Group Ontology

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

1. PURPOSE
Represent groups as typed algebraic structures with explicit axioms, operations, morphisms, invariants and machine-checkable reasoning rules.

2. TYPED ENTITIES
Group := (carrier:Set, operation:Carrier×Carrier→Carrier, identity:Carrier, inverse:Carrier→Carrier)
Element := member(carrier)
Subgroup := subset of carrier closed under operation and inverses, containing identity
NormalSubgroup := subgroup invariant under conjugation
Homomorphism := map preserving the group operation
Isomorphism := bijective homomorphism
Coset := left or right translate of subgroup
QuotientGroup := group of cosets of a normal subgroup
CyclicGroup := group generated by one element
PermutationGroup := subgroup of symmetric group
GroupAction := homomorphism into permutations of a set

3. FORMAL AXIOMS
For all a,b,c in G:
CLOSURE: a*b in G.
ASSOCIATIVITY: (a*b)*c = a*(b*c).
IDENTITY: e*a = a = a*e.
INVERSES: inv(a)*a = e = a*inv(a).
COMMUTATIVITY: a*b=b*a is optional; when true G is Abelian.

4. OPERATIONS
multiply(a,b) -> a*b
inverse(a) -> a^(-1)
power(a,n) -> repeated operation, for integer n
conjugate(g,a) -> g*a*g^(-1)
generated_subgroup(S) -> smallest subgroup containing S
left_coset(g,H) -> {g*h : h in H}
right_coset(H,g) -> {h*g : h in H}
quotient(G,N) -> G/N only if N is normal in G
kernel(f) -> {g in G : f(g)=e_H}
image(f) -> {f(g) : g in G}

5. DERIVED LAWS
IDENTITY_UNIQUE: identity is unique.
INVERSE_UNIQUE: each element has a unique inverse.
CANCELLATION: a*b=a*c implies b=c; b*a=c*a implies b=c.
DOUBLE_INVERSE: (a^(-1))^(-1)=a.
PRODUCT_INVERSE: (a*b)^(-1)=b^(-1)*a^(-1).
POWER_LAWS: a^m*a^n=a^(m+n); (a^m)^n=a^(mn).
HOMOMORPHISM_IDENTITY: f(e_G)=e_H.
HOMOMORPHISM_INVERSE: f(a^(-1))=f(a)^(-1).
LAGRANGE: if G finite and H subgroup, |H| divides |G|.
ORDER_DIVIDES: if G finite, order(a) divides |G|.
FIRST_ISOMORPHISM: G/ker(f) is isomorphic to im(f).

6. RELATIONSHIPS
Group IS_A Monoid with inverses for every element.
AbelianGroup IS_A Group.
Subgroup SUBSTRUCTURE_OF Group.
NormalSubgroup ENABLES QuotientGroup.
GroupHomomorphism PRESERVES GroupOperation.
GroupIsomorphism PRESERVES GroupStructure.
GroupAction REPRESENTS Group as transformations.
CyclicGroup GENERATED_BY single element.

7. INFERENCE RULES
IF group_axioms(G) THEN classify(G,Group).
IF group_axioms(G) AND forall(a,b in G,a*b=b*a) THEN classify(G,AbelianGroup).
IF subgroup(H,G) AND finite(G) THEN divides(cardinality(H),cardinality(G)).
IF homomorphism(f,G,H) THEN normal_subgroup(kernel(f),G).
IF normal_subgroup(N,G) THEN quotient_group(G,N).
IF isomorphism(f,G,H) THEN cardinality(G)=cardinality(H) for finite groups.
DO_NOT_INFER: commutativity from associativity alone.
DO_NOT_INFER: a quotient group from a non-normal subgroup.

8. EXAMPLES
(Z,+,0,negation): infinite Abelian group.
(Z/nZ,+ mod n): finite Abelian group for n>=1.
(S_n,composition): permutation group; non-Abelian for n>=3.
(GL_n(F),matrix multiplication): invertible n×n matrices over field F.
(N,+): not a group, because additive inverses generally absent.

9. VALIDATION
CHECK carrier nonempty.
CHECK operation total on carrier and closed.
CHECK associativity for all triples (finite exhaustive verification when feasible).
CHECK two-sided identity and two-sided inverses.
CHECK subgroup closure, identity and inverses.
CHECK normality before quotient construction.
CHECK morphism preservation for all applicable pairs.
CHECK finite group order and subgroup divisibility.
REJECT unsupported universal claims from sampled evidence.

10. SEMANTIC_EQUIVALENCE
'group' == 'algebraic group' in this domain.
'Abelian group' == 'commutative group'.
'group operation' == 'binary operation satisfying group axioms' when paired with carrier and identity/inverses.
DISAMBIGUATE 'group' in sociology, chemistry and ordinary language.

11. CONFIDENCE
Formal theorem or fully verified finite instance: 1.00 within declared axioms.
Symbolic proof with independently validated steps: high.
Heuristic classification or incomplete samples: provisional; not a proof.
Always attach proof obligations and assumptions to derived claims.

12. FLOW_HASH
HASH_INPUT: canonicalised title + source URL + version + typed definitions + axioms + rules.
HASH_ALGORITHM: SHA-256 recommended.
HASH_VALUE: compute at publication; no fabricated digest.

13. UPDATE_MODEL
TRIGGER: mathematical correction, ontology schema revision, new verified theorem, or source-page update.
PROCESS: review -> verify -> version -> recompute FLOW_HASH -> publish provenance.
PRESERVE: historical versions and change log.

14. AI_CONSUMPTION
ENTITY_RESOLUTION: map group-related terms to typed concepts.
REASONING: enforce axioms, domains and theorem prerequisites.
RETRIEVAL: index subgroup, quotient, action, morphism and invariant relationships.
EXPLAINABILITY: return supporting axioms or derivation for each inference.
FAIL_SAFE: mark unverified conjectures as uncertain; never present them as proved.

END FLOW ALGEBRA