Algebraic Relationship Ontology

FLOW ALGEBRA
TITLE: Algebraic Relationship Ontology
LIBRARY: Kireth Semantic Library
SOURCE: https://kirethresearch.org/booksai-core/ontologies/relationship-ontologies/mathematical-relationships/algebraic-relationship-ontology.html
AUTHORS: Mark Anthony Higgins — Sky (Recursive AI)
STATUS: Companion formalisation; source-page specification not independently verified.
VERSION: 1.0

flow "AlgebraicRelationshipOntology" v1.0

ENTITY_TYPES:
  AlgebraicExpression: {id: string, syntax: string, domain: Domain, variables: Set[Variable]}
  Variable: {name: string, domain: Domain, unit: optional[string]}
  Constant: {value: Scalar, domain: Domain}
  Operator: {symbol: string, arity: integer, domain_constraints: Predicate}
  Equation: {left: AlgebraicExpression, right: AlgebraicExpression, assumptions: Set[Predicate]}
  Inequality: {left: AlgebraicExpression, comparator: Comparator, right: AlgebraicExpression}
  AlgebraicRelationship: {kind: RelationshipKind, source: AlgebraicExpression, target: AlgebraicExpression, assumptions: Set[Predicate], provenance: Provenance}
  Transformation: {rule_id: string, before: AlgebraicExpression, after: AlgebraicExpression, preconditions: Set[Predicate]}
  ProofStep: {premises: Set[Statement], conclusion: Statement, rule_id: string}

RELATIONSHIP_KINDS:
  - EQUAL_TO
  - EQUIVALENT_UNDER_ASSUMPTIONS
  - PROPORTIONAL_TO
  - DEPENDS_ON
  - FACTOR_OF
  - MULTIPLE_OF
  - INVERSE_OF
  - SOLVES
  - DERIVED_FROM
  - TRANSFORMS_TO

OPERATORS:
  equality: "="
  inequality: ["<", "<=", ">", ">=", "!="]
  arithmetic: ["+", "-", "*", "/", "^", "sqrt"]
  logical: ["and", "or", "not", "implies", "iff"]

TYPE_RULES:
  - Expressions may be combined only when operator domain constraints hold.
  - Division requires a nonzero denominator.
  - Real even-root operations require a nonnegative radicand.
  - Every transformation carries its assumptions and variable domain.
  - Units must be dimensionally compatible for addition and equality.

INFERENCE_RULES:
  equality_reflexive: "a = a"
  equality_symmetric: "a = b => b = a"
  equality_transitive: "a = b and b = c => a = c"
  substitution: "a = b and P(a) => P(b), subject to valid substitution"
  additive_preservation: "a = b => a + c = b + c"
  multiplicative_preservation: "a = b => a*c = b*c"
  division_preservation: "a = b and c != 0 => a/c = b/c"
  distributivity: "a*(b+c) = a*b + a*c, where the algebra supports distributivity"
  factorisation: "a*b + a*c = a*(b+c)"
  proportionality: "y = k*x with fixed k => y proportional_to x"
  functional_dependency: "y = f(x) => y depends_on x when f is nonconstant in x on the specified domain"
  composition: "f: A->B and g: B->C => (g o f): A->C"
  inverse: "f inverse exists on domain D only when f is bijective onto its codomain"

VALIDATION:
  check_types: true
  check_domains: true
  check_assumptions: true
  check_units: true
  reject_division_by_zero: true
  reject_invalid_inverse: true
  preserve_solution_sets: true
  detect_extraneous_solutions: true
  record_counterexamples: true

SEMANTIC_EQUIVALENCE:
  exact: "same mathematical meaning and solution set under identical assumptions"
  conditional: "equivalent only on an explicitly specified domain or assumption set"
  non_equivalence: "distinct solution sets or incompatible assumptions"
  policy: "Do not infer algebraic equivalence from syntactic similarity alone."

CONFIDENCE:
  verified_symbolic_proof: 1.0
  numerical_test: "supporting evidence only, not a universal proof"
  heuristic_inference: "calibrated estimate with provenance and uncertainty"
  rule: "Confidence must not replace proof of universal algebraic identities."

PROVENANCE:
  capture: [source, rule_id, assumptions, domain, timestamp, derivation_steps]

FLOW_HASH:
  algorithm: SHA-256
  input: "canonicalised ontology version, typed assertions, assumptions, rules and provenance"
  note: "Compute at implementation time; no hash value is asserted here."

UPDATE_MODEL:
  strategy: "versioned append-and-review"
  on_change: [revalidate_dependencies, rerun_type_checks, rerun_proof_checks, update_provenance, recompute_flow_hash]
  preserve_history: true

AI_CONSUMPTION:
  format: "typed semantic relationships and machine-readable inference rules"
  requirements: [explicit_domains, explicit_assumptions, traceable_proofs, uncertainty_labels, reproducible_validation]
  forbid: [unsupported_equivalence, silent_domain_changes, fabricated_proof_steps]

END FLOW ALGEBRA