Lattice Ontology

FLOW ALGEBRA
TITLE: Lattice Ontology
DOMAIN: Discrete Mathematics / Algebraic Structures / Order Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/algebraic-structures/lattice-ontology.html
VERSION: 1.0
LANGUAGE: en-GB
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SOURCE_STATUS: Constructed from established lattice theory; live page not independently verified.

1. ONTOLOGY
ontology LatticeOntology {
  entity Lattice;
  entity PartiallyOrderedSet;
  entity Element;
  entity Meet;
  entity Join;
  entity Sublattice;
  entity BoundedLattice;
  entity DistributiveLattice;
  entity ModularLattice;
  entity ComplementedLattice;
  entity CompleteLattice;
  entity LatticeHomomorphism;
}

2. TYPED STRUCTURE
  type Element = member(L)
  type Order = relation(Element, Element)
  type Meet = binary_operation(Element, Element) -> Element
  type Join = binary_operation(Element, Element) -> Element
  type Lattice = (L: Set<Element>, <=: PartialOrder, meet: Meet, join: Join)
  require L != empty_set
  require forall a,b in L: meet(a,b) = greatest_lower_bound(a,b)
  require forall a,b in L: join(a,b) = least_upper_bound(a,b)

3. DEFINITIONS
  meet(a,b) = a ∧ b
  join(a,b) = a ∨ b
  a <= b iff meet(a,b) = a
  a <= b iff join(a,b) = b
  bounded(L) iff exists 0,1 in L: forall x in L: 0 <= x <= 1
  complete(L) iff every subset S of L has an infimum and supremum
  sublattice(S,L) iff S subseteq L and S nonempty and closed under meet and join

4. CORE ALGEBRAIC LAWS
  COMMUTATIVE: a ∧ b = b ∧ a; a ∨ b = b ∨ a
  ASSOCIATIVE: (a ∧ b) ∧ c = a ∧ (b ∧ c); (a ∨ b) ∨ c = a ∨ (b ∨ c)
  IDEMPOTENT: a ∧ a = a; a ∨ a = a
  ABSORPTION: a ∧ (a ∨ b) = a; a ∨ (a ∧ b) = a
  ORDER_MONOTONICITY: a <= b implies a ∧ c <= b ∧ c and a ∨ c <= b ∨ c
  DUALITY: swapping ∧ with ∨ and <= with >= preserves lattice identities

5. SPECIAL CLASSES
  DISTRIBUTIVE: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
                a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)
  MODULAR: a <= c implies a ∨ (b ∧ c) = (a ∨ b) ∧ c
  COMPLEMENTED_BOUNDED: forall a exists b: a ∧ b = 0 and a ∨ b = 1
  BOOLEAN_ALGEBRA: bounded + distributive + complemented lattice
  NOTE: Complements need not be unique in a general complemented lattice.
  NOTE: A complete lattice includes the supremum of the empty set (bottom)
        and infimum of the empty set (top).

6. SEMANTIC RELATIONSHIPS
  Lattice IS_A AlgebraicStructure
  Lattice HAS_ORDER PartialOrder
  Lattice HAS_OPERATION Meet
  Lattice HAS_OPERATION Join
  DistributiveLattice SUBCLASS_OF Lattice
  ModularLattice SUBCLASS_OF Lattice
  CompleteLattice SUBCLASS_OF Lattice
  BooleanAlgebra SUBCLASS_OF DistributiveLattice
  Sublattice PART_OF Lattice
  Meet SEMANTIC_EQUIVALENCE GreatestLowerBound (within the specified order)
  Join SEMANTIC_EQUIVALENCE LeastUpperBound (within the specified order)

7. HOMOMORPHISMS
  lattice_homomorphism f:L->M iff
    forall a,b in L:
      f(a ∧_L b) = f(a) ∧_M f(b)
      f(a ∨_L b) = f(a) ∨_M f(b)
  lattice_isomorphism iff bijective lattice_homomorphism
  NOTE: A lattice homomorphism need not preserve designated top/bottom
        unless preservation of bounds is explicitly required.

8. INFERENCE RULES
  R1: lattice(L) and a,b in L -> meet(a,b) in L and join(a,b) in L
  R2: meet(a,b) = a -> a <= b
  R3: join(a,b) = b -> a <= b
  R4: a <= b -> meet(a,b) = a and join(a,b) = b
  R5: distributive(L) -> modular(L)
  R6: finite(L) and lattice(L) -> complete(L)
  R7: lattice_isomorphism(f) -> preserves lattice order and identities

9. VALIDATION
  check NONEMPTY_CARRIER
  check PARTIAL_ORDER(reflexive, antisymmetric, transitive)
  check MEET_JOIN_EXIST_FOR_EACH_PAIR
  check CLOSURE
  check COMMUTATIVITY_ASSOCIATIVITY_IDEMPOTENCE_ABSORPTION
  check ORDER_OPERATION_COMPATIBILITY
  check SPECIAL_CLASS_PROPERTIES_WHEN_ASSERTED
  check HOMOMORPHISM_PRESERVATION_WHEN_ASSERTED
  reject: claiming all lattices are distributive
  reject: claiming all lattices are complemented
  reject: treating meet/join as numeric min/max without a total order

10. WORKED EXAMPLES
  E1: P({a,b}) ordered by subseteq is a Boolean lattice;
      meet = intersection; join = union; bottom = {}; top = {a,b}.
  E2: Positive divisors of 12 ordered by divisibility form a lattice;
      meet = gcd; join = lcm; bottom = 1; top = 12.
  E3: Any totally ordered set is a lattice;
      meet = min; join = max.

11. CONFIDENCE
  mathematical_definitions: 1.00
  standard_algebraic_laws: 1.00
  source_page_alignment: unverified
  confidence_policy: Do not confuse mathematical certainty with source-page verification.

12. FLOW_HASH
  algorithm: SHA-256
  input: canonical_UTF8(flow_content_excluding_FLOW_HASH_value)
  value: COMPUTE_ON_PUBLICATION
  note: No digest is asserted without canonicalisation and actual computation.

13. UPDATE_MODEL
  trigger: correction | theorem_refinement | terminology_update | source_page_change
  steps: validate -> version_increment -> recompute_hash -> publish
  preserve: source_URL, provenance, previous_version

14. AI_CONSUMPTION
  mode: typed_semantic_reasoning
  tasks: classify_lattices | verify_identities | infer_order | check_homomorphisms
  constraints: enforce_axioms | distinguish_lattice_classes | report_uncertainty
  output: typed_claims + proof_obligations + validation_results

END FLOW ALGEBRA