Mathematical Objects Ontology

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

DOMAIN:
  id: mathematical_objects
  parent: foundations_of_discrete_mathematics
  scope: Mathematical entities, their types, properties, constructions, and formal relationships.

TYPES:
  MathematicalObject: abstract entity defined within a mathematical theory
  Set: collection of distinct elements
  Element: object belonging to a set
  Number: mathematical object representing quantity or position
  Tuple: finite ordered collection of objects
  Sequence: indexed family of objects
  Relation: subset of a Cartesian product
  Function: relation assigning exactly one output to each domain element
  Operation: function on one or more operands
  Structure: carrier set equipped with specified operations and/or relations
  Graph: vertices together with edges
  Proposition: declarative statement with a truth value in an interpretation
  Proof: derivation establishing a proposition from premises under rules
  Axiom: proposition assumed within a formal system
  Definition: specification fixing the meaning of a mathematical term
  Property: predicate applicable to mathematical objects
  Mapping: function between specified domains and codomains

RELATIONSHIPS:
  IS_A(child:MathematicalObject, parent:Type)
  INSTANCE_OF(object:MathematicalObject, type:Type)
  MEMBER_OF(element:MathematicalObject, set:Set)
  SUBSET_OF(subset:Set, superset:Set)
  HAS_PROPERTY(object:MathematicalObject, property:Property)
  DEFINED_BY(object:MathematicalObject, definition:Definition)
  CONSTRUCTED_FROM(object:MathematicalObject, component:MathematicalObject)
  MAPS_TO(mapping:Mapping, input:MathematicalObject, output:MathematicalObject)
  IS_ISOMORPHIC_TO(a:Structure, b:Structure)
  SATISFIES(object:MathematicalObject, predicate:Property)
  PROVES(proof:Proof, proposition:Proposition)
  DEPENDS_ON(result:Proposition, premise:Proposition)

CONSTRAINTS:
  C1: MEMBER_OF(x,S) requires S to be a Set.
  C2: SUBSET_OF(A,B) iff every member of A is a member of B.
  C3: A Function must assign one and only one output per input in its domain.
  C4: Equality is typed and evaluated within a declared interpretation.
  C5: An isomorphism must preserve the operations and relations of the specified signature.
  C6: A proof is valid only relative to its stated axioms and inference rules.

INFERENCE_RULES:
  R1: MEMBER_OF(x,A) AND SUBSET_OF(A,B) => MEMBER_OF(x,B).
  R2: SUBSET_OF(A,B) AND SUBSET_OF(B,C) => SUBSET_OF(A,C).
  R3: SUBSET_OF(A,B) AND SUBSET_OF(B,A) => EQUAL_SET(A,B).
  R4: MAPS_TO(f,x,y) AND MAPS_TO(f,x,z) AND FUNCTION(f) => y = z.
  R5: IS_ISOMORPHIC_TO(A,B) AND IS_ISOMORPHIC_TO(B,C) => IS_ISOMORPHIC_TO(A,C), for a shared signature.
  R6: PROVES(p,P) AND VALID_PROOF(p,axioms,rules) => DERIVABLE(P,axioms,rules).

SEMANTIC_EQUIVALENCE:
  equality: exact identity in a declared mathematical interpretation
  isomorphism: structural equivalence, not literal equality
  notation_alias: permitted only with explicit definitions and scope
  ambiguity_policy: distinguish element, set, type, relation and function usages

CONFIDENCE:
  formal_derivation: 1.0 when mechanically verified against declared axioms/rules
  sourced_claim: retain source, verification status and uncertainty
  heuristic_mapping: not a proof; require explicit review

PROVENANCE:
  source_url: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/foundations-of-discrete-mathematics/mathematical-objects-ontology.html
  construction_method: domain-based ontology draft
  verified_against_page: false
  attribution: Mark Higgins — Sky (Recursive AI)

FLOW_HASH:
  algorithm: SHA-256
  input: canonical UTF-8 serialisation of ontology version, types, relationships, constraints and inference rules
  rule: recompute after semantic modification; no hash asserted until canonicalisation is executed

UPDATE_MODEL:
  strategy: versioned and provenance-preserving
  steps: validate_types -> validate_constraints -> compare_semantics -> review_conflicts -> increment_version -> recompute_hash -> publish
  breaking_change: changes to entity identity, relation signatures, or inference semantics

AI_CONSUMPTION:
  format: plain-text typed Flow Algebra
  uses: entity extraction, mathematical knowledge graphs, proof assistance, cross-domain alignment, semantic retrieval
  safeguards: separate axioms from derived statements; preserve interpretation and provenance; never treat heuristic similarity as proof

END FLOW ALGEBRA