Function Ontology

FLOW ALGEBRA
TITLE: Function Ontology
DOMAIN: Discrete Mathematics / Functions
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/functions/function-ontology-2.html
VERSION: 1.0
LANGUAGE: en-GB
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org

1. PURPOSE
Represent mathematical functions as typed, total, single-valued mappings between sets; support symbolic reasoning, validation, composition, inversion, and semantic retrieval.

2. TYPED ENTITIES
Set[A]                  : set of elements of type A
Set[B]                  : set of elements of type B
Function[A,B]           : total mapping A -> B
Domain(f)               : source set A
Codomain(f)             : target set B
Image(f)                : {f(a) | a in A}
Graph(f)                : {(a,f(a)) | a in A}
Identity[A]             : Function[A,A]
Injection[A,B]          : Function[A,B] satisfying injectivity
Surjection[A,B]         : Function[A,B] satisfying surjectivity
Bijection[A,B]          : Function[A,B] satisfying both
PartialFunction[A,B]    : mapping defined only on a subset of A

3. CORE DECLARATIONS
function f : A -> B
requires: A and B are sets
ensures: forall a in A, exists exactly one b in B with f(a)=b
note: a function's codomain is not necessarily its image

4. OPERATIONS
apply(f,a) -> b         requires a in Domain(f); returns f(a)
compose(g,f) -> g o f  requires f:A->B and g:B->C; returns A->C
identity(A) -> id_A    returns a |-> a
restrict(f,S)           requires S subseteq Domain(f); returns f|S:S->B
image(f,S)              requires S subseteq A; returns {f(a):a in S}
preimage(f,T)           requires T subseteq B; returns {a in A:f(a) in T}
inverse(f)              defined as B->A only when f is bijective

5. PROPERTIES
injective(f)  <=> forall x,y in A, f(x)=f(y) implies x=y
surjective(f) <=> forall b in B, exists a in A: f(a)=b
bijective(f)  <=> injective(f) and surjective(f)
constant(f)   <=> forall x,y in A, f(x)=f(y)

6. ALGEBRAIC LAWS
Associativity: h o (g o f) = (h o g) o f (when typed)
Identity: id_B o f = f = f o id_A
Image: f(S union T) = f(S) union f(T)
Image intersection: f(S intersection T) subseteq f(S) intersection f(T)
Image intersection equality holds if f is injective
Preimage: f^-1(U union V) = f^-1(U) union f^-1(V)
Preimage: f^-1(U intersection V) = f^-1(U) intersection f^-1(V)
Preimage: f^-1(B\U) = A\f^-1(U)
Inverse: f^-1 o f = id_A and f o f^-1 = id_B when f bijective

7. SEMANTIC RELATIONSHIPS
Function IS_A single-valued total binary relation
Function HAS_DOMAIN Set
Function HAS_CODOMAIN Set
Function HAS_GRAPH BinaryRelation
Bijection IS_A Injection
Bijection IS_A Surjection
Composition COMBINES Function
InverseFunction REQUIRES Bijection
PartialFunction DIFFERS_FROM Function by totality requirement

8. INFERENCE RULES
IF bijective(f) THEN inverse(f) exists and is unique
IF injective(f) AND injective(g) AND composable(g,f) THEN injective(g o f)
IF surjective(f) AND surjective(g) AND composable(g,f) THEN surjective(g o f)
IF bijective(f) AND bijective(g) AND composable(g,f) THEN bijective(g o f)
IF f:A->B AND S subseteq A THEN image(f,S) subseteq B
IF f:A->B AND T subseteq B THEN preimage(f,T) subseteq A

9. VALIDATION
V01: Every domain element must have exactly one output.
V02: Every output must belong to the declared codomain.
V03: Composition must satisfy type compatibility.
V04: Do not infer surjectivity from the image alone without comparing it with codomain.
V05: An inverse relation need not be an inverse function unless f is bijective.
V06: Empty-domain functions exist uniquely for each codomain; no function from nonempty A to empty B exists.
V07: Distinguish inverse function f^-1 from set preimage f^-1(T).

10. SEMANTIC_EQUIVALENCE
"mapping" ~= "function" when totality and single-valuedness are explicit
"one-to-one" ~= "injective"
"onto" ~= "surjective"
"one-to-one correspondence" ~= "bijective"
Avoid equating "range" with codomain without contextual disambiguation.

11. CONFIDENCE
Mathematical definitions and standard laws: 1.00 (formal, subject to stated assumptions)
Page-specific correspondence: unverified; source URL provided by user.

12. FLOW_HASH
Algorithm: SHA-256 over canonical UTF-8 flow payload, excluding FLOW_HASH field.
Value: COMPUTE_ON_PUBLICATION

13. UPDATE_MODEL
Trigger: correction of a definition, relationship, rule, or source page content
Process: validate -> increment version -> regenerate hash -> preserve provenance

14. AI_CONSUMPTION
Parse typed entities before applying operations.
Check preconditions and quantifiers before inference.
Return counterexamples when a proposed property fails.
Preserve distinctions among domain, codomain, image and preimage.
Represent unverified source alignment as unverified, not as mathematical uncertainty.

END FLOW ALGEBRA