Abelian Group Ontology

FLOW ALGEBRA
TITLE: Abelian Group Ontology
DOMAIN: Discrete Mathematics / Algebraic Structures
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/algebraic-structures/abelian-group-ontology-2.html
VERSION: 1.0
LANGUAGE: en-GB
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
SOURCE_STATUS: Mathematical model based on established abstract algebra; live page not independently verified.

1. ONTOLOGY DEFINITION
AbelianGroup := (G, ⊕, e, inv) where G is a set, ⊕: G × G → G is a binary operation, e ∈ G is an identity, and inv: G → G gives inverses. The operation is associative and commutative.

2. TYPED ENTITIES
GroupElement: element of G
CarrierSet: Set[GroupElement]
BinaryOperation: G × G → G
IdentityElement: GroupElement
InverseMap: G → G
Subgroup: subset H ⊆ G closed under operation and inverses, containing e
Homomorphism: map preserving group operation
Kernel: preimage of identity under homomorphism
Image: set of homomorphism outputs
Coset: a ⊕ H
QuotientGroup: G/H (defined for any subgroup H of an Abelian group)
DirectProduct: componentwise group product
CyclicGroup: group generated by one element

3. CORE AXIOMS
A1 Closure: ∀a,b∈G, a⊕b∈G.
A2 Associativity: ∀a,b,c∈G, (a⊕b)⊕c=a⊕(b⊕c).
A3 Identity: ∀a∈G, e⊕a=a⊕e=a.
A4 Inverse: ∀a∈G, a⊕inv(a)=inv(a)⊕a=e.
A5 Commutativity: ∀a,b∈G, a⊕b=b⊕a.

4. DERIVED LAWS
L1 Cancellation: a⊕b=a⊕c ⇒ b=c.
L2 Unique identity and unique inverse for every element.
L3 inv(inv(a))=a.
L4 inv(a⊕b)=inv(a)⊕inv(b).
L5 a⊕e=a; a⊕inv(a)=e.
L6 Integer multiples: 0·a=e, (m+n)·a=(m·a)⊕(n·a).
L7 n·(a⊕b)=(n·a)⊕(n·b) for n∈ℤ.
L8 All subgroups of Abelian groups are normal.

5. OPERATIONS
ADD(a,b): pre a,b∈G; return a⊕b.
NEGATE(a): pre a∈G; return inv(a).
SUBTRACT(a,b): return a⊕inv(b).
SCALAR_MULTIPLY(n,a): pre n∈ℤ and a∈G; return repeated group operation with signed inverse convention.
GENERATED_SUBGROUP(S): smallest subgroup of G containing S.
QUOTIENT(G,H): pre H≤G; return cosets G/H with (a+H)+(b+H)=(a+b)+H.
DIRECT_PRODUCT(G,H): return Abelian group G×H under componentwise operation.

6. SEMANTIC RELATIONSHIPS
AbelianGroup IS_A Group.
AbelianGroup IS_A CommutativeMonoidWithInverses.
AbelianGroup HAS CarrierSet, BinaryOperation, IdentityElement, InverseMap.
CyclicGroup IS_A AbelianGroup.
SubgroupOfAbelianGroup IS_A AbelianGroup.
QuotientOfAbelianGroup IS_A AbelianGroup.
DirectProductOfAbelianGroups IS_A AbelianGroup.
GroupHomomorphism PRESERVES BinaryOperation and IdentityElement and Inverses.

7. INFERENCE RULES
R1 If group axioms A1–A4 and commutativity A5 hold, classify (G,⊕) as AbelianGroup.
R2 If G is Abelian and H≤G, infer H is Abelian and normal in G.
R3 If G is Abelian and H≤G, infer G/H is Abelian.
R4 If G,H are Abelian, infer G×H is Abelian.
R5 If f:G→H is a group homomorphism and G is Abelian, infer im(f) is Abelian; do not infer H is Abelian unless f is surjective.
R6 If G is cyclic, infer G is Abelian; the converse does not generally hold.
R7 If f:G→H is an isomorphism and G is Abelian, infer H is Abelian.

8. EXAMPLES AND COUNTEREXAMPLES
E1 (ℤ,+,0) is an infinite cyclic Abelian group.
E2 (ℤ/nℤ,+,[0]) is a finite cyclic Abelian group for n≥1.
E3 (ℝ\{0},×,1) is an Abelian group.
E4 (ℤ²,+,(0,0)) is Abelian but not cyclic.
E5 (S₃,composition) is a group but not Abelian.
E6 (ℕ,+,0) is a commutative monoid, not a group (ordinary ℕ lacks additive inverses).

9. VALIDATION
V1 Confirm operation is total and closed on G.
V2 Check associativity and commutativity for all triples/pairs (or via accepted proof).
V3 Verify identity belongs to G and acts on both sides.
V4 Verify each element has an inverse in G.
V5 For finite G, validate operation table is closed, symmetric across diagonal, and meets axioms; symmetry alone is insufficient.
V6 For homomorphism, check f(a⊕b)=f(a)⊞f(b).
V7 Do not confuse Abelian with cyclic; do not assume every Abelian group is finite.

10. SEMANTIC_EQUIVALENCE
Abelian group ≡ commutative group.
Group written additively ≡ (G,+,0,−) when notation is consistent.
Commutativity alone ≠ Abelian group without remaining group axioms.

11. CONFIDENCE
AXIOMATIC_THEOREMS: 1.00 (relative to standard group axioms).
EXAMPLE_CLASSIFICATIONS: 1.00 (under stated definitions).
PAGE_ALIGNMENT: unverified; no confidence score asserted.

12. FLOW_HASH
METHOD: SHA-256 of canonical UTF-8 file content, computed by deployment pipeline.
VALUE: pending canonicalisation and hashing; do not treat as a computed digest.

13. UPDATE_MODEL
TRIGGER: revised source content, notation, mathematical correction, or ontology schema.
PROCESS: compare definitions → revalidate axioms and inferences → update version → recompute FLOW_HASH → publish provenance.

14. AI_CONSUMPTION
INPUT: mathematical query, group operation, carrier set, proof premises.
PIPELINE: parse → type-check → verify axioms → classify group → apply sound inference rules → validate results → return explanation and provenance.
OUTPUT: classification, formal statement, derivation, confidence category, validation status.
GUARDRAILS: no unsupported theorem claims; distinguish assumptions from proved facts; preserve notation and source provenance.