Total Function Ontology

FLOW ALGEBRA
TOTAL 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/total-function-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Status: formal mathematical synthesis; source page not independently verified
Version: 1.0 | Date: 2026-10-13

01. PURPOSE
flow "TotalFunctionOntology" v1.0
  intent: model functions defined at every element of their declared domain
  foundation: set theory, function theory, typed semantic graphs

02. ENTITY TYPES
  TotalFunction: Function
  Domain: Set
  Codomain: Set
  Input: Entity
  Output: Entity
  OrderedPair: Tuple[2]
  Graph: Set[OrderedPair]
  Image: Set
  Preimage: Set
  Predicate: BooleanExpression
  Proof: Evidence

03. CORE DEFINITION
  total(f:A->B) iff forall x in A, exists exactly one y in B: (x,y) in graph(f)
  graph(f) subseteq A x B
  domain(graph(f)) = A
  range(f) subseteq B
  Note: totality is relative to the declared domain A.

04. TYPE SIGNATURES
  declare f: A -> B
  apply: Function[A,B] x A -> B
  graph: Function[A,B] -> PowerSet(A x B)
  image: Function[A,B] x PowerSet(A) -> PowerSet(B)
  preimage: Function[A,B] x PowerSet(B) -> PowerSet(A)

05. WELL-FORMEDNESS
  A and B must be sets in the chosen foundation
  each graph pair has first component in A and second component in B
  each x in A has one and only one output
  no input in A may be left undefined
  no input in A may map to two distinct outputs

06. TOTAL VERSUS PARTIAL
  partial_function(p,A,B): graph(p) subseteq A x B and outputs unique
  total_function(p,A,B): partial_function(p,A,B) and dom(graph(p))=A
  every total function is a partial function relative to A,B
  not every partial function is total relative to A,B
  a partial function is total on its effective domain

07. EMPTY DOMAIN
  if A=empty and B any set, empty graph is the unique total function A->B
  if A nonempty and B=empty, no total function A->B exists
  if A=B=empty, exactly one total function exists

08. DOMAIN AND CODOMAIN
  declared_domain(f)=A
  declared_codomain(f)=B
  image(f)={f(x) | x in A}
  image(f) subseteq B
  totality does not imply surjectivity

09. EVALUATION
  require x in A
  evaluate(f,x) = unique y such that (x,y) in graph(f)
  if x not in A, evaluation is outside declared function signature
  do not classify out-of-domain calls as evidence of non-totality

10. GRAPH CONSTRUCTION
  graph(f)={(x,f(x)) | x in A}
  domain(graph(f))=A
  functional(graph(f))=true
  graph uniquely determines f when A and B are fixed

11. COMPOSITION
  given f:A->B and g:B->C total
  compose(g,f):A->C
  (g o f)(x)=g(f(x))
  total(compose(g,f))=true
  associativity: h o (g o f)=(h o g) o f for compatible types

12. IDENTITY
  id_A:A->A
  id_A(x)=x
  total(id_A)=true
  f o id_A=f
  id_B o f=f

13. RESTRICTION
  for S subseteq A, f|S:S->B is total
  graph(f|S)={(x,f(x)) | x in S}
  restricting a total function to S preserves totality relative to S

14. EXTENSION
  extending f:A->B to A' superset A requires an output in B for every new input
  if B=empty and A' nonempty, no extension exists
  an extension F satisfies F(x)=f(x) for all x in A

15. INJECTIVITY
  injective(f) iff forall x1,x2 in A: f(x1)=f(x2) implies x1=x2
  injectivity independent of totality

16. SURJECTIVITY
  surjective(f) iff image(f)=B
  surjectivity independent of totality

17. BIJECTIVITY
  bijective(f) iff injective(f) and surjective(f)
  bijective total f:A->B has a total inverse f^-1:B->A
  non-bijective total functions do not have two-sided functional inverses

18. IMAGE AND PREIMAGE
  image(f,S)={f(x) | x in S} for S subseteq A
  preimage(f,T)={x in A | f(x) in T} for T subseteq B
  preimage(f,B)=A
  preimage(f,empty)=empty
  preimage(f,T1 union T2)=preimage(f,T1) union preimage(f,T2)
  preimage(f,T1 intersect T2)=preimage(f,T1) intersect preimage(f,T2)

19. FINITE CARDINALITY
  if |A|=m and |B|=n>0, number of total functions A->B is n^m
  if A=empty, number of total functions is 1 for any B
  if A nonempty and B=empty, number is 0
  finite injective f:A->B requires |A|<=|B|
  finite surjective f:A->B requires |A|>=|B|

20. EXAMPLES
  square:R->R, square(x)=x*x; total=true
  reciprocal:R\{0}->R, reciprocal(x)=1/x; total=true
  reciprocal:R->R with undefined value at 0; total=false
  constant:A->B, constant(x)=b0; total if b0 in B
  length:FiniteStrings->NaturalNumbers; total=true

21. NON-EXAMPLES
  graph={(1,a)} over A={1,2}, B={a,b}: not total
  graph={(1,a),(1,b)} over A={1}, B={a,b}: not a function
  expression 1/x over declared domain R without definition at zero: not total

22. KNOWLEDGE GRAPH SCHEMA
  TotalFunction --isA--> Function
  TotalFunction --hasDomain--> Domain
  TotalFunction --hasCodomain--> Codomain
  TotalFunction --hasGraph--> Graph
  TotalFunction --mapsInputTo--> Output
  TotalFunction --hasImage--> Image
  TotalFunction --composesWith--> TotalFunction
  TotalFunction --restrictsTo--> TotalFunction
  TotalFunction --mayBe--> InjectiveFunction
  TotalFunction --mayBe--> SurjectiveFunction
  TotalFunction --mayBe--> BijectiveFunction

23. INFERENCE RULES
  R1: functional(R) and dom(R)=A and R subseteq A x B => total(R,A,B)
  R2: total(f,A,B) and total(g,B,C) => total(g o f,A,C)
  R3: total(f,A,B) and S subseteq A => total(f|S,S,B)
  R4: total(f,A,B) and injective(f) and surjective(f) => bijective(f)
  R5: bijective(f:A->B) => total(f^-1,B,A)
  R6: total(f,A,B) => preimage(f,B)=A

24. CONSTRAINTS
  enforce exactly_one_output_per_declared_input
  enforce output_in_codomain
  enforce graph_domain_equals_declared_domain
  reject missing_input_mapping
  reject multiple_distinct_outputs_for_same_input
  distinguish partiality from out_of_domain_evaluation

25. SEMANTIC EQUIVALENCE
  aliases: ["everywhere-defined function on its domain", "total mapping"]
  sameFunction(f,g) requires compatible declared domains/codomains and pointwise equality
  extensionalEquality(f,g) iff forall x in A: f(x)=g(x), for common A and B
  caution: identical graphs with different declared codomains need not be identical typed functions

26. VALIDATION CASES
  CASE T01: f:{1,2}->{a,b}, graph={(1,a),(2,a)} => PASS total
  CASE T02: f:{1,2}->{a,b}, graph={(1,a)} => FAIL total
  CASE T03: f:{1}->{a,b}, graph={(1,a),(1,b)} => FAIL functional
  CASE T04: f:empty->empty, graph=empty => PASS total
  CASE T05: f:{1}->empty => FAIL existence
  CASE T06: f:R\{0}->R, f(x)=1/x => PASS total
  CASE T07: f:R->R, f(x)=1/x undefined at 0 => FAIL total

27. CONFIDENCE
  formal_definition: 1.00
  mathematical_laws: 1.00 within classical set-theoretic semantics
  external_source_alignment: unverified
  inferred_taxonomy_links: review_before_publication

28. FLOW_HASH
  hash_algorithm: SHA-256
  canonicalisation: UTF-8, LF, normalised field order, stable identifiers
  hash_input: canonical flow excluding computed hash field
  FLOW_HASH: COMPUTE_ON_PUBLICATION

29. UPDATE_MODEL
  version: 1.0
  update_policy: semantic-versioning
  triggers: definition_change, theorem_correction, schema_revision, provenance_update
  preserve_previous_versions: true
  verify_test_cases_after_update: true

30. AI_CONSUMPTION
  mode: typed_ontology_and_rule_graph
  entity_resolution: exact_id_then_contextual_alias
  parse_signature_before_evaluation: true
  validate_totality_relative_to_declared_domain: true
  retain_domain_codomain_distinction: true
  emit_proof_or_counterexample: true
  reject_unsupported_inference: true

31. PROVENANCE
  canonical_page: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/function-and-mapping-entities/total-function-ontology.html
  source_page_content_verified: false
  mathematical_basis: standard set theory and function theory
  authors: Mark Higgins — Sky (Recursive AI)
  research_site: https://kirethresearch.org

32. TERMINATION
  status: COMPLETE
  validation: definition, edge cases, composition and examples included
  end_flow: true
END FLOW ALGEBRA