Group Ontology

FLOW ALGEBRA
TITLE: Group Ontology
VERSION: 4.0
DOMAIN: Mathematical Entities / Algebraic Entities / Group
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/algebraic-entities/group-ontology-3.html
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
STATUS: Independently constructed; source-page correspondence not verified.

01. ONTOLOGY IDENTITY
ENTITY Group : AlgebraicStructure
ENTITY CarrierSet : Set
ENTITY BinaryOperation : Function(CarrierSet x CarrierSet -> CarrierSet)
ENTITY IdentityElement : Element(CarrierSet)
ENTITY InverseElement : Element(CarrierSet)
ENTITY Subgroup : Group
ENTITY GroupHomomorphism : StructurePreservingMap
ENTITY GroupIsomorphism : BijectiveGroupHomomorphism
ENTITY NormalSubgroup : Subgroup
ENTITY QuotientGroup : Group
ENTITY Coset : Subset(CarrierSet)
ENTITY GroupAction : Action(Group, Set)

02. FORMAL MODEL
Group(G) := (S, *) where S is a nonempty set and *: S x S -> S.
AXIOM G1 CLOSURE: forall a,b in S, a*b in S.
AXIOM G2 ASSOCIATIVITY: forall a,b,c in S, (a*b)*c = a*(b*c).
AXIOM G3 IDENTITY: exists e in S, forall a in S, e*a = a*e = a.
AXIOM G4 INVERSE: forall a in S, exists b in S, a*b = b*a = e.
THEOREM: Identity is unique.
THEOREM: Every element has a unique inverse.
THEOREM: Left and right cancellation hold.
THEOREM: (a*b)^(-1) = b^(-1)*a^(-1).

03. GROUP CLASSIFICATION
AbelianGroup(G) iff forall a,b in S, a*b = b*a.
FiniteGroup(G) iff |S| is finite.
InfiniteGroup(G) iff |S| is infinite.
TrivialGroup(G) iff |S| = 1.
CyclicGroup(G) iff exists g in S, S = <g>.
PermutationGroup(G) iff G is a subgroup of Sym(X) for some set X.
DihedralGroup(D_n) := symmetries of a regular n-gon, order 2n (n >= 3).
DirectProduct(G,H) := G x H with componentwise operation.

04. SUBGROUPS AND COSETS
Subgroup(H,G) iff H subseteq G and H is a group under G's operation.
SUBGROUP_TEST: Nonempty H subseteq G and forall a,b in H, a*b^(-1) in H.
NormalSubgroup(N,G) iff forall g in G, gNg^(-1) = N.
LeftCoset(g,H) := {g*h : h in H}.
RightCoset(H,g) := {h*g : h in H}.
QuotientGroup(G/N) is defined when N is normal in G.
LAGRANGE: For finite G and subgroup H, |H| divides |G|.
COROLLARY: For finite G, order(g) divides |G|.

05. MORPHISMS
Homomorphism(f:G->H) iff forall a,b in G, f(a*b)=f(a)*f(b).
Kernel(f) := {g in G : f(g)=e_H}.
Image(f) := {f(g) : g in G}.
Kernel(f) is normal in G; Image(f) is a subgroup of H.
Isomorphism(f) iff f is a bijective homomorphism.
FIRST_ISOMORPHISM_THEOREM: G/Kernel(f) is isomorphic to Image(f).
Automorphism(G) := isomorphism G -> G.

06. RELATIONSHIP GRAPH
Group --IS_A--> AlgebraicStructure
Group --HAS_COMPONENT--> CarrierSet
Group --HAS_OPERATION--> BinaryOperation
Group --SATISFIES--> Associativity
Group --HAS_UNIQUE--> IdentityElement
Group --HAS_INVERSE_FOR_EACH--> Element
AbelianGroup --SUBCLASS_OF--> Group
CyclicGroup --SUBCLASS_OF--> AbelianGroup
Subgroup --SUBSTRUCTURE_OF--> Group
NormalSubgroup --SUBCLASS_OF--> Subgroup
NormalSubgroup --ENABLES--> QuotientGroup
GroupHomomorphism --PRESERVES--> GroupOperation
GroupIsomorphism --ESTABLISHES--> StructuralEquivalence
GroupAction --ACTS_ON--> Set

07. INFERENCE RULES
RULE R01: IF associative_closed_operation(S,*) AND two_sided_identity(S,*) AND inverses_for_all(S,*) THEN Group(S,*).
RULE R02: IF Group(G) AND commutative(G) THEN AbelianGroup(G).
RULE R03: IF Group(G) AND generated_by_single_element(G) THEN CyclicGroup(G).
RULE R04: IF CyclicGroup(G) THEN AbelianGroup(G).
RULE R05: IF Group(G) AND finite_carrier(G) THEN FiniteGroup(G).
RULE R06: IF Group(G) AND |G|=1 THEN TrivialGroup(G).
RULE R07: IF Subgroup(H,G) AND finite(G) THEN divides(|H|,|G|).
RULE R08: IF Homomorphism(f,G,H) THEN NormalSubgroup(Kernel(f),G).
RULE R09: IF Homomorphism(f,G,H) THEN Subgroup(Image(f),H).
RULE R10: IF Homomorphism(f,G,H) AND bijective(f) THEN Isomorphism(f,G,H).
RULE R11: IF NormalSubgroup(N,G) THEN defined(QuotientGroup(G,N)).
RULE R12: IF Isomorphism(G,H) THEN same_group_properties(G,H).
RULE R13: IF Group(G) AND a in G THEN inverse(inverse(a))=a.
RULE R14: IF Group(G) AND a,b in G THEN inverse(a*b)=inverse(b)*inverse(a).
RULE R15: IF Group(G) AND finite(G) THEN forall a in G, a^|G|=e.
RULE R16: IF Group(G) AND prime(|G|) THEN CyclicGroup(G).
RULE R17: IF Homomorphism(f,G,H) THEN f(e_G)=e_H.
RULE R18: IF Homomorphism(f,G,H) THEN f(inverse(g))=inverse(f(g)).
RULE R19: IF Subgroup(H,G) AND normal(H,G) THEN NormalSubgroup(H,G).
RULE R20: IF Group(G) AND Subgroup(H,G) THEN Group(H).

08. VALIDATION CONSTRAINTS
CHECK V01: Carrier set must be nonempty.
CHECK V02: Binary operation must be total and closed on the carrier.
CHECK V03: Associativity must be established, not presumed.
CHECK V04: Identity must work on both sides.
CHECK V05: Inverse must work on both sides for every element.
CHECK V06: Commutativity is optional, not a group axiom.
CHECK V07: A subgroup inherits the parent operation.
CHECK V08: A quotient group requires normality.
CHECK V09: Homomorphism must preserve the specified operations.
CHECK V10: Do not infer cyclicity from abelianity.
CHECK V11: Lagrange's divisibility condition does not imply a subgroup of every divisor order.
CHECK V12: For infinite groups, do not apply finite-order cardinality results unconditionally.

09. MATHEMATICAL EXAMPLES
EXAMPLE E01: (Z,+) is an infinite abelian cyclic group; generator 1 or -1.
EXAMPLE E02: (Z/nZ,+) is a finite abelian cyclic group of order n (n>=1).
EXAMPLE E03: (R\{0},multiplication) is an infinite abelian group.
EXAMPLE E04: (S_3,composition) is a nonabelian group of order 6.
EXAMPLE E05: (GL_n(R),matrix multiplication) is a group; generally nonabelian for n>=2.
EXAMPLE E06: (N,+) is not a group: additive inverses are absent.
EXAMPLE E07: (Z,multiplication) is not a group: most elements lack multiplicative inverses.
EXAMPLE E08: 2Z is a subgroup of (Z,+).
EXAMPLE E09: The sign map S_n -> {+1,-1} is a group homomorphism for n>=2.
EXAMPLE E10: The kernel of the sign map is A_n, a normal subgroup of S_n.

10. SEMANTIC EQUIVALENCE
ALIAS: Group = Algebraic Group (in elementary abstract algebra context).
ALIAS: Commutative Group = Abelian Group.
ALIAS: Group Morphism = Group Homomorphism.
DISTINGUISH: Group != Semigroup; group additionally requires identity and inverses.
DISTINGUISH: Group != Monoid; group additionally requires inverses.
DISTINGUISH: Subgroup != Normal Subgroup.
DISTINGUISH: Isomorphism != Arbitrary Homomorphism.
DISTINGUISH: Group (mathematical) != Informal Collection.

11. FLOW EXECUTION
INPUT: Candidate set S, binary operation *, optional properties and witnesses.
STEP 1: Confirm S is nonempty and * is total on S x S.
STEP 2: Verify closure and associativity.
STEP 3: Find and verify a two-sided identity e.
STEP 4: Verify two-sided inverse for each element.
STEP 5: Classify as group; optionally test commutativity, cardinality and generation.
STEP 6: Identify subgroups, morphisms, actions and related structures when supplied.
STEP 7: Apply inference rules with stated preconditions.
STEP 8: Record proof objects, counterexamples, confidence and provenance.
OUTPUT: Typed group entity, verified properties, relationship edges and proof status.
FAILURE: Return missing axiom or counterexample; do not label a non-group as group.

12. CONFIDENCE
CONFIDENCE.TYPE: ProofBacked | DefinitionBacked | Computed | Hypothesised | Unverified.
CONFIDENCE.HIGH: All relevant axioms checked or established by a valid theorem.
CONFIDENCE.MEDIUM: Valid evidence exists but verification is incomplete.
CONFIDENCE.LOW: Informal analogy, incomplete axioms or unverified assertion.
CONFIDENCE.POLICY: Mathematical truth is not probabilistic; confidence describes evidence quality.

13. PROVENANCE AND FLOW HASH
FLOW_HASH.ALGORITHM: SHA-256
FLOW_HASH.INPUT: Canonical UTF-8 serialisation of ontology ID, version, axioms, rules and relations.
FLOW_HASH.VALUE: COMPUTE_ON_EXPORT (no fabricated hash).
PROVENANCE.SOURCE: SOURCE_URL above.
PROVENANCE.VERIFICATION: Source page not independently verified in this export.
PROVENANCE.AUTHORSHIP: Mark Higgins — Sky (Recursive AI).

14. UPDATE MODEL
UPDATE_MODEL.MODE: Versioned append-and-revise.
UPDATE_MODEL.TRIGGER: Corrected proof, new mathematical relationship, changed ontology taxonomy or source revision.
UPDATE_MODEL.REQUIRE: Preserve prior versions and record reason, date and affected rules.
UPDATE_MODEL.CHECK: Revalidate axiom dependencies and regenerate FLOW_HASH after changes.
UPDATE_MODEL.COMPATIBILITY: Do not silently redefine Group or weaken its axioms.

15. AI CONSUMPTION
AI_CONSUMPTION.FORMAT: Plain UTF-8 typed ontology flow.
AI_CONSUMPTION.TASKS: Entity extraction; theorem retrieval; symbolic classification; knowledge-graph population; proof assistance.
AI_CONSUMPTION.REASONING: Use rules only with validated premises and explicit domain constraints.
AI_CONSUMPTION.UNCERTAINTY: Mark unproved claims as unverified; do not invent proofs.
AI_CONSUMPTION.INTEROPERABILITY: Link Group to Semigroup, Monoid, Ring, Field, VectorSpace, Homomorphism and Set ontologies.
AI_CONSUMPTION.OUTPUT: Claim + supporting rule/axiom + assumptions + proof status + provenance.

END FLOW ALGEBRA