Mathematical Structure Ontology

FLOW ALGEBRA
TITLE: Mathematical Structure Ontology
VERSION: 1.0
SOURCE: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/foundations-of-discrete-mathematics/mathematical-structure-ontology.html
STATUS: Domain-based formalisation; source page not independently verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

FLOW "MathematicalStructureOntology" v1.0

DOMAIN:
  name: Mathematical Structure
  parent: Foundations of Discrete Mathematics
  scope: Formal objects composed of carrier sets, operations, relations, constants and axioms.

TYPES:
  MathematicalObject: entity
  CarrierSet: Set[MathematicalObject]
  Operation: Function[CarrierSet^n -> CarrierSet]
  Relation: Subset[CarrierSet^n]
  Constant: Element[CarrierSet]
  Axiom: LogicalFormula
  Signature: Tuple[OperationSymbols, RelationSymbols, ConstantSymbols, Arities]
  Structure: Tuple[Signature, CarrierSet, Interpretations]
  Homomorphism: StructurePreservingMap
  Isomorphism: BijectiveHomomorphismWithHomomorphicInverse
  Substructure: Structure
  Property: Predicate[Structure]
  Proof: Derivation

RELATIONSHIPS:
  has_carrier(Structure, CarrierSet)
  has_signature(Structure, Signature)
  interprets(Structure, Symbol, Operation|Relation|Constant)
  satisfies(Structure, Axiom)
  models(Structure, Theory)
  substructure_of(Substructure, Structure)
  homomorphic_to(Structure, Structure)
  isomorphic_to(Structure, Structure)
  preserves(Homomorphism, Operation|Relation)
  has_property(Structure, Property)

CONSTRAINTS:
  C1: Every interpreted operation has the arity specified by its signature.
  C2: Every operation is closed over the appropriate carrier sort(s).
  C3: Relation interpretations have the signature-declared arity and sorts.
  C4: A substructure is closed under inherited operations and contains inherited constants.
  C5: Isomorphism requires bijectivity and preservation and reflection of structure.
  C6: Axioms are not assumed true without model satisfaction or proof.

INFERENCE_RULES:
  R1: isomorphic_to(A,B) AND has_property(A,P) AND isomorphism_invariant(P) => has_property(B,P)
  R2: isomorphic_to(A,B) => isomorphic_to(B,A)
  R3: isomorphic_to(A,B) AND isomorphic_to(B,C) => isomorphic_to(A,C)
  R4: substructure_of(A,B) AND substructure_of(B,C) => substructure_of(A,C)
  R5: models(S,T) AND entails(T,phi) => satisfies(S,phi)
  R6: homomorphic_to(A,B) does NOT generally imply isomorphic_to(A,B)

EXAMPLES:
  - Graph: vertices as carrier; adjacency as binary relation.
  - Group: carrier; binary operation; identity; inverse operation; group axioms.
  - Poset: carrier; partial-order relation; reflexive, antisymmetric, transitive axioms.
  - Ring: carrier; addition and multiplication operations; ring axioms.

CONFIDENCE:
  value_range: [0.0,1.0]
  semantics: Epistemic confidence in an asserted fact or mapping; not mathematical truth.
  require_evidence: true

SEMANTIC_EQUIVALENCE:
  canonicalise_terms: ["mathematical structure", "relational structure", "algebraic structure"]
  caution: Related terms are not universally interchangeable; preserve signature and axioms.
  equivalence_test: Compare signatures, interpretations and declared notion of equivalence.

PROVENANCE:
  source_url: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/foundations-of-discrete-mathematics/mathematical-structure-ontology.html
  derivation: Domain-based draft using standard mathematical definitions.
  verification_status: unverified_source_content
  attribution: Mark Higgins — Sky (Recursive AI)

FLOW_HASH:
  algorithm: SHA-256
  input: Canonical UTF-8 serialisation of version, types, constraints and inference rules.
  rule: Recalculate after every substantive update; never treat placeholder as computed hash.

UPDATE_MODEL:
  strategy: versioned_append_and_validate
  on_change: [revalidate_types, check_axioms, rerun_inference, update_provenance, recompute_flow_hash]
  backward_compatibility: Preserve stable entity identifiers or record mappings.

AI_CONSUMPTION:
  format: text/plain; charset=utf-8
  entrypoint: MathematicalStructureOntology
  tasks: [ontology_alignment, structure_classification, symbolic_reasoning, theorem_support, knowledge_graph_integration]
  safeguards: [type_check, axiom_check, distinguish_proof_from_hypothesis, provenance_check]
  output: Typed claims with derivations, confidence and source attribution.

END FLOW