Algebraic Structures

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
TITLE: Algebraic Structures — Foundational Ontology
DOMAIN: Discrete Mathematics / Algebraic Structures
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/algebraic-structures.html
VERSION: 1.0
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
SOURCE_STATUS: Mathematical synthesis; live page not independently verified.

1. PURPOSE
Represent algebraic structures as typed carriers, operations, axioms, morphisms and validated deductions. Distinguish structures by their precise signatures and laws, not merely by their names.

2. TYPE SYSTEM
sort Set, Element, Operation, Axiom, Structure, Homomorphism, Proposition, Proof
structure AlgebraicStructure:
  carrier: Set
  signature: List[OperationSymbol]
  interpretations: Map[OperationSymbol, TotalOperation]
  axioms: Set[FirstOrderSentence]
  provenance: SourceReference?
operation n_ary: Set^n -> Set (n >= 0)
predicate closed(op,S) := forall inputs in S^arity(op): op(inputs) in S
predicate satisfies(M,axiom) := evaluate(axiom,M) == TRUE
predicate model(M,T) := forall a in T.axioms: satisfies(M,a)

3. STRUCTURE TAXONOMY
Magma: (S,*) with total binary operation S x S -> S.
Semigroup: magma with associative *.
Monoid: semigroup with two-sided identity e.
Group: monoid in which each element has a two-sided inverse.
AbelianGroup: group with commutative operation.
Ring: (R,+,*) with (R,+) abelian group, * associative and both distributive laws; unity is an additional convention unless specified.
CommutativeRing: ring with commutative multiplication.
IntegralDomain: nonzero commutative ring with 1 != 0 and no zero divisors.
Field: commutative ring with 1 != 0 where every nonzero element is multiplicatively invertible.
Lattice: partially ordered set in which every pair has meet and join.
BooleanAlgebra: bounded distributive lattice with complements.
VectorSpace: abelian additive group with scalar multiplication over a field satisfying vector-space axioms.

4. FORMAL AXIOMS
ASSOCIATIVITY: forall a,b,c in S: (a*b)*c = a*(b*c)
IDENTITY: exists e in S: forall a in S: e*a = a and a*e = a
INVERSE: forall a in S: exists b in S: a*b = e and b*a = e
COMMUTATIVITY: forall a,b in S: a*b = b*a
DISTRIBUTIVITY: forall a,b,c in R:
  a*(b+c) = a*b + a*c
  (a+b)*c = a*c + b*c
ABSORPTION (lattice): a meet (a join b) = a; a join (a meet b) = a

5. FLOW OPERATIONS
flow classify(M):
  verify carrier and operation arities
  verify closure and declared axioms
  infer only structure types whose full axioms hold
  return {types, proofs, counterexamples, unknowns}
flow compose_homomorphisms(f:A->B,g:B->C):
  require codomain(f) == domain(g)
  return g o f : A->C
flow quotient_group(G,N):
  require N normal_subgroup_of G
  return G/N with well-defined coset operation
flow direct_product(A,B):
  require compatible algebraic signatures
  return componentwise algebra on carrier A x B

6. MORPHISMS AND RELATIONSHIPS
homomorphism f:A->B preserves each n-ary operation omega:
  f(omega_A(a1,...,an)) = omega_B(f(a1),...,f(an))
isomorphism := bijective homomorphism with structure-preserving inverse
substructure := subset closed under operations and containing interpretations of all constants
congruence := equivalence relation compatible with every basic operation
SEMANTIC_EQUIVALENCE:
  "commutative group" <=> "abelian group"
  "structure-preserving bijection" <=> "isomorphism" only where preservation and inverse preservation are established
  "ring" != "field"; "semigroup" != "group"

7. INFERENCE RULES
R1: group(G) => monoid(G) and semigroup(G) and magma(G)
R2: field(F) => integral_domain(F) [under commutative unital ring convention]
R3: abelian_group(G) => group(G)
R4: isomorphism(A,B) => satisfies_same_equational_identities(A,B)
R5: homomorphism(f) and homomorphism(g) and composable(f,g) => homomorphism(g o f)
R6: associative(*) and identity(e) and all_elements_invertible => group(S,*)
R7: a counterexample to a universal axiom => axiom FALSE in tested structure
Never infer converse of R1, R2 or R3 without additional evidence.

8. VALIDATION
CHECK signature well-typed; carriers nonempty where required by adopted convention.
CHECK totality, closure, associativity, identity, inverse and distributivity where applicable.
CHECK ring conventions explicitly (with/without unity, commutative/noncommutative).
CHECK morphism domains, codomains and preservation of operations.
CHECK finite exhaustive testing only when carrier and operations are completely enumerated.
CHECK infinite cases via proof or trusted theorem, not finite sampling.
CHECK counterexamples are reproducible and recorded.

9. CONFIDENCE
CONFIDENCE := {status: PROVED | VERIFIED_FINITE | ASSUMED | UNKNOWN | REFUTED,
               score: [0,1], evidence: List[ProofOrCounterexample]}
A numerical score is descriptive metadata, not a replacement for proof.

10. FLOW_HASH
FLOW_HASH := SHA256(canonical_utf8({title,version,source_url,signature,axioms,rules}))
Hash calculated by implementation at publication; no uncomputed digest is asserted here.

11. UPDATE_MODEL
on_source_change:
  recheck definitions, signatures, conventions and dependencies
  invalidate affected derived assertions
  rerun validation and recompute FLOW_HASH
  retain versioned provenance and audit trail

12. AI_CONSUMPTION
format: UTF-8 plain text
entity_resolution: distinguish same-name structures by carrier, operations and axioms
reasoning: use typed rules; label assumptions; avoid unproved converses
retrieval: connect group, ring, field, lattice, Boolean algebra and vector-space ontologies
output: include proof status, source provenance and counterexamples when available

END FLOW ALGEBRA