Intersection Ontology

FLOW ALGEBRA
TITLE: Intersection Ontology
VERSION: 1.0
DOMAIN: Discrete Mathematics / Set Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/set-theory/intersection-ontology-2.html
STATUS: Domain-based formalisation; source page not independently verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

1. PURPOSE
Model set intersection as a typed, deterministic operation, including membership, identities, laws, proof obligations, and machine-consumable reasoning.

2. TYPED ENTITIES
  Element<T>: an object in the universe of discourse U<T>.
  Set<T>: a collection of distinct elements of type T.
  Universe<T>: the declared universe of discourse.
  Intersection<T>: a derived Set<T> with operands A and B.
  MembershipClaim<T>: proposition x IN A with a truth value and provenance.
  SetExpression<T>: well-typed symbolic expression built from set operators.
  Proof: derivation of a claim from axioms or established rules.
  Context: chosen universe, type system, assumptions, and version.

3. TYPED RELATIONSHIPS
  MEMBER_OF: Element<T> x Set<T> -> Boolean
  INTERSECTS_WITH: Set<T> x Set<T> -> Boolean [true iff intersection nonempty]
  INTERSECTION_OF: Intersection<T> x (Set<T>,Set<T>)
  SUBSET_OF: Set<T> x Set<T> -> Boolean
  DISJOINT_WITH: Set<T> x Set<T> -> Boolean
  EQUIVALENT_SET: Set<T> x Set<T> -> Boolean [extensional equality]
  DERIVED_FROM: Claim x Proof

4. CORE OPERATION
  intersection(A:Set<T>, B:Set<T>) -> Set<T>
  A INTERSECT B = {x:T | x IN A AND x IN B}
  For every x:T: x IN (A INTERSECT B) IFF (x IN A AND x IN B).
  Type constraint: operands share compatible element types and context.

5. ALGEBRAIC LAWS
  Commutativity: A INTERSECT B = B INTERSECT A.
  Associativity: (A INTERSECT B) INTERSECT C = A INTERSECT (B INTERSECT C).
  Idempotence: A INTERSECT A = A.
  Identity: A INTERSECT U = A, provided A SUBSET_OF U.
  Annihilator: A INTERSECT EMPTY = EMPTY.
  Absorption: A INTERSECT (A UNION B) = A.
  Distributivity: A INTERSECT (B UNION C) = (A INTERSECT B) UNION (A INTERSECT C).
  De Morgan: complement(A INTERSECT B) = complement(A) UNION complement(B), relative to U.
  Subset: (A INTERSECT B) SUBSET_OF A AND (A INTERSECT B) SUBSET_OF B.
  Disjointness: DISJOINT_WITH(A,B) IFF A INTERSECT B = EMPTY.

6. FINITE CARDINALITY
  If A and B are finite:
    |A INTERSECT B| = |A| + |B| - |A UNION B|.
    0 <= |A INTERSECT B| <= min(|A|, |B|).
  Do not apply finite-cardinality arithmetic to infinite sets without a specified cardinal framework.

7. INFERENCE RULES
  R1: MEMBER_OF(x,A) AND MEMBER_OF(x,B) => MEMBER_OF(x,intersection(A,B)).
  R2: MEMBER_OF(x,intersection(A,B)) => MEMBER_OF(x,A) AND MEMBER_OF(x,B).
  R3: SUBSET_OF(A,B) => intersection(A,B) = A.
  R4: intersection(A,B) = EMPTY => DISJOINT_WITH(A,B).
  R5: intersection(A,B) = intersection(B,A) [commutativity].
  R6: A = B => intersection(A,C) = intersection(B,C) [substitution].
  All rules require well-typed inputs and an explicit reasoning context.

8. EXAMPLE EXECUTION
  INPUT: A={1,2,3,5}; B={2,3,4}; universe={1,2,3,4,5}.
  EVALUATE: intersection(A,B).
  OUTPUT: {2,3}.
  PROOF: 2 and 3 occur in both operands; 1,4,5 fail at least one membership test.

9. CONFIDENCE
  Exact symbolic results with verified premises: confidence=1.0.
  Claims inferred from incomplete, noisy, or externally extracted data: confidence must be explicitly calibrated; never silently treat unknown membership as false.
  Separate mathematical proof validity from confidence in input data.

10. SEMANTIC_EQUIVALENCE
  Canonical operator: SET_INTERSECTION.
  Accepted notation aliases: INTERSECT, intersection, ∩.
  Equivalent expressions require shared universe, type, and extensional semantics.
  Distinguish intersection from union, symmetric difference, and relational joins.

11. PROVENANCE
  Record source URL, operand provenance, context, proof rule IDs, transformation history, author, and revision timestamp.
  No claim of independent source-page extraction is made.

12. FLOW_HASH
  Canonicalise schema, operands, expression trees, and context before hashing.
  FLOW_HASH := SHA256(canonical_serialisation(flow_definition, version, context)).
  Recompute after semantic changes; hash is an integrity identifier, not a proof of truth.

13. UPDATE_MODEL
  Validate new operands and schema; invalidate cached intersections when operand content, type, universe, or assumptions change.
  Preserve version history and provenance; rerun dependent inference rules.

14. AI_CONSUMPTION
  Expose typed entities, operation signature, axioms, proof traces, confidence, provenance, and FLOW_HASH.
  Require type checking, contradiction detection, and explicit unknown handling before automated downstream use.
  Output contract: {operation, operands, result, context, proof, confidence, provenance, flow_hash}.

END FLOW ALGEBRA