Preimage Ontology

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

01. PURPOSE
Define the preimage of a subset under a function, its mathematical laws, typed knowledge representation, and safe AI inference.

02. CORE TYPES
Set[A] := collection of elements of type A.
Function[A,B] := total single-valued mapping f:A -> B.
Subset[A] := S with S subseteq A.
Preimage[A,B] := inverse-image construction (f,S) -> subset of A.
Predicate[A] := A -> Boolean.

03. DEFINITIONS
For f:A -> B and S subseteq B:
  preimage(f,S) = {x in A | f(x) in S}.
For y in B:
  fibre(f,y) = preimage(f,{y}).
The preimage is a SET, not generally an inverse function.
Notation: f^(-1)[S], f^(-1)(S) when context is clear.

04. TYPE SIGNATURES
preimage : Function[A,B] x Subset[B] -> Subset[A]
fibre    : Function[A,B] x B -> Subset[A]
contains : Set[A] x A -> Boolean
image    : Function[A,B] x Subset[A] -> Subset[B]

05. MEMBERSHIP RULE
x in preimage(f,S) iff x in A and f(x) in S.
If x not in A, f(x) is not defined under the declared function type.

06. EMPTY AND WHOLE SETS
preimage(f,empty_B) = empty_A.
preimage(f,B) = A.
These statements require f to be total on A.

07. UNION PRESERVATION
preimage(f,union_i S_i) = union_i preimage(f,S_i).
Holds for arbitrary indexed families, including empty families.

08. INTERSECTION PRESERVATION
preimage(f,intersection_i S_i) = intersection_i preimage(f,S_i).
For empty families, use B as the intersection on the left and A on the right.

09. COMPLEMENT PRESERVATION
preimage(f,B \ S) = A \ preimage(f,S).
Complements must be interpreted relative to the declared domain and codomain.

10. DIFFERENCE AND SYMMETRIC DIFFERENCE
preimage(f,S \ T) = preimage(f,S) \ preimage(f,T).
preimage(f,S delta T) = preimage(f,S) delta preimage(f,T).

11. MONOTONICITY
S subseteq T implies preimage(f,S) subseteq preimage(f,T).

12. COMPOSITION
For f:A -> B, g:B -> C, U subseteq C:
preimage(g compose f,U) = preimage(f,preimage(g,U)).
Composition direction is explicit: (g compose f)(x)=g(f(x)).

13. IDENTITY
For id_A:A -> A and S subseteq A:
preimage(id_A,S)=S.

14. IMAGE-PREIMAGE ADJUNCTION
For T subseteq A and S subseteq B:
image(f,T) subseteq S iff T subseteq preimage(f,S).
Always T subseteq preimage(f,image(f,T)).
Equality holds when T is saturated under the fibres of f.
Always image(f,preimage(f,S)) = S intersection image(f,A).
Equality with S holds when S subseteq image(f,A).

15. INJECTIVE AND SURJECTIVE CASES
If f is injective, preimage(f,image(f,T))=T for every T subseteq A.
If f is surjective, image(f,preimage(f,S))=S for every S subseteq B.
Neither property is required to define a preimage.

16. FIBRES AND PARTITIONS
Distinct nonempty fibres are pairwise disjoint.
The family of nonempty fibres partitions A for total f.
x ~_f x' iff f(x)=f(x') defines an equivalence relation on A.

17. PREDICATE PULLBACK
Given predicate P:B -> Boolean, define Q:A -> Boolean by Q(x)=P(f(x)).
preimage(f,{y in B | P(y)}) = {x in A | Q(x)}.

18. TOPOLOGICAL INTERPRETATION
For topological spaces A,B, f is continuous iff preimage(f,U) is open in A for every open U in B.
Equivalently, preimages of closed subsets of B are closed in A.
No continuity assumption is needed for the set-theoretic definition.

19. MEASURABLE INTERPRETATION
For measurable spaces (A,Sigma_A), (B,Sigma_B):
f is measurable iff preimage(f,S) in Sigma_A for every S in Sigma_B.
This is the foundational pullback rule for measurable sets.

20. KNOWLEDGE GRAPH ENTITY MODEL
ENTITY PreimageConstruction:
  function: Function[A,B]
  target_subset: Subset[B]
  result_subset: Subset[A]
  domain: Set[A]
  codomain: Set[B]
  assumptions: [totality, typing]
  provenance: SourceReference
RELATIONS:
  hasDomain(function,domain)
  hasCodomain(function,codomain)
  hasTargetSubset(construction,target_subset)
  hasResultSubset(construction,result_subset)
  isPreimageOf(result_subset,target_subset,function)
  isFibreOf(result_subset,value,function)

21. SEMANTIC EQUIVALENCE
Equivalent expressions: {x in A | f(x) in S} and f^(-1)[S].
Do not equate preimage with an inverse FUNCTION without bijectivity and suitable typing.
Canonical identity depends on f, S, domain A, and codomain B.

22. INFERENCE RULES
RULE MEMBER_PREIMAGE:
  IF x in A AND f(x) in S THEN x in preimage(f,S).
RULE NONMEMBER_PREIMAGE:
  IF x in A AND f(x) notin S THEN x notin preimage(f,S).
RULE PREIMAGE_UNION:
  REWRITE preimage(f,S union T) -> preimage(f,S) union preimage(f,T).
RULE PREIMAGE_COMPOSITION:
  REWRITE preimage(g compose f,U) -> preimage(f,preimage(g,U)).
RULE IMAGE_PREIMAGE:
  REWRITE image(f,preimage(f,S)) -> S intersection image(f,A).

23. VALIDATION
Check that f is typed A -> B and S subseteq B.
Check that result is a subset of A.
Reject inverse-function inference from preimage notation alone.
Check composition types and direction.
Check complement universes explicitly.
Distinguish total functions from partial functions: for partial f, use its defined domain D subseteq A and adapt whole-set rules accordingly.

24. TEST CASES
A={1,2,3,4}; B={0,1}; f(x)=x mod 2.
preimage(f,{0})={2,4}.
preimage(f,{1})={1,3}.
preimage(f,empty) = empty.
preimage(f,B)=A.
preimage(f,{0}) intersection preimage(f,{1})=empty.
For f:R -> R, f(x)=x^2:
preimage(f,{4})={-2,2}; an inverse function is not implied.
preimage(f,[-1,4])=[-2,2].
preimage(f,(-infinity,0))=empty.

25. CONFIDENCE
Mathematical identities: deductive confidence 1.0 under stated axioms and types.
Implementation claims: require independent verification.
Page-specific correspondence: unverified until source text is inspected.

26. FLOW_HASH
Canonicalise typed schema, ordered rules, assumptions, and version.
Compute FLOW_HASH = SHA256(canonical_UTF8_payload) at publication time.
Do not invent a digest before canonicalisation and hashing.

27. UPDATE_MODEL
Trigger review when source ontology, function typing, mathematical conventions, or inference rules change.
Increment version; preserve prior versions and provenance.
Revalidate examples and regenerate FLOW_HASH after changes.

28. AI_CONSUMPTION
Provide typed facts, precise mathematical semantics, explicit domain/codomain, and traceable proof rules.
Prefer symbolic exactness to approximate numeric matching.
Expose assumptions and distinguish proven facts from inferred or unverified claims.
Avoid conflating preimages, inverse functions, and images.

29. PROVENANCE AND AUTHORSHIP
Project: Kireth Semantic Library
Authors: Mark Higgins — Sky (Recursive AI)
Website: https://kirethresearch.org
Source URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/function-and-mapping-entities/preimage-ontology.html
Basis: Standard set theory, function theory, topology and measure theory.

30. TERMINATION
STATUS: COMPLETE
END FLOW ALGEBRA