Homomorphism Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of homomorphisms within abstract algebra. A homomorphism is a structure-preserving mapping between 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 = "Homomorphism Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Algebra"
 SPECIALISATION = "Abstract Algebra"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/homomorphism-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Homomorphism
 SYMBOLIC_FORMS = ["f: A -> B", "phi: G -> H"]
 ENTITY_CLASS = StructurePreservingMap
 PRIMARY_CONTEXT = AbstractAlgebra
 DEFINITION = "A homomorphism is a mapping between algebraic structures that preserves their defining operations."
 GENERAL_FORM = "f(omega_A(x1,...,xn))=omega_B(f(x1),...,f(xn))"
}

SCOPE {
 INCLUDE = [
  Homomorphism, GroupHomomorphism, RingHomomorphism,
  ModuleHomomorphism, LinearMap, AlgebraHomomorphism,
  Kernel, Image, Injectivity, Surjectivity, Isomorphism,
  Endomorphism, Automorphism, Composition,
  IdentityHomomorphism, QuotientStructure,
  FirstIsomorphismTheorem
 ]
 EXCLUDE_AS_PRIMARY = [ArbitraryFunction, NonStructurePreservingMap, Homeomorphism, Homotopy]
 DISAMBIGUATION_RULE = "Interpret homomorphism as an algebraic structure-preserving map when context includes groups, rings, modules, vector spaces, algebras, kernels, images, quotients or isomorphisms."
}

CORE_CONCEPTS {
 Homomorphism {
  definition = "A mapping preserving all operations required by the relevant algebraic signature."
 }
 GroupHomomorphism {
  preservation = "phi(x*y)=phi(x)*phi(y)"
  identity = "phi(e_G)=e_H"
  inverse = "phi(x^-1)=phi(x)^-1"
 }
 RingHomomorphism {
  preservation = ["f(a+b)=f(a)+f(b)", "f(ab)=f(a)f(b)"]
  note = "Preservation of multiplicative identity depends on the stated convention for ring homomorphisms."
 }
 ModuleHomomorphism {
  definition = "A map preserving module addition and compatible scalar multiplication."
 }
 LinearMap {
  definition = "A vector-space homomorphism preserving vector addition and scalar multiplication."
  equivalence = "VectorSpaceHomomorphism ≡ LinearTransformation"
 }
 Kernel {
  general_role = "Preimage of the distinguished identity or zero."
  group_form = "ker(phi)={g in G | phi(g)=e_H}"
  additive_form = "ker(f)={x | f(x)=0}"
 }
 Image {
  definition = "The attained target elements."
  form = "im(f)={f(x) | x in Domain(f)}"
 }
 Isomorphism {
  definition = "A bijective homomorphism."
 }
 Endomorphism {
  definition = "A homomorphism from an algebraic structure to itself."
 }
 Automorphism {
  definition = "An isomorphism from an algebraic structure to itself."
 }
}

RELATIONS {
 Homomorphism --maps_between--> AlgebraicStructures
 Homomorphism --preserves--> AlgebraicOperations
 Homomorphism --has_kernel--> Kernel
 Homomorphism --has_image--> Image
 Kernel --is_substructure_of--> Domain
 Image --is_substructure_of--> Codomain
 Homomorphism --can_be--> Injective
 Homomorphism --can_be--> Surjective
 BijectiveHomomorphism --is--> Isomorphism
 Endomorphism --is_special_case_of--> Homomorphism
 Automorphism --is_special_case_of--> Isomorphism
 LinearTransformation --is_special_case_of--> Homomorphism
 Homomorphism --participates_in--> FirstIsomorphismTheorem
 Kernel --induces--> QuotientStructure
}

OPERATORS {
 IS_HOMOMORPHISM(f,A,B) := PRESERVES_REQUIRED_OPERATIONS(f,A,B)
 KERNEL(f) := PREIMAGE(f,distinguished_identity_or_zero)
 IMAGE(f) := {f(x) | x in Domain(f)}
 IS_INJECTIVE(f) := FOR_ALL x,y: f(x)=f(y) => x=y
 IS_SURJECTIVE(f) := IMAGE(f)==Codomain(f)
 IS_ISOMORPHISM(f) := IS_HOMOMORPHISM(f) AND IS_INJECTIVE(f) AND IS_SURJECTIVE(f)
 COMPOSE(g,f) := g o f
 IS_ENDOMORPHISM(f) := IS_HOMOMORPHISM(f) AND Domain(f)==Codomain(f)
 IS_AUTOMORPHISM(f) := IS_ENDOMORPHISM(f) AND IS_ISOMORPHISM(f)
}

DEFINITIONS {
 D1 = "A homomorphism preserves the defining operations of its algebraic structures."
 D2 = "For groups, phi(xy)=phi(x)phi(y)."
 D3 = "Group homomorphisms preserve identity and inverses."
 D4 = "Ring homomorphisms preserve addition and multiplication."
 D5 = "A vector-space homomorphism is a linear map."
 D6 = "The kernel is the preimage of the distinguished identity or zero."
 D7 = "The image is the set of attained outputs."
 D8 = "A bijective homomorphism is an isomorphism."
 D9 = "An endomorphism maps a structure to itself."
 D10 = "An automorphism is an isomorphism from a structure to itself."
 D11 = "Composition of compatible homomorphisms is a homomorphism."
}

AXIOMS {
 A1:
  IF f is a homomorphism
  THEN f preserves every operation required by the relevant algebraic signature
 A2:
  IF phi: G -> H is a group homomorphism
  THEN phi(e_G)=e_H
 A3:
  IF phi: G -> H is a group homomorphism
  THEN phi(g^-1)=phi(g)^-1
 A4:
  IF f is a homomorphism
  THEN ker(f) is an appropriate substructure of Domain(f)
 A5:
  IF f is a homomorphism
  THEN im(f) is an appropriate substructure of Codomain(f)
 A6:
  IF f: A -> B and g: B -> C are compatible homomorphisms
  THEN g o f is a homomorphism
 A7:
  IdentityMap(A) is a homomorphism
}

REASONING {
 HOMOMORPHISM_TEST {
  INPUT = [f,A,B]
  IDENTIFY = required_operations(A,B)
  VERIFY = preservation_of_each_required_operation
  IF all_preserved RETURN Homomorphism
  ELSE RETURN NotHomomorphism
 }
 KERNEL_ANALYSIS {
  INPUT = f
  IDENTIFY = target_identity_or_zero
  SOLVE = "f(x)=target_identity_or_zero"
  RETURN = ker(f)
 }
 IMAGE_ANALYSIS {
  INPUT = f
  COMPUTE = attainable_outputs
  RETURN = im(f)
 }
 INJECTIVITY_TEST {
  INPUT = f
  TEST = direct_injectivity_definition
  IF applicable_kernel_criterion
   THEN also_test(trivial_kernel)
  RETURN = InjectivityStatus
 }
 SURJECTIVITY_TEST {
  COMPUTE = im(f)
  IF im(f)==Codomain(f) RETURN Surjective
  ELSE RETURN NotSurjective
 }
 ISOMORPHISM_TEST {
  REQUIRE = homomorphism
  IF injective AND surjective RETURN Isomorphism
  ELSE RETURN NotIsomorphism
 }
 FIRST_ISOMORPHISM_REASONING {
  INPUT = f
  CONSTRUCT = Domain(f)/Kernel(f)
  RELATE = Image(f)
  RETURN = "Domain(f)/ker(f) is isomorphic to im(f) in the applicable algebraic setting."
 }
}

CONSTRAINTS {
 C1 = "Source and target must possess compatible algebraic signatures."
 C2 = "Required preservation laws depend on the homomorphism type."
 C3 = "Kernel semantics depend on the distinguished identity or zero."
 C4 = "Image is contained in the codomain but need not equal it."
 C5 = "Bijectivity alone does not establish isomorphism without structure preservation."
 C6 = "For rings, multiplicative-identity preservation follows the convention explicitly adopted."
 C7 = "Composition requires compatible source and target structures."
 C8 = "Homomorphism must not be confused with homeomorphism."
}

SEMANTIC_EQUIVALENCE {
 "Homomorphism" <=> "StructurePreservingMap"
 "VectorSpaceHomomorphism" <=> "LinearMap"
 "BijectiveHomomorphism" <=> "Isomorphism"
 "SelfHomomorphism" <=> "Endomorphism"
 "SelfIsomorphism" <=> "Automorphism"
 "Image(f)" <=> "Range(f)"
}

VALIDATION {
 V1 = "Group homomorphisms preserve the group operation."
 V2 = "Group homomorphisms preserve identity."
 V3 = "Group homomorphisms preserve inverses."
 V4 = "Kernel is an appropriate substructure of the domain."
 V5 = "Image is an appropriate substructure of the codomain."
 V6 = "Compatible homomorphisms are closed under composition."
 V7 = "Identity mappings are homomorphisms."
 V8 = "A bijective homomorphism is an isomorphism."
 V9 = "A vector-space homomorphism is a linear map."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 HOMOMORPHISM_DEFINITION = 0.99
 OPERATION_PRESERVATION = 0.99
 KERNEL_RELATION = 0.99
 IMAGE_RELATION = 0.99
 ISOMORPHISM_RELATION = 0.99
 COMPOSITION_RELATION = 0.99
 FIRST_ISOMORPHISM_RELATION = 0.98
 POLICY = "Reduce confidence when source structure, target structure or required operations are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, AbstractAlgebraReasoning,
  EntityResolution, StructurePreservationReasoning,
  KernelReasoning, ImageReasoning, KnowledgeGraphConstruction,
  RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "homomorphism" -> Homomorphism
  "structure-preserving map" -> Homomorphism
  "group homomorphism" -> GroupHomomorphism
  "ring homomorphism" -> RingHomomorphism
  "module homomorphism" -> ModuleHomomorphism
  "vector-space homomorphism" -> LinearMap
  "bijective homomorphism" -> Isomorphism
  "self-homomorphism" -> Endomorphism
  "self-isomorphism" -> Automorphism
 }
 INTERPRETATION_ORDER = [
  identify_source_structure,
  identify_target_structure,
  identify_algebraic_signature,
  identify_mapping_rule,
  test_required_operation_preservation,
  establish_homomorphism_type,
  derive_kernel,
  derive_image,
  evaluate_injectivity,
  evaluate_surjectivity,
  evaluate_isomorphism,
  apply_isomorphism_theorems_when_applicable,
  return_semantic_result
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("homomorphism")
  THEN activate(HomomorphismOntology)
 E2:
  IF group_context
  THEN require_preservation(group_operation)
 E3:
  IF ring_context
  THEN require_preservation([addition,multiplication])
  AND inspect_identity_convention
 E4:
  IF vector_space_context
  THEN route_to(LinearTransformationOntology)
 E5:
  IF request=="find kernel"
  THEN solve(preimage_of_identity_or_zero)
 E6:
  IF request=="find image"
  THEN compute(attainable_outputs)
 E7:
  IF request=="test isomorphism"
  THEN verify_homomorphism
  AND verify_injectivity
  AND verify_surjectivity
 E8:
  IF quotient_and_kernel_context
  THEN consider(FirstIsomorphismTheorem)
}

RECURSION {
 ENTRY = Homomorphism
 EXPAND(Homomorphism) -> [
  AlgebraicStructure, Operation, GroupHomomorphism,
  RingHomomorphism, ModuleHomomorphism, LinearMap,
  Kernel, Image, Injectivity, Surjectivity,
  Isomorphism, Endomorphism, Automorphism
 ]
 EXPAND(Kernel) -> [
  IdentityElement, ZeroElement, NormalSubgroup,
  Ideal, Submodule, NullSpace
 ]
 EXPAND(Image) -> [
  Range, Subgroup, Subring, Submodule, Subspace
 ]
 EXPAND(Isomorphism) -> [
  Bijectivity, InverseHomomorphism, StructuralEquivalence
 ]
 EXPAND(FirstIsomorphismTheorem) -> [
  Kernel, QuotientStructure, Image, Isomorphism
 ]
 STOP_CONDITION = "Stop when the homomorphism type or requested property is resolved with sufficient confidence."
}

ROUTING {
 PARENT_ONTOLOGY = "Algebra Ontology"
 IMMEDIATE_CONTEXTS = ["Abstract Algebra Ontology", "Transformation Ontology"]
 RELATED_ONTOLOGIES = [
  "Transformation Ontology",
  "Linear Transformation Ontology",
  "Kernel Ontology",
  "Image Ontology",
  "Group Ontology",
  "Ring Ontology",
  "Vector Space Ontology",
  "Basis Ontology",
  "Dimension Ontology"
 ]
 ROUTE_RULES = [
  "Homomorphism -> Kernel for identity-preimage reasoning",
  "Homomorphism -> Image for attained-substructure reasoning",
  "Homomorphism -> Linear Transformation for vector-space homomorphisms",
  "Homomorphism -> Group for group-operation preservation",
  "Homomorphism -> Ring for ring-operation preservation",
  "Homomorphism -> Isomorphism for bijective structure-preserving maps",
  "Homomorphism -> Quotient Structure for first-isomorphism 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 structure-preserving semantics.",
  "Preserve operation-specific interpretation by algebraic signature.",
  "Preserve kernel as preimage of the distinguished identity or zero.",
  "Preserve image as attained target substructure.",
  "Preserve bijective-homomorphism/isomorphism relation.",
  "Preserve composition closure for compatible homomorphisms.",
  "Explicitly distinguish homomorphism from homeomorphism."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A homomorphism is a mapping between algebraic structures that preserves their defining operations."
 GENERAL_PRESERVATION_SUMMARY = "f(omega_A(x1,...,xn))=omega_B(f(x1),...,f(xn))"
 GROUP_SUMMARY = "phi(xy)=phi(x)phi(y)"
 KERNEL_SUMMARY = "ker(f)=preimage of target identity or zero"
 IMAGE_SUMMARY = "im(f)={f(x) | x in Domain(f)}"
 ISOMORPHISM_SUMMARY = "isomorphism = bijective homomorphism"
 FIRST_ISOMORPHISM_SUMMARY = "Domain(f)/ker(f) ≅ im(f)"
}

FLOW {
 INPUT
 -> detect_term("homomorphism")
 -> identify_source_algebraic_structure
 -> identify_target_algebraic_structure
 -> identify_required_operations
 -> identify_mapping_rule
 -> verify_operation_preservation
 -> establish_homomorphism_type
 -> derive_kernel
 -> derive_image
 -> evaluate_injectivity
 -> evaluate_surjectivity
 -> evaluate_isomorphism
 -> classify_endomorphism_or_automorphism_when_applicable
 -> apply_isomorphism_theorems_when_applicable
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "A homomorphism is a structure-preserving map between algebraic structures. Its precise preservation laws depend on the structures involved. Homomorphisms have kernels and images, compose naturally, and become isomorphisms when bijective. Group, ring, module and vector-space homomorphisms are specialised instances of this general algebraic concept."
}

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

}

END_FLOW