Endomorphism Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of endomorphisms within abstract algebra and linear algebra. An endomorphism is a structure-preserving mapping from a mathematical structure to itself. 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 = "Endomorphism 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/endomorphism-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Endomorphism
 SYMBOLIC_FORMS = ["f: A -> A", "T: V -> V", "End(A)", "End(V)"]
 ENTITY_CLASS = SelfHomomorphism
 PRIMARY_CONTEXT = AbstractAlgebra
 SECONDARY_CONTEXT = LinearAlgebra
 DEFINITION = "An endomorphism is a homomorphism from an algebraic structure to itself."
 LINEAR_FORM = "A linear endomorphism is a linear transformation T: V -> V."
}

SCOPE {
 INCLUDE = [
  Endomorphism, Homomorphism, SelfMap,
  LinearEndomorphism, EndomorphismMonoid,
  EndomorphismRing, Kernel, Image, Rank, Nullity,
  Composition, IdentityEndomorphism,
  Invertibility, Automorphism, MatrixRepresentation,
  Eigenvalue, Eigenvector, InvariantSubspace
 ]
 EXCLUDE_AS_PRIMARY = [
  ArbitrarySelfFunction, NonStructurePreservingSelfMap,
  Homeomorphism, UnrelatedEverydayUse
 ]
 DISAMBIGUATION_RULE = "Interpret endomorphism as a structure-preserving self-map when context includes groups, rings, modules, vector spaces, homomorphisms, matrices, eigenvalues, kernels or images."
}

CORE_CONCEPTS {

 Endomorphism {
  definition = "A homomorphism whose domain and codomain are the same algebraic structure."
  canonical_form = "f: A -> A"
  defining_conditions = [Homomorphism, SameDomainAndCodomain]
 }

 LinearEndomorphism {
  definition = "A linear transformation from a vector space to itself."
  canonical_form = "T: V -> V"
 }

 EndomorphismMonoid {
  definition = "The collection of endomorphisms of a structure under composition."
  identity = "Identity endomorphism"
  operation = "Composition"
 }

 EndomorphismRing {
  definition = "For an abelian group or module, endomorphisms can carry ring structure under pointwise addition and composition, under the applicable algebraic setting."
 }

 IdentityEndomorphism {
  definition = "The identity homomorphism id_A: A -> A."
  form = "id_A(a)=a"
 }

 Automorphism {
  definition = "An invertible endomorphism, equivalently an isomorphism from a structure to itself."
 }

 Kernel {
  definition = "The preimage of the distinguished zero or identity under an endomorphism."
 }

 Image {
  definition = "The substructure consisting of outputs attained by the endomorphism."
 }

 InvariantSubspace {
  definition = "For a linear endomorphism T: V -> V, a subspace W is T-invariant when T(W) is contained in W."
 }
}

RELATIONS {
 Endomorphism --is_special_case_of--> Homomorphism
 Endomorphism --has_same--> DomainAndCodomain
 Endomorphism --has_kernel--> Kernel
 Endomorphism --has_image--> Image
 Endomorphism --can_be_composed_with--> Endomorphism
 Endomorphism --belongs_to--> EndomorphismMonoid
 IdentityEndomorphism --is--> Endomorphism
 InvertibleEndomorphism --is--> Automorphism
 Automorphism --is_special_case_of--> Endomorphism
 LinearEndomorphism --is_special_case_of--> LinearTransformation
 LinearEndomorphism --can_be_represented_by--> SquareMatrix
 LinearEndomorphism --can_have--> Eigenvalues
 LinearEndomorphism --can_have--> Eigenvectors
 LinearEndomorphism --can_preserve--> InvariantSubspace
}

OPERATORS {
 IS_ENDOMORPHISM(f,A) :=
  IS_HOMOMORPHISM(f,A,A)

 IS_LINEAR_ENDOMORPHISM(T,V) :=
  IS_LINEAR(T) AND Domain(T)==V AND Codomain(T)==V

 KERNEL(f) :=
  PREIMAGE(f,distinguished_identity_or_zero)

 IMAGE(f) :=
  {f(x) | x in Domain(f)}

 COMPOSE(g,f) :=
  g o f

 IDENTITY_ENDOMORPHISM(A) :=
  id_A

 IS_AUTOMORPHISM(f) :=
  IS_ENDOMORPHISM(f) AND IS_BIJECTIVE(f)

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

 IS_INVARIANT(W,T) :=
  T(W) subset_of W

 MATRIX_OF(T,B) :=
  matrix representation of T relative to basis B
}

DEFINITIONS {
 D1 = "An endomorphism is a homomorphism f: A -> A."
 D2 = "A linear endomorphism is a linear transformation T: V -> V."
 D3 = "The identity map is an endomorphism."
 D4 = "The composition of endomorphisms is an endomorphism."
 D5 = "Endomorphisms of a structure form a monoid under composition."
 D6 = "An invertible endomorphism is an automorphism."
 D7 = "For finite-dimensional V, a linear endomorphism is represented by a square matrix relative to a basis."
 D8 = "The kernel of an endomorphism is a substructure of its domain."
 D9 = "The image of an endomorphism is a substructure of the same ambient structure."
 D10 = "A T-invariant subspace W satisfies T(W) subset_of W."
}

AXIOMS {
 A1_SELF_MAP:
  IF f is an endomorphism of A
  THEN Domain(f)=A AND Codomain(f)=A

 A2_STRUCTURE_PRESERVATION:
  IF f is an endomorphism
  THEN f preserves all operations required by the algebraic structure

 A3_IDENTITY:
  id_A is an endomorphism of A

 A4_COMPOSITION:
  IF f and g are endomorphisms of A
  THEN g o f is an endomorphism of A

 A5_AUTOMORPHISM:
  IF f is an endomorphism AND f is bijective
  THEN f is an automorphism

 A6_LINEAR_MATRIX:
  IF T: V -> V is a linear endomorphism
  AND V is finite-dimensional
  THEN relative to any ordered basis of V, T has a square matrix representation
}

REASONING {

 R1_ENDOMORPHISM_TEST {
  INPUT = [mapping f, structure A]
  TEST_1 = Domain(f)==A
  TEST_2 = Codomain(f)==A
  TEST_3 = verify_structure_preservation
  IF TEST_1 AND TEST_2 AND TEST_3
   RETURN Endomorphism
  ELSE
   RETURN NotEndomorphism
 }

 R2_LINEAR_ENDOMORPHISM_TEST {
  INPUT = [T,V]
  VERIFY = T is linear
  VERIFY = Domain(T)==V
  VERIFY = Codomain(T)==V
  IF all_verified
   RETURN LinearEndomorphism
 }

 R3_KERNEL {
  INPUT = endomorphism f
  SOLVE = "f(x)=identity_or_zero"
  RETURN = ker(f)
 }

 R4_IMAGE {
  INPUT = endomorphism f
  COMPUTE = attainable_outputs
  RETURN = im(f)
 }

 R5_AUTOMORPHISM_TEST {
  INPUT = endomorphism f
  TEST = bijectivity(f)
  IF bijective
   RETURN Automorphism
  ELSE
   RETURN NonInvertibleEndomorphism
 }

 R6_ITERATION {
  INPUT = [endomorphism f, positive integer n]
  COMPUTE = repeated_composition(f,n)
  RETURN = f^n
 }

 R7_MATRIX_REPRESENTATION {
  INPUT = [linear endomorphism T, ordered basis B]
  FOR each basis vector b_i
   compute T(b_i)
   express T(b_i) in B
   place coordinates as column i
  RETURN = square_matrix
 }

 R8_INVARIANT_SUBSPACE {
  INPUT = [linear endomorphism T, subspace W]
  TEST = "T(w) in W for every w in W"
  IF true
   RETURN InvariantSubspace
  ELSE
   RETURN NotInvariant
 }

 R9_EIGEN_REASONING {
  INPUT = linear endomorphism T
  SOLVE = "T(v)=lambda*v with v != 0"
  RETURN = eigenvalue_eigenvector_pairs
 }
}

CONSTRAINTS {
 C1 = "An endomorphism must have identical source and target structures."
 C2 = "A self-map is not necessarily an endomorphism unless it preserves the required structure."
 C3 = "An automorphism is an invertible endomorphism."
 C4 = "Composition of endomorphisms is defined because source and target structures coincide."
 C5 = "For finite-dimensional linear endomorphisms, matrix representations are square."
 C6 = "Matrix entries depend on the chosen ordered basis."
 C7 = "Kernel and image remain substructures of the same ambient structure."
 C8 = "Do not equate endomorphism with automorphism unless invertibility is established."
}

SEMANTIC_EQUIVALENCE {
 "Endomorphism" <=> "SelfHomomorphism"
 "LinearEndomorphism" <=> "LinearTransformationFromVToV"
 "InvertibleEndomorphism" <=> "Automorphism"
 "IdentityHomomorphismOnA" <=> "IdentityEndomorphismOfA"
}

VALIDATION {
 V1 = "Every endomorphism is a homomorphism."
 V2 = "Domain and codomain of an endomorphism coincide."
 V3 = "Identity map is an endomorphism."
 V4 = "Composition of endomorphisms is an endomorphism."
 V5 = "Endomorphisms form a monoid under composition."
 V6 = "An invertible endomorphism is an automorphism."
 V7 = "A finite-dimensional linear endomorphism has a square matrix representation."
 V8 = "Kernel and image are appropriate substructures."
 V9 = "A linear invariant subspace satisfies T(W) subset_of W."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ENDOMORPHISM_DEFINITION = 0.99
 SELF_MAP_CONDITION = 0.99
 STRUCTURE_PRESERVATION = 0.99
 COMPOSITION_RELATION = 0.99
 AUTOMORPHISM_RELATION = 0.99
 LINEAR_MATRIX_RELATION = 0.99
 INVARIANT_SUBSPACE_RELATION = 0.99
 POLICY = "Reduce confidence when the source structure, target structure or preservation laws are underspecified."
}

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

 NORMALISE {
  "endomorphism" -> Endomorphism
  "self-homomorphism" -> Endomorphism
  "linear endomorphism" -> LinearEndomorphism
  "linear operator" -> LinearEndomorphism WHEN vector_space_self_map_context
  "invertible endomorphism" -> Automorphism
  "endomorphism monoid" -> EndomorphismMonoid
  "endomorphism ring" -> EndomorphismRing
 }

 INTERPRETATION_ORDER = [
  identify_algebraic_structure,
  identify_mapping_rule,
  verify_same_domain_and_codomain,
  verify_structure_preservation,
  establish_endomorphism,
  derive_kernel,
  derive_image,
  evaluate_invertibility,
  classify_automorphism_when_applicable,
  analyse_iteration_when_requested,
  derive_matrix_representation_for_linear_case,
  analyse_invariant_subspaces_or_eigenstructure_when_requested,
  return_semantic_result
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("endomorphism")
  THEN activate(EndomorphismOntology)

 E2:
  IF query_mentions("linear operator")
  AND vector_space_self_map_context
  THEN activate(EndomorphismOntology)

 E3:
  IF Domain(f)!=Codomain(f)
  THEN conclude(NotEndomorphism)

 E4:
  IF request=="test endomorphism"
  THEN verify_same_domain_codomain
  AND verify_homomorphism

 E5:
  IF request=="test automorphism"
  THEN verify_endomorphism
  AND verify_bijectivity

 E6:
  IF request=="iterate"
  THEN apply_repeated_composition

 E7:
  IF linear_endomorphism_context
  THEN permit_square_matrix_reasoning

 E8:
  IF request=="test invariant subspace"
  THEN verify(T(W) subset_of W)

 E9:
  IF request=="find eigenvalues"
  THEN route_to_eigenvalue_reasoning
}

RECURSION {
 ENTRY = Endomorphism

 EXPAND(Endomorphism) -> [
  Homomorphism, Domain, Codomain,
  Kernel, Image, Composition,
  IdentityEndomorphism, Automorphism,
  EndomorphismMonoid
 ]

 EXPAND(LinearEndomorphism) -> [
  LinearTransformation, Matrix, Basis,
  Kernel, Image, Rank, Nullity,
  Eigenvalue, Eigenvector, InvariantSubspace
 ]

 EXPAND(Automorphism) -> [
  Isomorphism, Bijectivity, InverseHomomorphism
 ]

 EXPAND(EndomorphismMonoid) -> [
  Composition, IdentityEndomorphism, Associativity
 ]

 STOP_CONDITION =
  "Stop when endomorphism status or the requested derived property is resolved with sufficient confidence."
}

ROUTING {
 PARENT_ONTOLOGY = "Homomorphism Ontology"

 IMMEDIATE_CONTEXTS = [
  "Isomorphism Ontology",
  "Linear Transformation Ontology",
  "Transformation Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Homomorphism Ontology",
  "Isomorphism Ontology",
  "Transformation Ontology",
  "Linear Transformation Ontology",
  "Kernel Ontology",
  "Image Ontology",
  "Basis Ontology",
  "Dimension Ontology",
  "Rank Ontology",
  "Matrix Ontology",
  "Vector Space Ontology"
 ]

 ROUTE_RULES = [
  "Endomorphism -> Homomorphism for structure-preservation reasoning",
  "Endomorphism -> Isomorphism for invertibility and automorphism reasoning",
  "Endomorphism -> Linear Transformation for vector-space self-maps",
  "Endomorphism -> Kernel for identity-preimage reasoning",
  "Endomorphism -> Image for attained-substructure reasoning",
  "Endomorphism -> Matrix for finite-dimensional linear representation",
  "Endomorphism -> Eigenstructure for linear self-map spectral 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 endomorphism as a structure-preserving self-map.",
  "Preserve equality of domain and codomain.",
  "Preserve closure under composition.",
  "Preserve identity endomorphism.",
  "Preserve invertible-endomorphism/automorphism relation.",
  "Preserve square-matrix representation for finite-dimensional linear endomorphisms.",
  "Explicitly distinguish arbitrary self-maps from endomorphisms."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "An endomorphism is a homomorphism from an algebraic structure to itself."

 SYMBOLIC_SUMMARY =
  "f: A -> A"

 LINEAR_SUMMARY =
  "T: V -> V"

 AUTOMORPHISM_SUMMARY =
  "automorphism = invertible endomorphism"

 COMPOSITION_SUMMARY =
  "f,g in End(A) => g o f in End(A)"

 MATRIX_SUMMARY =
  "A finite-dimensional linear endomorphism has a square matrix representation relative to a basis."

 INVARIANT_SUBSPACE_SUMMARY =
  "W is T-invariant <=> T(W) subset_of W"
}

FLOW {
 INPUT
 -> detect_term("endomorphism")
 -> identify_algebraic_structure
 -> identify_mapping_rule
 -> verify_domain_equals_codomain
 -> verify_structure_preservation
 -> establish_endomorphism
 -> derive_kernel
 -> derive_image
 -> evaluate_composition_or_iteration
 -> evaluate_invertibility
 -> classify_automorphism_when_applicable
 -> identify_linear_endomorphism_context
 -> derive_square_matrix_representation_when_applicable
 -> analyse_invariant_subspaces_or_eigenstructure_when_requested
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "An endomorphism is a structure-preserving map from a mathematical structure to itself. Endomorphisms are closed under composition and include the identity map, forming an endomorphism monoid. An invertible endomorphism is an automorphism. In linear algebra, endomorphisms are linear self-maps represented by square matrices and provide the natural setting for eigenvalues, eigenvectors and invariant subspaces."
}

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

}

END_FLOW