Inverse Function Ontology

FLOW ALGEBRA
TITLE: Inverse Function Ontology
DOMAIN: Discrete Mathematics / Functions
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/functions/inverse-function-ontology.html
VERSION: 1.0
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
STATUS: Mathematical reference model; live source page not independently verified

1. PURPOSE
Represent inverse functions, their existence conditions, typed domains, compositional identities, proofs, and computational checks in a machine-readable semantic flow.

2. TYPED ENTITIES
Set[A], Set[B] : mathematical sets
Function[f] : A -> B
InverseFunction[g] : B -> A
Identity[id_A] : A -> A
Identity[id_B] : B -> B
Image[f(A)] : subset of B
Domain, Codomain, Range, OrderedPair, Bijection, Injection, Surjection, Restriction, Composition, Proof, Counterexample

3. FORMAL DEFINITIONS
INVERSE(f,g) := (g ∘ f = id_A) AND (f ∘ g = id_B).
INVERTIBLE(f) := EXISTS g:B->A SUCH_THAT INVERSE(f,g).
BIJECTIVE(f) := INJECTIVE(f) AND SURJECTIVE(f).
THEOREM: INVERTIBLE(f) IFF BIJECTIVE(f).
UNIQUENESS: INVERSE(f,g) AND INVERSE(f,h) IMPLIES g=h.
POINTWISE: g(b)=a IFF f(a)=b, provided f is bijective.
GRAPH_INVERSE: graph(g) = {(b,a) | (a,b) in graph(f)}.

4. OPERATIONS
inverse(f): requires bijective(f); returns unique g:B->A.
compose(g,f): A->A, equals id_A when g=inverse(f).
compose(f,g): B->B, equals id_B when g=inverse(f).
restrict(f,D): D->B for D subset A; may become injective but not necessarily onto B.
corestrict(f,f(A)): A->f(A); invertible iff f is injective.
preimage(f,S): {a in A | f(a) in S} for S subset B; does not require invertibility.

5. ALGEBRAIC LAWS
(f^-1)^-1 = f for bijective f.
(g ∘ f)^-1 = f^-1 ∘ g^-1 for bijections f:A->B and g:B->C.
(id_A)^-1 = id_A.
f^-1(f(a))=a for all a in A.
f(f^-1(b))=b for all b in B.
For bijective f and S subset A: f^-1[f[S]]=S, where f^-1[.] denotes set preimage.
For bijective f and T subset B: f[f^-1[T]]=T.

6. IMPORTANT DISTINCTIONS
Inverse function f^-1:B->A exists as a total two-sided inverse only when f is bijective.
An injective but non-surjective f:A->B has an inverse from f(A) to A, not necessarily from all B.
A surjective non-injective function need not admit a single-valued inverse function.
The inverse relation {(b,a)|(a,b) in graph(f)} always exists, but need not be a function B->A.
Set preimage f^-1[S] is defined for any function f and must not be confused with an inverse function.
Left inverses and right inverses are distinct from two-sided inverses; existence of a right inverse for arbitrary sets may depend on choice principles.

7. SEMANTIC RELATIONSHIPS
InverseFunction --inverseOf--> Function
InverseFunction --hasDomain--> original Codomain
InverseFunction --hasCodomain--> original Domain
Bijection --enables--> TwoSidedInverse
Injectivity --guarantees--> uniquePreimageForImageElement
Surjectivity --guarantees--> everyCodomainElementHasPreimage
FunctionComposition --verifies--> InverseIdentity
InverseRelation --reverses--> OrderedPairs

8. INFERENCE RULES
R1: bijective(f) => exists_unique(inverse(f)).
R2: inverse(f)=g => injective(f) AND surjective(f).
R3: inverse(f)=g => inverse(g)=f.
R4: inverse(f)=g AND inverse(g)=h => h=f.
R5: bijective(f) AND bijective(g) AND codomain(f)=domain(g) => inverse(g∘f)=inverse(f)∘inverse(g).
R6: injective(f) => corestrict(f,image(f)) is bijective.
R7: NOT injective(f) OR NOT surjective(f) => no two-sided inverse B->A.

9. VALIDATION
V1: Verify f is a total single-valued map A->B.
V2: Check distinct domain elements never share an output (injectivity).
V3: Check every b in B is attained (surjectivity).
V4: Construct inverse by swapping ordered pairs only after bijectivity is established.
V5: Verify both compositions equal their respective identity maps.
V6: Check source and target types before composing functions.
V7: Reject claims that preimage notation alone proves invertibility.
V8: For infinite sets, use symbolic proof or certified reasoning; finite enumeration alone is insufficient.

10. EXAMPLES
E1: f:Z->Z, f(n)=n+3; inverse f^-1(m)=m-3.
E2: f:R->R, f(x)=x^2; not invertible (not injective and not surjective onto R).
E3: f:[0,infinity)->[0,infinity), f(x)=x^2; inverse f^-1(y)=sqrt(y).
E4: f:Z->Z, f(n)=2n; injective but not surjective, so no inverse Z->Z.

11. SEMANTIC_EQUIVALENCE
"invertible function" == "bijective function" (for total set functions with two-sided inverse).
"inverse mapping" == "inverse function" only when the inverse relation is a function on the stated codomain.
"one-to-one correspondence" == "bijection".
"preimage of a set" != "inverse function".

12. CONFIDENCE
mathematical_definitions: 1.00
formal_theorems: 1.00 (within classical set theory)
source_page_alignment: unverified
semantic_mapping: 0.98
Rule: confidence describes mathematical validity, not live-page verification.

13. FLOW_HASH
algorithm: SHA-256
scope: canonical UTF-8 representation of the flow excluding the FLOW_HASH digest field
value: TO_BE_COMPUTED_AT_PUBLICATION

14. UPDATE_MODEL
trigger: change to formal definitions, scope, ontology relationships, source text, or validation logic
procedure: version_increment -> theorem_check -> type_check -> semantic_review -> regenerate_hash -> publish
preserve: prior versions and provenance

15. AI_CONSUMPTION
entity_resolution: Function, InverseFunction, Bijection, Domain, Codomain
reasoning: type-safe inference with proof obligations
retrieval_keys: inverse function; bijective mapping; inverse relation; left inverse; right inverse; function composition
output_constraints: distinguish inverse function from inverse relation and set preimage; do not assert an inverse without proving bijectivity
provenance: SOURCE_URL and VERSION

END FLOW ALGEBRA