Image Ontology

FLOW ALGEBRA
IMAGE ONTOLOGY — FUNCTION AND MAPPING ENTITIES
Version: 1.0
Date: 2026-10-13
Authors: Mark Higgins — Sky (Recursive AI)
Publisher: Kireth Research | https://kirethresearch.org
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/function-and-mapping-entities/image-ontology-3.html
Source status: Formal mathematical specification; source page content not independently verified.

01. PURPOSE
flow "ImageOntology" v1.0
  objective: represent and reason about images of elements and subsets under functions.
  scope: set theory, mappings, algebra, topology, knowledge representation.

02. ENTITY TYPES
  Set, Element, Function, Domain, Codomain, Image, Preimage, Subset,
  OrderedPair, MappingRule, Restriction, Composition, Surjection.

03. FORMAL SIGNATURE
  f: A -> B
  A = domain(f)
  B = codomain(f)
  image(f) = f[A] = {f(a) | a in A} subseteq B
  for S subseteq A: f[S] = {f(s) | s in S}
  for a in A: image_of_element(f,a) = f(a) in B
  for T subseteq B: preimage(f,T) = {a in A | f(a) in T}

04. TYPED SCHEMA
  Function:
    id: URI
    domain: Set
    codomain: Set
    mapping_rule: -Expression | Relation
    total: Boolean
  Image:
    function_ref: Function
    source_subset: Subset(domain(function_ref))
    result_subset: Subset(codomain(function_ref))
    derivation: exact | symbolic | enumerated | bounded
    proof_status: verified | conditional | unknown

05. WELL-FORMEDNESS
  require f is a single-valued total mapping on A for the default schema.
  require S subseteq A before evaluating f[S].
  require each f(s) in B.
  image(f) may be a proper subset of B.
  empty domain implies empty image for a total function.

06. MEMBERSHIP SEMANTICS
  y in f[S] iff exists x in S such that f(x)=y.
  y notin f[S] iff for all x in S, f(x)!=y.
  f[S] subseteq B.
  f[A] = range(f) under the convention 'range' means attained outputs.
  distinguish range(f) terminology when authors use 'range' for codomain.

07. BASIC LAWS
  f[empty_set] = empty_set.
  S subseteq T implies f[S] subseteq f[T].
  f[S union T] = f[S] union f[T].
  f[S intersection T] subseteq f[S] intersection f[T].
  equality in intersection law holds when f is injective on S union T.
  f[S \ T] superseteq f[S] \ f[T].
  equality in difference law holds when f is injective on S union T.

08. IMAGE AND PREIMAGE
  S subseteq preimage(f,f[S]).
  equality iff S is saturated with respect to f (union of fibres).
  f[preimage(f,T)] = T intersection f[A].
  preimage(f,f[S]) = S when f is injective on A.
  f[preimage(f,T)] = T when f is surjective onto B.

09. COMPOSITION
  given f:A->B and g:B->C:
    (g compose f)[S] = g[f[S]] for S subseteq A.
    image(g compose f) = g[f[A]].
  composition requires codomain/domain compatibility or a justified restriction.

10. INJECTIVITY AND SURJECTIVITY
  injective(f) iff for all x1,x2 in A, f(x1)=f(x2) implies x1=x2.
  surjective(f) iff f[A]=B.
  bijective(f) iff injective(f) and surjective(f).
  if f injective: |f[S]|=|S| for finite S.
  if S finite: |f[S]|<=|S|.

11. FIBRES AND EQUIVALENCE
  fibre(f,y) = {x in A | f(x)=y}.
  y in f[A] iff fibre(f,y) is nonempty.
  define x ~f x' iff f(x)=f(x').
  ~f is an equivalence relation on A.
  quotient A/~f is in bijection with f[A].

12. REPRESENTATION
  canonical form: Image(function_id, subset_id).
  preserve domain, codomain, and mapping-rule references.
  canonicalise extensional finite sets by unique members, order-independent.
  do not assume distinct inputs yield distinct outputs.
  support symbolic comprehensions for infinite sets.

13. EXAMPLES
  f: Z -> Z, f(x)=x^2.
  f[{-2,-1,0,1,2}] = {0,1,4}.
  f[Z] = {n^2 | n in Z}.
  f is not injective and not surjective onto Z.
  g: R -> R, g(x)=x^2.
  g[R] = [0,infinity).
  codomain(g)=R, image(g)=[0,infinity).
  h: R -> [0,infinity), h(x)=x^2 is surjective.

14. KNOWLEDGE GRAPH RELATIONSHIPS
  Image --derivedFrom--> Function
  Image --usesSourceSubset--> Subset
  Image --subsetOf--> Codomain
  Function --hasDomain--> Domain
  Function --hasCodomain--> Codomain
  Function --mapsElementTo--> Element
  Image --hasMember--> Element
  Image --relatedTo--> Preimage
  Image --relatedTo--> Fibre
  Surjection --hasImageEqualTo--> Codomain

15. INFERENCE RULES
  R1: Function(f,A,B) & Member(x,A) -> Member(f(x),B).
  R2: Subset(S,A) & Member(x,S) -> Member(f(x),Image(f,S)).
  R3: Member(y,Image(f,S)) -> Exists(x, Member(x,S) & f(x)=y).
  R4: Subset(S,T) -> Subset(Image(f,S),Image(f,T)).
  R5: Surjective(f,A,B) -> Equal(Image(f,A),B).
  R6: Injective(f) & Equal(f(x),f(y)) -> Equal(x,y).
  R7: Empty(S) -> Empty(Image(f,S)).
  R8: Compatible(f,g) -> Equal(Image(g∘f,S),Image(g,Image(f,S))).

16. SEMANTIC EQUIVALENCE
  f[S] == {f(x) | x in S}.
  image_of_f_on_S == direct_image(f,S).
  image(f) == f[domain(f)].
  avoid conflating image(f), codomain(f), inverse image, or graph(f).
  distinguish image of an element from image of a subset.

17. VALIDATION
  V1: reject undefined function values in total-function mode.
  V2: reject source elements outside domain.
  V3: check resulting members belong to codomain.
  V4: deduplicate finite image members.
  V5: verify union and composition identities where computable.
  V6: flag image=codomain claims lacking surjectivity proof.
  V7: preserve symbolic conditions for infinite images.
  V8: report undecidable or computationally intractable membership as unknown.

18. CONFIDENCE
  proven: 1.00 for verified derivations under explicit axioms.
  conditional: retain assumptions and proof obligations.
  estimated: report numerical uncertainty separately from logical truth.
  unknown: do not infer falsehood from missing evidence.

19. FLOW HASH
  hash_input: canonical_schema + typed_entities + axioms + inference_rules + source_version.
  algorithm: SHA-256 recommended.
  FLOW_HASH: compute on publication; do not invent a digest.

20. UPDATE MODEL
  trigger: change to domain, codomain, mapping rule, source subset, or axioms.
  invalidate: dependent images, surjectivity conclusions, and cached memberships.
  regenerate: canonical form, validations, derivations, and FLOW_HASH.
  retain: prior version and provenance for auditability.

21. AI CONSUMPTION
  mode: typed semantic reasoning.
  inputs: function, domain, codomain, source subset, assumptions.
  outputs: symbolic image, membership claims, derivation, confidence, provenance.
  guardrails: never replace codomain with image without surjectivity evidence.
  preserve: set-theoretic context, quantifiers, and unresolved conditions.

22. TEST CASES
  T1: f(x)=x^2, S={-2,2} -> f[S]={4}.
  T2: f(x)=x^2, S={-2,0}, T={0,2} -> f[S∩T]={0}; f[S]∩f[T]={0,4}.
  T3: f(x)=x+1 on Z, f[empty_set]=empty_set.
  T4: f(x)=x^2 on R, image(f)=[0,infinity), not R.
  T5: identity_A[S]=S for every S subseteq A.
  T6: constant c:A->B with nonempty A -> image(c)={c}.
  T7: g∘f image matches g applied to f-image when types align.

23. PROVENANCE
  ontology: Image Ontology (Function and Mapping Entities).
  publisher: Kireth Research.
  authors: Mark Higgins — Sky (Recursive AI).
  verification: independently constructed from established mathematics.
  source_page_review: not verified.

24. TERMINATION
  status: COMPLETE
  end_flow: ImageOntology v1.0