Multivalued Function Ontology

FLOW ALGEBRA
MULTIVALUED FUNCTION ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Function and Mapping Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/function-and-mapping-entities/multivalued-function-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13 | Locale: en-GB
Status: Formal ontology synthesis based on established mathematics; source page not independently verified.

01. PURPOSE
flow "MultivaluedFunctionOntology" v1.0
  intent: represent mathematical correspondences assigning zero, one or several outputs to an input, with explicit semantics.
  scope: set-valued maps, relations, branches, inverse correspondences, domains, graphs and validation.
  exclusion: do not silently equate a multivalued correspondence with an ordinary single-valued function.

02. CORE TYPES
  Set: collection of elements.
  Domain X: set of admissible inputs.
  Codomain Y: set of potential outputs.
  PowerSet(Y): collection of all subsets of Y.
  NonemptyPowerSet(Y): PowerSet(Y) excluding empty set.
  Correspondence F: X -> PowerSet(Y).
  TotalNonemptyCorrespondence: F(x) != empty for every x in X.
  PartialCorrespondence: some x in X may have F(x) = empty.
  SingleValuedFunction f: X -> Y, equivalent to singleton-valued F(x) = {f(x)}.
  Relation R: subset of X x Y, representing graph of F.

03. FORMAL SEMANTICS
  F(x) = {y in Y : (x,y) in R}.
  graph(F) = {(x,y) in X x Y : y in F(x)}.
  effective_domain(F) = {x in X : F(x) != empty}.
  range(F) = union over x in X of F(x).
  fibre_at_y(F) = {x in X : y in F(x)}.
  typed distinction: range(F) subset_of Y, while F(x) subset_of Y.

04. ONTOLOGY ENTITIES
  MultivaluedFunction, SetValuedMap, Correspondence, Relation, Branch,
  Input, Output, OutputSet, Domain, Codomain, EffectiveDomain,
  Graph, Selection, InverseCorrespondence, Constraint, ProvenanceRecord.

05. ENTITY ATTRIBUTES
  id: unique stable identifier.
  label: human-readable term.
  source_domain: set identifier.
  target_codomain: set identifier.
  output_set_rule: symbolic expression or extensional lookup.
  definedness_policy: nonempty_total | may_be_empty.
  branch_policy: labelled | unlabelled | contextual.
  semantic_context: mathematics | analysis | optimisation | complex_analysis | other.
  evidence: proof | definition | verified_computation | assertion.

06. TYPED RELATIONSHIPS
  has_domain(F,X)
  has_codomain(F,Y)
  maps_input_to_set(F,x,F(x))
  has_graph(F,R)
  has_branch(F,b)
  has_selection(F,s)
  is_inverse_correspondence_of(G,f)
  is_equivalent_to(F,G,context)
  depends_on(F,assumption)
  is_subrelation_of(R,S)

07. WELL-FORMEDNESS
  require X and Y are sets.
  require for each x in X: F(x) subset_of Y.
  require every graph pair (x,y) has x in X and y in Y.
  require graph(F) subset_of X x Y.
  if total_nonempty: require all F(x) nonempty.
  if singleton-valued: ordinary function can be extracted uniquely.
  if empty-valued inputs exist: preserve definedness status explicitly.

08. EXAMPLES
  square_root_correspondence: X = nonnegative_reals; Y = reals;
    F(x) = {y : y*y = x}; F(4) = {-2,2}; F(0) = {0}.
  inverse_of_square: f: reals -> nonnegative_reals, f(x)=x*x;
    f_inverse_as_correspondence(y) = {x : x*x=y}.
  finite_example: X={a,b}, Y={1,2,3}; F(a)={1,2}; F(b)={3}.
  empty_example: F(x)={y in reals : y*y=x} on X=reals; F(-1)=empty.

09. FUNCTION VERSUS CORRESPONDENCE
  ordinary_function: exactly one output for each input in its domain.
  correspondence: each input receives a set of possible outputs.
  multivalued: at least one input has more than one output.
  totality and single-valuedness are independent properties.
  term "multivalued function" is conventional; use typed set-valued semantics.

10. SELECTIONS AND BRANCHES
  selection s: D -> Y where D subset_of effective_domain(F).
  require for all x in D: s(x) in F(x).
  global_selection: D = effective_domain(F).
  branch: locally defined selection subject to context and regularity conditions.
  branch properties: continuous | differentiable | analytic | unspecified.
  branch existence is not assumed without proof or applicable selection theorem.

11. INVERSE CORRESPONDENCE
  for f: X -> Y define f^(-1)[y] = {x in X : f(x)=y}.
  inverse relation R^(-1) = {(y,x) : (x,y) in R}.
  injective f implies each inverse fibre contains at most one element.
  surjective f implies each inverse fibre is nonempty.
  bijective f implies inverse correspondence is singleton-valued and total.

12. COMPOSITION
  F: X -> PowerSet(Y), G: Y -> PowerSet(Z).
  (G o F)(x) = union {G(y) : y in F(x)}.
  empty input image yields empty composed image.
  graph(G o F) = graph(G) relationally_composed_with graph(F).
  composition is associative for well-typed correspondences.
  identity I_X(x) = {x}; F o I_X = F and I_Y o F = F.

13. IMAGE AND PREIMAGE OF SETS
  F[A] = union {F(x) : x in A} for A subset_of X.
  existential_preimage(B) = {x in X : F(x) intersects B is nonempty}.
  universal_preimage(B) = {x in X : F(x) subset_of B}.
  distinguish existential and universal semantics; they are not interchangeable.
  empty-valued inputs satisfy subset_of B but not intersection_nonempty B.

14. ALGEBRAIC OPERATIONS
  pointwise_union(F,G)(x) = F(x) union G(x).
  pointwise_intersection(F,G)(x) = F(x) intersection G(x).
  restriction(F,A)(x) = F(x) for x in A subset_of X.
  pointwise_subset(F,G) iff for every x: F(x) subset_of G(x).
  equality(F,G) iff same typed domain/codomain and all output sets equal.
  if codomains differ, use extensional equivalence with explicit coercion policy.

15. COMPLEX ANALYSIS CONTEXT
  complex logarithm has values log|z| + i(arg(z) + 2*pi*k), k in integers, z != 0.
  complex roots may have multiple distinct branches.
  principal branch is a selected convention, not the complete correspondence.
  analytic continuation and branch cuts require explicit domain restrictions.

16. OPTIMISATION CONTEXT
  argmin correspondence F(x) = {y in feasible(x) : objective(x,y) minimal}.
  multiple minimisers imply a genuinely multivalued output.
  no minimiser implies empty output unless an existence theorem applies.
  regularity (upper/lower hemicontinuity) requires separately stated assumptions.

17. LOGICAL REPRESENTATION
  member_of_output(y,F,x) iff y in F(x).
  exists_output(x) iff exists y: member_of_output(y,F,x).
  unique_output(x) iff exists exactly one y: member_of_output(y,F,x).
  multi_output(x) iff exists y1,y2: y1 != y2 and both are outputs.
  deterministic_selection(s,F) iff for every x in domain(s): s(x) in F(x).

18. INFERENCE RULES
  R(x,y) => y in F(x).
  y in F(x) => (x,y) in graph(F).
  F(x)={y} => selected_value(x)=y (when singleton extraction requested).
  y in F(x) and z in G(y) => z in (G o F)(x).
  x in A and y in F(x) => y in F[A].
  F(x)=empty => x not_in effective_domain(F).
  multi_output(x) => F is not single_valued.

19. SEMANTIC EQUIVALENCE
  exact: F(x)=G(x) for all x under aligned domains and codomains.
  graph_equivalence: graph(F)=graph(G) with compatible ambient sets.
  notation_equivalence: F(x) and {y:R(x,y)} where R is graph(F).
  selection_is_not_equivalence: s(x) in F(x) does not imply {s(x)}=F(x).
  inverse_relation_is_not_ordinary_inverse unless bijectivity holds.

20. KNOWLEDGE GRAPH MODEL
  node F type MultivaluedFunction.
  edge F --hasDomain--> X.
  edge F --hasCodomain--> Y.
  edge F --hasGraph--> R.
  edge F --hasSelection--> s.
  reified_output_assertion: (F,x,y) with membership evidence.
  store quantifier scope and context with every inference.
  prohibit inference of unique output from existence alone.

21. VALIDATION
  check domain and codomain membership.
  check graph-output consistency.
  check totality policy against empty outputs.
  check branch outputs satisfy parent correspondence.
  check composition intermediate type alignment.
  check distinction between range, codomain and individual output set.
  check that inverse correspondence does not assert bijectivity automatically.

22. CONFIDENCE
  CONFIDENCE:
    formal_definition: 1.0 when directly derived from definitions.
    theorem_application: proof_dependent.
    numerical_evaluation: method_and_error_dependent.
    sourced_claim: provenance_dependent.
  confidence does not replace mathematical proof.

23. FLOW HASH
  FLOW_HASH:
    algorithm: SHA-256
    canonicalisation: UTF-8; LF newlines; stable key order; exclude hash field.
    value: compute_on_publish
    note: do not invent a digest before canonical bytes are final.

24. UPDATE MODEL
  UPDATE_MODEL:
    version_policy: semantic_versioning
    trigger: corrected theorem, new formalisation, taxonomy change, source revision.
    immutable_history: true
    preserve_provenance: true
    revalidate_dependent_flows: true

25. AI CONSUMPTION
  AI_CONSUMPTION:
    parse_mode: typed_semantic_flow
    extraction: entities, relationships, definitions, axioms, rules, examples.
    reasoning: sound_under_stated_assumptions
    missing_value: unknown_or_empty_as_explicitly_typed
    disambiguation: correspondence_vs_single_valued_function
    output: explainable claims with provenance and confidence.
    safety: do not infer branch existence or uniqueness without justification.

26. TEST CASES
  TEST T1: F(4)={-2,2} => multi_output(4)=true.
  TEST T2: F(0)={0} => unique_output(0)=true.
  TEST T3: F(-1)=empty for real square roots => no real output.
  TEST T4: f(x)=x*x on reals => inverse correspondence of 4 = {-2,2}.
  TEST T5: singleton-valued F => extraction produces an ordinary function.
  TEST T6: F(a)={1,2}; G(1)={p}; G(2)={q} => (G o F)(a)={p,q}.
  TEST T7: selection s(4)=2 is valid, but not equivalent to F(4)={-2,2}.
  TEST T8: empty output does not imply an unknown output.

27. TERMINATION
  status: COMPLETE
  integrity: all numbered sections 01-27 present.
  final_rule: preserve set-valued semantics, types and proof conditions.
end flow