Finite Mathematics Ontology

FLOW ALGEBRA
TITLE: Finite Mathematics Ontology
VERSION: 1.0
DOMAIN: Discrete Mathematics / Foundations / Finite Mathematics
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/foundations-of-discrete-mathematics/finite-mathematics-ontology.html
STATUS: Domain-based formalisation; source-page specification not independently verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

1. PURPOSE
Represent finite mathematical objects, operations, constraints and verifiable inferences as typed, provenance-aware semantic flows. Distinguish exact mathematical truths from uncertain interpretations of source data.

2. TYPE SYSTEM
TYPE FiniteSet<T> = {elements: Set<T>, cardinality: Natural}
TYPE FiniteSequence<T> = {elements: List<T>, length: Natural}
TYPE FiniteRelation<A,B> = Set<(A,B)>
TYPE FiniteFunction<A,B> = {domain: FiniteSet<A>, codomain: FiniteSet<B>, mapping: A -> B}
TYPE FiniteGraph<V,E> = {vertices: FiniteSet<V>, edges: FiniteSet<E>, incidence: FiniteRelation<V,E>}
TYPE FiniteStructure = {carrier: FiniteSet, operations: Set, relations: Set, axioms: Set}
TYPE FiniteModel = {variables: FiniteSet, domains: Map, constraints: Set, solutions: Set}
TYPE Proof = {premises: Set, rule: Identifier, conclusion: Proposition, validity: Boolean}
TYPE Provenance = {source: URI?, method: String, timestamp: ISO8601?, version: String}
TYPE Confidence = Real[0,1]

3. ENTITIES
ENTITY FiniteMathematics
ENTITY FiniteSet
ENTITY Cardinality
ENTITY FiniteSequence
ENTITY Permutation
ENTITY Combination
ENTITY FiniteRelation
ENTITY FiniteFunction
ENTITY FiniteGraph
ENTITY FiniteStructure
ENTITY FiniteProbabilitySpace
ENTITY FiniteStateSystem
ENTITY FiniteModel
ENTITY Constraint
ENTITY Enumeration
ENTITY Proof

4. RELATIONSHIPS
RELATION hasElement(FiniteSet, Object)
RELATION hasCardinality(FiniteSet, Natural)
RELATION subsetOf(FiniteSet, FiniteSet)
RELATION mapsTo(FiniteFunction, Object, Object)
RELATION relates(FiniteRelation, Object, Object)
RELATION hasVertex(FiniteGraph, Object)
RELATION hasEdge(FiniteGraph, Object)
RELATION satisfies(FiniteModel, Constraint)
RELATION enumerates(Enumeration, FiniteSet)
RELATION proves(Proof, Proposition)
RELATION derivedFrom(Object, Provenance)

5. FORMAL OPERATORS
OP CARDINALITY(S: FiniteSet<T>) -> Natural = |S|
OP UNION(A: FiniteSet<T>, B: FiniteSet<T>) -> FiniteSet<T> = A ∪ B
OP INTERSECTION(A: FiniteSet<T>, B: FiniteSet<T>) -> FiniteSet<T> = A ∩ B
OP DIFFERENCE(A: FiniteSet<T>, B: FiniteSet<T>) -> FiniteSet<T> = A \\ B
OP PRODUCT(A: FiniteSet<A>, B: FiniteSet<B>) -> FiniteSet<(A,B)> = A × B
OP POWERSET(A: FiniteSet<T>) -> FiniteSet<FiniteSet<T>> = P(A)
OP FACTORIAL(n: Natural) -> Natural = n!
OP PERMUTATIONS(n: Natural, r: Natural) -> Natural = n!/(n-r)! WHERE 0 <= r <= n
OP COMBINATIONS(n: Natural, r: Natural) -> Natural = n!/(r!(n-r)!) WHERE 0 <= r <= n
OP ENUMERATE(M: FiniteModel) -> Set<Assignment> = {a | a satisfies every constraint in M}

6. INFERENCE RULES
RULE FiniteUnionCardinality:
  IF finite(A) AND finite(B)
  THEN |A ∪ B| = |A| + |B| - |A ∩ B|
RULE FiniteCartesianProduct:
  IF finite(A) AND finite(B)
  THEN |A × B| = |A| * |B|
RULE FinitePowerSet:
  IF finite(A)
  THEN |P(A)| = 2^|A|
RULE FinitePigeonhole:
  IF n objects assigned to m containers AND n > m AND m > 0
  THEN at least one container has at least 2 objects
RULE FiniteFunctionCount:
  IF |A| = n AND |B| = m
  THEN number_of_functions(A,B) = m^n
RULE FiniteDecision:
  IF search_space is finite AND every candidate is decidably testable
  THEN exhaustive_enumeration terminates

7. CONSTRAINTS AND VALIDATION
CONSTRAINT finite(S) => CARDINALITY(S) is a nonnegative integer
CONSTRAINT function(f) => each element of domain(f) has exactly one image
CONSTRAINT bijection(f) => injective(f) AND surjective(f)
CONSTRAINT probability(P) => all weights >= 0 AND sum(weights) = 1
CONSTRAINT enumeration(E,S) => sound(E,S) AND complete(E,S)
VALIDATE types, domains, preconditions, cardinality consistency, and proof steps before accepting a result.

8. SEMANTIC_EQUIVALENCE
SEMANTIC_EQUIVALENCE: "finite set" <=> "set with finite cardinality" [context: set theory]
SEMANTIC_EQUIVALENCE: "number of elements" <=> "cardinality" [context: finite sets]
SEMANTIC_EQUIVALENCE: "one-to-one correspondence" <=> "bijection" [context: mappings]
NOTE: Do not equate finite mathematics with all discrete mathematics; infinite discrete structures also exist.

9. CONFIDENCE
CONFIDENCE:
  formal_proof_validated: 1.0
  exact_computation_with_verified_inputs: 1.0
  source_interpretation: calibrated_estimate
  unverified_claim: unresolved
RULE: Confidence never substitutes for mathematical proof. Preserve uncertainty about data and interpretation separately from deductive validity.

10. PROVENANCE
PROVENANCE:
  source_url: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/foundations-of-discrete-mathematics/finite-mathematics-ontology.html
  author_credit: Mark Higgins — Sky (Recursive AI)
  source_verification: not_verified
  transformation: topic_based_formalisation
  record_inputs: true
  record_assumptions: true
  record_derivation: true

11. FLOW_HASH
FLOW_HASH:
  algorithm: SHA-256
  input: canonical_UTF8(normalised_types + entities + relations + operators + rules + constraints + version)
  output: lowercase_hex_digest
  note: Compute at ingestion; no digest is asserted in this draft.

12. UPDATE_MODEL
UPDATE_MODEL:
  trigger: source_revision | formal_correction | ontology_alignment_change
  actions:
    - preserve_prior_version
    - revalidate_types_and_axioms
    - recompute_dependent_inferences
    - update_provenance
    - regenerate_FLOW_HASH
    - publish_versioned_change_log
  invariant: never silently overwrite previously verified proofs

13. AI_CONSUMPTION
AI_CONSUMPTION:
  representation: typed_semantic_flow
  parsing: deterministic
  retrieval_keys: [finite_set, cardinality, combinatorics, finite_function, finite_graph, enumeration]
  reasoning_mode: constraint_checked_symbolic_inference
  execution_policy: validate_preconditions_before_execution
  output_requirements: [result, proof_or_derivation, provenance, confidence, version]
  hallucination_guard: do_not_invent_source_axioms_or_verified_results

14. EXAMPLE FLOW
INPUT A = {a,b,c}; B = {1,2}
ASSERT finite(A) AND finite(B)
DERIVE |A| = 3; |B| = 2
APPLY FiniteCartesianProduct
OUTPUT |A × B| = 6
PROOF: enumerate ordered pairs or apply the product cardinality theorem
CONFIDENCE: 1.0 [verified finite inputs and valid derivation]

END FLOW ALGEBRA