Boolean Algebra Ontology

FLOW ALGEBRA
TITLE: Boolean Algebra Ontology
DOMAIN: Discrete Mathematics / Algebraic Structures
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/algebraic-structures/boolean-algebra-ontology-3.html
VERSION: 1.0
LANGUAGE: en-GB
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
SOURCE_STATUS: Formal mathematical synthesis; live page not independently verified.

1. ONTOLOGY
flow BooleanAlgebraOntology {
  entity BooleanAlgebra : AlgebraicStructure
  entity CarrierSet : Set
  entity Meet : BinaryOperation
  entity Join : BinaryOperation
  entity Complement : UnaryOperation
  entity Bottom : Constant
  entity Top : Constant
  entity BooleanElement : Element
  entity BooleanHomomorphism : Mapping
  entity BooleanIdeal : Subset
  entity BooleanFilter : Subset
  relation HAS_CARRIER(BooleanAlgebra, CarrierSet)
  relation HAS_OPERATION(BooleanAlgebra, Meet | Join | Complement)
  relation HAS_BOUND(BooleanAlgebra, Bottom | Top)
  relation MEMBER_OF(BooleanElement, CarrierSet)
}

2. TYPED SIGNATURE
B : nonempty Set
meet : B × B -> B
join : B × B -> B
not : B -> B
zero, one : B
BA := (B, meet, join, not, zero, one)
forall a,b,c in B.

3. AXIOMS
COMMUTATIVE: meet(a,b)=meet(b,a); join(a,b)=join(b,a)
ASSOCIATIVE: meet(meet(a,b),c)=meet(a,meet(b,c)); join(join(a,b),c)=join(a,join(b,c))
DISTRIBUTIVE: meet(a,join(b,c))=join(meet(a,b),meet(a,c))
              join(a,meet(b,c))=meet(join(a,b),join(a,c))
IDENTITY: meet(a,one)=a; join(a,zero)=a
COMPLEMENT: meet(a,not(a))=zero; join(a,not(a))=one
NONTRIVIAL_OPTIONAL: zero != one (only for nontrivial Boolean algebras)

4. DERIVED LAWS
IDEMPOTENCE: meet(a,a)=a; join(a,a)=a
ABSORPTION: meet(a,join(a,b))=a; join(a,meet(a,b))=a
ANNIHILATION: meet(a,zero)=zero; join(a,one)=one
INVOLUTION: not(not(a))=a
DE_MORGAN: not(meet(a,b))=join(not(a),not(b))
           not(join(a,b))=meet(not(a),not(b))
BOUND_COMPLEMENTS: not(zero)=one; not(one)=zero
COMPLEMENT_UNIQUENESS: complement of every element is unique

5. ORDER AND SEMANTIC RELATIONS
ORDER(a,b) iff meet(a,b)=a iff join(a,b)=b
BOTTOM: forall a, ORDER(zero,a)
TOP: forall a, ORDER(a,one)
ORDER_REVERSAL: ORDER(a,b) => ORDER(not(b),not(a))
SEMANTIC_EQUIVALENCE:
  meet <-> logical conjunction <-> set intersection (powerset interpretation)
  join <-> logical disjunction <-> set union (powerset interpretation)
  not <-> logical negation <-> relative complement (powerset interpretation)
  zero <-> false <-> empty set (powerset interpretation)
  one <-> true <-> universe (powerset interpretation)
NOTE: These correspondences depend on a chosen interpretation; arbitrary Boolean algebras need not literally be sets of subsets.

6. CONSTRUCTIONS AND EXAMPLES
EXAMPLE_2: ({false,true}, AND, OR, NOT, false, true)
POWERSET: (P(U), intersection, union, U-relative-complement, empty, U)
PRODUCT: componentwise product of Boolean algebras is a Boolean algebra
SUBALGEBRA: subset containing bounds and closed under meet, join, complement
HOMOMORPHISM h:B->C:
  h(meet(a,b))=meet(h(a),h(b))
  h(join(a,b))=join(h(a),h(b))
  h(not(a))=not(h(a)); h(zero)=zero; h(one)=one
ISOMORPHISM: bijective Boolean-algebra homomorphism
FINITE_CARDINALITY: every finite Boolean algebra has 2^n elements for some n>=0

7. EXECUTABLE INFERENCE RULES
RULE BA_MEET_ELIM: x=meet(a,b) => ORDER(x,a) and ORDER(x,b)
RULE BA_JOIN_INTRO: x=join(a,b) => ORDER(a,x) and ORDER(b,x)
RULE BA_DOUBLE_NEG: not(not(a)) => a
RULE BA_DE_MORGAN_MEET: not(meet(a,b)) => join(not(a),not(b))
RULE BA_DE_MORGAN_JOIN: not(join(a,b)) => meet(not(a),not(b))
RULE BA_ABSORB: meet(a,join(a,b)) => a
RULE BA_COMPLEMENT: join(a,not(a)) => one
RULE BA_DUALITY: interchange meet/join and zero/one in any valid identity to obtain its dual

8. VALIDATION
VALIDATE_CARRIER: B is nonempty and operations are total on B
VALIDATE_AXIOMS: verify all identities in section 3
VALIDATE_ORDER: induced relation is reflexive, antisymmetric, transitive
VALIDATE_DERIVED: verify identities in section 4
VALIDATE_FINITE: enumerate all tuples for finite B; otherwise require symbolic proof or trusted theorem
COUNTEREXAMPLE: provide concrete elements and failing axiom when validation fails
STATUS: VALID | INVALID | UNVERIFIED

9. CONFIDENCE
CONFIDENCE:
  mathematical_axioms: 1.0 (definition-level)
  derived_laws: 1.0 (under axioms)
  source_page_alignment: UNVERIFIED
  implementation_results: UNVERIFIED
Do not interpret confidence values as experimental measurements.

10. FLOW_HASH
FLOW_HASH:
  algorithm: SHA-256
  canonicalisation: UTF-8, LF line endings, stable field order
  value: COMPUTE_ON_PUBLICATION
  note: no hash asserted without computing canonical published content

11. UPDATE_MODEL
UPDATE_MODEL:
  version: 1.0
  update_triggers: corrected theorem, changed ontology scope, changed source content
  preserve: source URL, version history, mathematical proof obligations
  revalidate: axioms, rules, equivalences, examples
  recompute: FLOW_HASH after canonical publication

12. AI_CONSUMPTION
AI_CONSUMPTION:
  parse: typed entities, signatures, axioms, derived rules
  reason: use validated rewrite and inference rules only
  distinguish: definition vs derived theorem vs example
  enforce: source verification and proof obligations
  applications: knowledge representation, digital logic, query optimisation, set reasoning, formal verification
  outputs: proof trace, normalised expression, validation result, provenance

END FLOW ALGEBRA