Automorphism Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of automorphisms within abstract algebra and linear algebra. An automorphism is an isomorphism from a mathematical structure to itself, equivalently an invertible endomorphism. Automorphisms encode internal symmetries of algebraic structures. This record is designed for semantic retrieval, recursive mathematical reasoning, ontology linking, knowledge-graph construction and AI consumption within the Kireth Research Semantic Library.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Automorphism Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Algebra"
 SPECIALISATION = "Abstract Algebra / Linear Algebra"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/automorphism-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Automorphism
 SYMBOLIC_FORMS = ["f: A -> A", "Aut(A)", "Aut(G)", "Aut(V)"]
 ENTITY_CLASS = SelfIsomorphism
 PRIMARY_CONTEXT = AbstractAlgebra
 SECONDARY_CONTEXT = LinearAlgebra
 DEFINITION = "An automorphism is an isomorphism from a mathematical structure to itself."
 EQUIVALENT_DEFINITION = "An automorphism is an invertible endomorphism."
 STRUCTURAL_ROLE = "Represents an internal symmetry of an algebraic structure."
}

SCOPE {
 INCLUDE = [
  Automorphism, Isomorphism, Endomorphism,
  SelfIsomorphism, Invertibility, Bijectivity,
  InverseAutomorphism, IdentityAutomorphism,
  Composition, AutomorphismGroup,
  InnerAutomorphism, OuterAutomorphism,
  LinearAutomorphism, MatrixRepresentation,
  StructuralSymmetry
 ]
 EXCLUDE_AS_PRIMARY = [
  ArbitraryPermutation,
  NonStructurePreservingSelfMap,
  Homeomorphism,
  PhysicalSymmetryWithoutAlgebraicContext
 ]
 DISAMBIGUATION_RULE = "Interpret automorphism as a structure-preserving bijective self-map when context includes groups, rings, modules, vector spaces, homomorphisms, endomorphisms, isomorphisms or algebraic symmetry."
}

CORE_CONCEPTS {

 Automorphism {
  definition = "A bijective homomorphism from a structure A to itself."
  canonical_form = "f: A -> A"
  conditions = [Endomorphism, Bijectivity]
 }

 SelfIsomorphism {
  equivalence = "SelfIsomorphism ≡ Automorphism"
 }

 Invertibility {
  definition = "An automorphism possesses a two-sided structure-preserving inverse."
  laws = ["f^-1 o f = id_A", "f o f^-1 = id_A"]
 }

 IdentityAutomorphism {
  definition = "The identity map id_A: A -> A."
  role = "Identity element of the automorphism group."
 }

 InverseAutomorphism {
  definition = "The inverse of an automorphism, itself an automorphism."
 }

 AutomorphismGroup {
  definition = "The set Aut(A) of all automorphisms of A under composition."
  operation = "Composition"
  identity = "id_A"
  inverse = "inverse automorphism"
 }

 LinearAutomorphism {
  definition = "An invertible linear endomorphism T: V -> V."
  finite_matrix_criterion = "For finite-dimensional V, T is an automorphism iff its representing matrix is invertible."
 }

 InnerAutomorphism {
  group_form = "phi_g(x)=g*x*g^-1"
  definition = "For a group, an automorphism induced by conjugation by a group element."
 }

 OuterAutomorphism {
  definition = "An automorphism not represented by an inner automorphism; formally studied via Aut(G)/Inn(G)."
 }
}

RELATIONS {
 Automorphism --is_special_case_of--> Isomorphism
 Automorphism --is_special_case_of--> Endomorphism
 Automorphism --requires--> Bijectivity
 Automorphism --has_inverse--> Automorphism
 Automorphism --preserves--> AlgebraicStructure
 Automorphism --represents--> StructuralSymmetry
 IdentityAutomorphism --is--> Automorphism
 Automorphism --belongs_to--> AutomorphismGroup
 AutomorphismGroup --is_group_under--> Composition
 InnerAutomorphism --is_special_case_of--> Automorphism
 OuterAutomorphism --is_classified_relative_to--> InnerAutomorphism
 LinearAutomorphism --is_special_case_of--> LinearEndomorphism
 LinearAutomorphism --can_be_represented_by--> InvertibleMatrix
}

OPERATORS {
 IS_AUTOMORPHISM(f,A) :=
  IS_HOMOMORPHISM(f,A,A)
  AND IS_BIJECTIVE(f)

 IS_AUTOMORPHISM_ALT(f,A) :=
  IS_ENDOMORPHISM(f,A)
  AND IS_INVERTIBLE(f)

 AUTOMORPHISM_GROUP(A) :=
  {f | IS_AUTOMORPHISM(f,A)}

 COMPOSE(g,f) :=
  g o f

 INVERSE(f) :=
  f^-1 IF IS_AUTOMORPHISM(f)

 IDENTITY_AUTOMORPHISM(A) :=
  id_A

 IS_LINEAR_AUTOMORPHISM(T,V) :=
  IS_LINEAR(T)
  AND Domain(T)==V
  AND Codomain(T)==V
  AND IS_BIJECTIVE(T)

 INNER_AUTOMORPHISM(g,G) :=
  map x -> g*x*g^-1

 ITERATE(f,n) :=
  f composed with itself n times
}

DEFINITIONS {
 D1 = "An automorphism is an isomorphism from a structure to itself."
 D2 = "An automorphism is equivalently an invertible endomorphism."
 D3 = "The identity map is an automorphism."
 D4 = "The inverse of an automorphism is an automorphism."
 D5 = "The composition of automorphisms is an automorphism."
 D6 = "Aut(A) forms a group under composition."
 D7 = "A linear automorphism is an invertible linear transformation T: V -> V."
 D8 = "For finite-dimensional vector spaces, a linear automorphism is represented by an invertible square matrix."
 D9 = "For groups, conjugation by an element defines an inner automorphism."
 D10 = "Automorphisms describe internal structural symmetries."
}

AXIOMS {
 A1_SELF_ISOMORPHISM:
  IF f is an automorphism of A
  THEN f: A -> A is an isomorphism

 A2_STRUCTURE_PRESERVATION:
  IF f is an automorphism
  THEN f preserves all operations defining A

 A3_BIJECTIVITY:
  IF f is an automorphism
  THEN f is injective AND surjective

 A4_IDENTITY:
  id_A is an automorphism

 A5_INVERSE:
  IF f is an automorphism
  THEN f^-1 is an automorphism

 A6_COMPOSITION:
  IF f and g are automorphisms of A
  THEN g o f is an automorphism of A

 A7_GROUP_STRUCTURE:
  Aut(A) with composition satisfies closure, associativity, identity and inverse axioms
}

REASONING {

 R1_AUTOMORPHISM_TEST {
  INPUT = [mapping f, structure A]
  TEST_1 = Domain(f)==A
  TEST_2 = Codomain(f)==A
  TEST_3 = verify_structure_preservation
  TEST_4 = verify_injectivity
  TEST_5 = verify_surjectivity
  IF all_tests_true
   RETURN Automorphism
  ELSE
   RETURN NotAutomorphism
 }

 R2_ENDOMORPHISM_ROUTE {
  INPUT = endomorphism f
  TEST = invertibility(f)
  IF invertible
   RETURN Automorphism
  ELSE
   RETURN NonInvertibleEndomorphism
 }

 R3_INVERSE {
  INPUT = automorphism f
  CONSTRUCT = inverse_function(f)
  VERIFY = structure_preservation
  RETURN = inverse_automorphism
 }

 R4_COMPOSITION {
  INPUT = automorphisms f,g
  COMPUTE = g o f
  RETURN = Automorphism
 }

 R5_LINEAR_AUTOMORPHISM {
  INPUT = linear endomorphism T: V -> V
  IF finite_dimensional
   COMPUTE = representing_matrix_A
   TEST = invertibility(A)
   IF invertible RETURN LinearAutomorphism
   ELSE RETURN NotAutomorphism
 }

 R6_GROUP_INNER_AUTOMORPHISM {
  INPUT = [group G, element g]
  DEFINE = "phi_g(x)=g*x*g^-1"
  VERIFY = homomorphism_and_bijection
  RETURN = InnerAutomorphism
 }

 R7_AUTOMORPHISM_GROUP {
  INPUT = structure A
  COLLECT = all_structure_preserving_bijections_A_to_A
  OPERATION = composition
  RETURN = Aut(A)
 }

 R8_ITERATION {
  INPUT = [automorphism f, integer n]
  IF n>=0
   RETURN repeated_composition(f,n)
  ELSE
   RETURN repeated_composition(f^-1,-n)
 }
}

CONSTRAINTS {
 C1 = "An automorphism must map a structure to itself."
 C2 = "A self-bijection is not necessarily an automorphism unless it preserves the required structure."
 C3 = "Every automorphism is an endomorphism, but not every endomorphism is an automorphism."
 C4 = "Every automorphism is an isomorphism, but an isomorphism between distinct structures is not an automorphism of either unless source and target coincide."
 C5 = "Automorphism composition is generally not commutative."
 C6 = "For finite-dimensional linear automorphisms, the representing matrix must be invertible."
 C7 = "Inner and outer automorphism terminology depends on the algebraic structure, especially group context."
 C8 = "Do not confuse algebraic automorphism with unrelated uses of automatic transformation."
}

SEMANTIC_EQUIVALENCE {
 "Automorphism" <=> "SelfIsomorphism"
 "Automorphism" <=> "InvertibleEndomorphism"
 "LinearAutomorphism" <=> "InvertibleLinearEndomorphism"
 "Aut(A)" <=> "AutomorphismGroupOfA"
 "IdentityAutomorphism" <=> "IdentityIsomorphismOnA"
}

VALIDATION {
 V1 = "Every automorphism is an isomorphism."
 V2 = "Every automorphism is an endomorphism."
 V3 = "Every automorphism is bijective."
 V4 = "Identity map is an automorphism."
 V5 = "Inverse of an automorphism is an automorphism."
 V6 = "Composition of automorphisms is an automorphism."
 V7 = "Aut(A) forms a group under composition."
 V8 = "An invertible endomorphism is an automorphism."
 V9 = "A finite-dimensional linear automorphism has an invertible square matrix representation."
 V10 = "Group conjugation defines an inner automorphism."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 AUTOMORPHISM_DEFINITION = 0.99
 SELF_ISOMORPHISM_RELATION = 0.99
 INVERTIBLE_ENDOMORPHISM_RELATION = 0.99
 AUTOMORPHISM_GROUP_RELATION = 0.99
 INVERSE_RELATION = 0.99
 LINEAR_MATRIX_CRITERION = 0.99
 INNER_AUTOMORPHISM_RELATION = 0.99

 POLICY = "Reduce confidence when the algebraic structure, preservation laws or self-map condition are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  AbstractAlgebraReasoning,
  LinearAlgebraReasoning,
  EntityResolution,
  SymmetryReasoning,
  GroupReasoning,
  MatrixReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "automorphism" -> Automorphism
  "self-isomorphism" -> Automorphism
  "invertible endomorphism" -> Automorphism
  "linear automorphism" -> LinearAutomorphism
  "automorphism group" -> AutomorphismGroup
  "Aut(A)" -> AutomorphismGroup
  "inner automorphism" -> InnerAutomorphism
  "outer automorphism" -> OuterAutomorphism
 }

 INTERPRETATION_ORDER = [
  identify_algebraic_structure,
  identify_mapping_rule,
  verify_self_map,
  verify_structure_preservation,
  verify_injectivity,
  verify_surjectivity,
  establish_automorphism,
  construct_inverse,
  analyse_composition,
  classify_linear_case_when_applicable,
  classify_inner_or_outer_context_when_applicable,
  derive_automorphism_group_relations,
  return_semantic_result
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("automorphism")
  THEN activate(AutomorphismOntology)

 E2:
  IF request=="test automorphism"
  THEN verify_self_map
  AND verify_homomorphism
  AND verify_bijectivity

 E3:
  IF endomorphism_known
  THEN test_invertibility

 E4:
  IF request=="find inverse"
  AND automorphism_established
  THEN construct_inverse

 E5:
  IF request=="compose automorphisms"
  THEN compose
  AND classify_result(Automorphism)

 E6:
  IF linear_endomorphism_context
  AND finite_dimensional
  THEN test_matrix_invertibility

 E7:
  IF group_context
  AND mapping_has_form("g*x*g^-1")
  THEN classify(InnerAutomorphism)

 E8:
  IF request=="automorphism group"
  THEN collect_all_automorphisms
  AND use_composition_as_group_operation
}

RECURSION {
 ENTRY = Automorphism

 EXPAND(Automorphism) -> [
  Isomorphism,
  Endomorphism,
  Homomorphism,
  Bijectivity,
  InverseAutomorphism,
  IdentityAutomorphism,
  Composition,
  AutomorphismGroup,
  StructuralSymmetry
 ]

 EXPAND(LinearAutomorphism) -> [
  LinearEndomorphism,
  LinearTransformation,
  Matrix,
  Determinant,
  Rank,
  Kernel,
  Eigenstructure
 ]

 EXPAND(AutomorphismGroup) -> [
  Group,
  Composition,
  IdentityAutomorphism,
  InverseAutomorphism,
  InnerAutomorphism,
  OuterAutomorphism
 ]

 EXPAND(InnerAutomorphism) -> [
  Group,
  Conjugation,
  NormalSubgroup,
  Centre
 ]

 STOP_CONDITION =
  "Stop when automorphism status or the requested symmetry/group property is resolved with sufficient confidence."
}

ROUTING {
 PARENT_ONTOLOGY = "Isomorphism Ontology"

 IMMEDIATE_CONTEXTS = [
  "Endomorphism Ontology",
  "Homomorphism Ontology",
  "Linear Transformation Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Isomorphism Ontology",
  "Endomorphism Ontology",
  "Homomorphism Ontology",
  "Transformation Ontology",
  "Linear Transformation Ontology",
  "Matrix Ontology",
  "Determinant Ontology",
  "Rank Ontology",
  "Kernel Ontology",
  "Image Ontology",
  "Vector Space Ontology",
  "Group Ontology"
 ]

 ROUTE_RULES = [
  "Automorphism -> Isomorphism for structural-equivalence reasoning",
  "Automorphism -> Endomorphism for self-map reasoning",
  "Automorphism -> Homomorphism for operation-preservation reasoning",
  "Automorphism -> Linear Transformation for vector-space automorphisms",
  "Automorphism -> Matrix for finite-dimensional representation",
  "Automorphism -> Determinant or Rank for matrix invertibility reasoning",
  "Automorphism -> Group for Aut(A), inner and outer automorphism reasoning"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible concepts, operators or relations"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve automorphism as self-isomorphism.",
  "Preserve equivalence with invertible endomorphism.",
  "Preserve closure, identity and inverses under composition.",
  "Preserve automorphism-group semantics.",
  "Preserve internal-symmetry interpretation.",
  "Preserve matrix invertibility criterion for finite-dimensional linear automorphisms.",
  "Explicitly distinguish automorphisms from arbitrary self-bijections."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "An automorphism is an isomorphism from a mathematical structure to itself, equivalently an invertible endomorphism."

 SYMBOLIC_SUMMARY =
  "f: A -> A"

 GROUP_SUMMARY =
  "Aut(A) = group of automorphisms of A under composition"

 INVERSE_SUMMARY =
  "f in Aut(A) => f^-1 in Aut(A)"

 LINEAR_SUMMARY =
  "Linear automorphism = invertible linear transformation T: V -> V"

 MATRIX_SUMMARY =
  "For finite-dimensional V, T is a linear automorphism iff its representing matrix is invertible."

 INNER_SUMMARY =
  "For groups, phi_g(x)=g*x*g^-1 defines an inner automorphism."
}

FLOW {
 INPUT
 -> detect_term("automorphism")
 -> identify_algebraic_structure
 -> identify_mapping_rule
 -> verify_domain_equals_codomain
 -> verify_structure_preservation
 -> verify_injectivity
 -> verify_surjectivity
 -> establish_self_isomorphism
 -> classify_automorphism
 -> derive_inverse
 -> analyse_composition
 -> connect_to_automorphism_group
 -> classify_linear_case_when_applicable
 -> classify_inner_or_outer_context_when_applicable
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "An automorphism is an isomorphism from a mathematical structure to itself and therefore an invertible endomorphism. Automorphisms represent internal symmetries, are closed under composition, possess inverse automorphisms and collectively form the automorphism group Aut(A). In linear algebra they correspond to invertible linear self-maps and, in finite dimensions, to invertible square matrices."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-ALGEBRA-AUTOMORPHISM-ONTOLOGY-V1"
}

}

END_FLOW