R(a)=class_R(b)
  guard: never infer R(a,c) from two edges without transitivity or composition semantics

14. VALIDATION
  check arity and argument type for every relation instance
  check declared carrier set before evaluating reflexivity/order properties
  check domain/codomain compatibility before composition
  check proof obligations or produce explicit counterexamples
  distinguish open-world UNKNOWN from closed-world false
  preserve proof system and assumptions

15. UNCERTAINTY AND CONFIDENCE
  CONFIDENCE: {score: Real[0,1], method, evidence_refs, calibration_status}
  formal_theorem: proof_status takes precedence over probabilistic confidence
  empirical_relation: confidence estimates evidential support, not logical truth
  avoid assigning confidence=1 without a valid proof or stipulated axiom

16. SEMANTIC_EQUIVALENCE
  equivalent_relation_descriptions(R,S) requires:
    compatible typed domains AND ∀arguments: (R(arguments) ↔ S(arguments))
  alias labels alone do not establish extensional equivalence
  isomorphism of relation structures does not imply identical carriers

17. PROVENANCE
  record source_uri, source_version, derivation, assumptions, observation_time
  distinguish AXIOM, DEFINITION, THEOREM, EMPIRICAL_ASSERTION, HYPOTHESIS
  record counterexamples and revision history

18. FLOW_HASH
  FLOW_HASH := SHA256(canonical_serialisation({
    ontology_id, version, types, axioms, definitions, rules, dependencies
  }))
  exclude volatile timestamps from canonical payload unless semantically required
  canonicalise ordering, whitespace and Unicode before hashing

19. UPDATE_MODEL
  on update:
    validate_types()
    recheck_affected_properties()
    invalidate_dependent_proofs_if_axioms_change()
    regenerate_equivalence_classes_if_relation_changes()
    recompute_FLOW_HASH()
    record_version_and_provenance()

20. AI_CONSUMPTION
  output:
    ontology: MathematicalRelationOntology
    entity_types: [MathematicalRelation, RelationInstance, RelationProperty,
                   EquivalenceClass, PartialOrder]
    inference_mode: proof-aware, typed, provenance-preserving
    ambiguity_policy: request domain and relation definition when missing
    truth_policy: distinguish PROVED, DISPROVED, UNKNOWN, INCONSISTENT
    semantic_interoperability: graph, predicate, set-theoretic representation

21. EXECUTION EXAMPLE
  let A = {1,2,3}
  let R = {(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}
  assert reflexive(R) = true
  assert antisymmetric(R) = true
  assert transitive(R) = true
  assert total_order(R) = true
  assert symmetric(R) = false
  counterexample symmetric(R): R(1,2) ∧ ¬R(2,1)

22. SAFETY AND LIMITATIONS
  symbolic inference requires sound axioms and valid proof rules
  undecidable or infinite-domain checks may not terminate
  incomplete observations must not be treated as definitive negatives
  source webpage may define narrower terminology than this expansion

END FLOW ALGEBRA