Isomorphism Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of isomorphisms within abstract algebra and linear algebra. An isomorphism is a bijective structure-preserving mapping and expresses structural equivalence between mathematical objects. 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 = "Isomorphism 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/isomorphism-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = Isomorphism
 SYMBOLIC_FORMS = ["f: A -> B", "A ≅ B", "V ≅ W"]
 ENTITY_CLASS = BijectiveStructurePreservingMap
 PRIMARY_CONTEXT = AbstractAlgebra
 DEFINITION = "An isomorphism is a bijective homomorphism between algebraic structures."
 STRUCTURAL_MEANING = "Two structures are isomorphic when they have the same algebraic structure up to relabelling of elements."
}

SCOPE {
 INCLUDE = [
  Isomorphism, Homomorphism, Bijectivity, Injectivity,
  Surjectivity, InverseHomomorphism, StructuralEquivalence,
  GroupIsomorphism, RingIsomorphism, ModuleIsomorphism,
  VectorSpaceIsomorphism, LinearIsomorphism,
  Automorphism, IsomorphicStructures,
  FirstIsomorphismTheorem
 ]
 EXCLUDE_AS_PRIMARY = [
  Homeomorphism, Diffeomorphism, HomotopyEquivalence,
  ArbitraryBijection, VisualSimilarity
 ]
 DISAMBIGUATION_RULE = "Interpret isomorphism as algebraic structural equivalence when context includes groups, rings, modules, vector spaces, homomorphisms, kernels, images, quotients, bases or dimensions."
}

CORE_CONCEPTS {

 Isomorphism {
  definition = "A homomorphism that is both injective and surjective."
  conditions = [StructurePreservation, Injectivity, Surjectivity]
  consequence = "The inverse map exists and is structure-preserving."
 }

 StructuralEquivalence {
  definition = "Equivalence of mathematical structure independent of the particular names or representations of elements."
  symbolic_form = "A ≅ B"
 }

 Injectivity {
  definition = "Distinct source elements have distinct images."
  form = "f(x)=f(y) => x=y"
 }

 Surjectivity {
  definition = "Every element of the codomain has at least one preimage."
  form = "im(f)=Codomain(f)"
 }

 Bijectivity {
  definition = "A mapping that is both injective and surjective."
 }

 InverseIsomorphism {
  definition = "The inverse of an isomorphism, which is itself an isomorphism."
  laws = ["f^-1(f(a))=a", "f(f^-1(b))=b"]
 }

 Automorphism {
  definition = "An isomorphism from a mathematical structure to itself."
 }

 VectorSpaceIsomorphism {
  definition = "A bijective linear transformation between vector spaces."
  finite_dimension_criterion = "Finite-dimensional vector spaces over the same field are isomorphic iff they have equal dimension."
 }

 GroupIsomorphism {
  definition = "A bijective group homomorphism."
 }

 RingIsomorphism {
  definition = "A bijective ring homomorphism under the adopted ring-homomorphism convention."
 }
}

RELATIONS {
 Isomorphism --is_special_case_of--> Homomorphism
 Isomorphism --requires--> Injectivity
 Isomorphism --requires--> Surjectivity
 Isomorphism --implies--> Bijectivity
 Isomorphism --has_inverse--> InverseIsomorphism
 InverseIsomorphism --is--> Isomorphism
 Isomorphism --establishes--> StructuralEquivalence
 Automorphism --is_special_case_of--> Isomorphism
 GroupIsomorphism --is_special_case_of--> Isomorphism
 RingIsomorphism --is_special_case_of--> Isomorphism
 VectorSpaceIsomorphism --is_special_case_of--> Isomorphism
 Isomorphism --preserves--> AlgebraicInvariants
 Homomorphism --participates_in--> FirstIsomorphismTheorem
}

OPERATORS {
 IS_ISOMORPHISM(f,A,B) :=
  IS_HOMOMORPHISM(f,A,B)
  AND IS_INJECTIVE(f)
  AND IS_SURJECTIVE(f)

 ISOMORPHIC(A,B) :=
  EXISTS f: A -> B SUCH_THAT IS_ISOMORPHISM(f,A,B)

 INVERSE_ISOMORPHISM(f) :=
  f^-1 IF IS_ISOMORPHISM(f)

 IS_AUTOMORPHISM(f,A) :=
  IS_ISOMORPHISM(f,A,A)

 IS_VECTOR_SPACE_ISOMORPHISM(T,V,W) :=
  IS_LINEAR(T)
  AND IS_BIJECTIVE(T)

 FINITE_VECTOR_SPACE_ISOMORPHIC(V,W) :=
  same_scalar_field(V,W)
  AND dim(V)==dim(W)
}

DEFINITIONS {
 D1 = "An isomorphism is a bijective homomorphism."
 D2 = "A ≅ B means there exists an isomorphism between A and B."
 D3 = "The inverse of an isomorphism is an isomorphism."
 D4 = "The identity map is an isomorphism."
 D5 = "The composition of isomorphisms is an isomorphism."
 D6 = "Isomorphism is reflexive, symmetric and transitive on structures of a fixed type."
 D7 = "An automorphism is an isomorphism from a structure to itself."
 D8 = "A vector-space isomorphism is a bijective linear transformation."
 D9 = "Finite-dimensional vector spaces over the same field are isomorphic iff they have the same dimension."
 D10 = "Isomorphic structures share all properties invariant under the relevant notion of structure."
}

AXIOMS {
 A1_HOMOMORPHISM:
  IF f is an isomorphism
  THEN f is a homomorphism

 A2_INJECTIVITY:
  IF f is an isomorphism
  THEN f is injective

 A3_SURJECTIVITY:
  IF f is an isomorphism
  THEN f is surjective

 A4_INVERSE:
  IF f: A -> B is an isomorphism
  THEN f^-1: B -> A is an isomorphism

 A5_IDENTITY:
  IdentityMap(A) is an isomorphism A -> A

 A6_COMPOSITION:
  IF f: A -> B and g: B -> C are isomorphisms
  THEN g o f: A -> C is an isomorphism

 A7_EQUIVALENCE:
  IsomorphicRelation is reflexive, symmetric and transitive
}

REASONING {

 R1_ISOMORPHISM_TEST {
  INPUT = [f,A,B]
  TEST_1 = verify_structure_preservation
  TEST_2 = verify_injectivity
  TEST_3 = verify_surjectivity
  IF TEST_1 AND TEST_2 AND TEST_3
   RETURN Isomorphism
  ELSE
   RETURN NotIsomorphism
 }

 R2_INVERSE {
  INPUT = isomorphism f
  CONSTRUCT = inverse_function(f)
  VERIFY = inverse_preserves_structure
  RETURN = f^-1
 }

 R3_STRUCTURAL_EQUIVALENCE {
  INPUT = [A,B]
  SEEK = explicit_isomorphism
  IF found
   RETURN "A ≅ B"
  ELSE
   RETURN "Isomorphism not established"
 }

 R4_VECTOR_SPACE_TEST {
  INPUT = finite-dimensional vector spaces V,W over same field
  COMPARE = [dim(V),dim(W)]
  IF dim(V)==dim(W)
   RETURN "V ≅ W"
  ELSE
   RETURN "V and W are not isomorphic as vector spaces"
 }

 R5_GROUP_TEST {
  INPUT = candidate map phi: G -> H
  VERIFY = group_homomorphism
  VERIFY = bijection
  RETURN = GroupIsomorphismStatus
 }

 R6_FIRST_ISOMORPHISM {
  INPUT = homomorphism f
  CONSTRUCT = Domain(f)/Kernel(f)
  RELATE = Image(f)
  RETURN = "Domain(f)/ker(f) ≅ im(f) under the applicable theorem."
 }

 R7_INVARIANT_REJECTION {
  INPUT = [A,B,invariant I]
  IF I(A) != I(B)
   RETURN "A and B cannot be isomorphic"
  ELSE
   RETURN "Invariant does not rule out isomorphism"
 }
}

CONSTRAINTS {
 C1 = "A bijection is not necessarily an isomorphism unless it preserves the required structure."
 C2 = "The source and target must be structures of compatible type."
 C3 = "The precise preservation laws depend on the algebraic category."
 C4 = "An isomorphism must possess a two-sided inverse."
 C5 = "Isomorphic does not mean identical; structures may have different underlying elements or representations."
 C6 = "An invariant mismatch can disprove isomorphism, but matching invariants do not always prove it."
 C7 = "For vector spaces, equal finite dimension implies isomorphism only when scalar fields are compatible."
 C8 = "Do not confuse isomorphism with homeomorphism or arbitrary similarity."
}

SEMANTIC_EQUIVALENCE {
 "Isomorphism" <=> "BijectiveHomomorphism"
 "A ≅ B" <=> "A and B are structurally isomorphic"
 "VectorSpaceIsomorphism" <=> "BijectiveLinearTransformation"
 "SelfIsomorphism" <=> "Automorphism"
 "InverseOfIsomorphism" <=> "IsomorphismInReverseDirection"
}

VALIDATION {
 V1 = "Every isomorphism is a homomorphism."
 V2 = "Every isomorphism is injective."
 V3 = "Every isomorphism is surjective."
 V4 = "The inverse of an isomorphism is an isomorphism."
 V5 = "Identity maps are isomorphisms."
 V6 = "Composition of isomorphisms is an isomorphism."
 V7 = "Isomorphism defines an equivalence relation."
 V8 = "A vector-space isomorphism preserves dimension."
 V9 = "Finite-dimensional vector spaces over the same field with equal dimension are isomorphic."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ISOMORPHISM_DEFINITION = 0.99
 BIJECTIVE_HOMOMORPHISM_RELATION = 0.99
 INVERSE_RELATION = 0.99
 STRUCTURAL_EQUIVALENCE = 0.99
 COMPOSITION_RELATION = 0.99
 VECTOR_SPACE_DIMENSION_CRITERION = 0.99
 FIRST_ISOMORPHISM_RELATION = 0.98
 POLICY = "Reduce confidence when the relevant algebraic structures, operations or category are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, AbstractAlgebraReasoning,
  LinearAlgebraReasoning, EntityResolution,
  StructuralEquivalenceReasoning, InvariantReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "isomorphism" -> Isomorphism
  "isomorphic" -> StructuralEquivalence
  "bijective homomorphism" -> Isomorphism
  "linear isomorphism" -> VectorSpaceIsomorphism
  "vector-space isomorphism" -> VectorSpaceIsomorphism
  "group isomorphism" -> GroupIsomorphism
  "ring isomorphism" -> RingIsomorphism
  "self-isomorphism" -> Automorphism
 }

 INTERPRETATION_ORDER = [
  identify_source_structure,
  identify_target_structure,
  identify_required_operations,
  identify_candidate_mapping,
  verify_homomorphism,
  verify_injectivity,
  verify_surjectivity,
  establish_isomorphism,
  construct_inverse_when_required,
  compare_invariants,
  apply_isomorphism_theorems_when_applicable,
  return_structural_equivalence_result
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("isomorphism")
  OR query_mentions("isomorphic")
  THEN activate(IsomorphismOntology)

 E2:
  IF request=="test isomorphism"
  THEN verify_structure_preservation
  AND verify_injectivity
  AND verify_surjectivity

 E3:
  IF invariant_mismatch_detected
  THEN conclude(NotIsomorphic)

 E4:
  IF vector_space_context
  AND finite_dimensional
  AND same_scalar_field
  THEN compare_dimensions

 E5:
  IF request=="find inverse"
  AND isomorphism_established
  THEN construct_inverse_isomorphism

 E6:
  IF self_map AND isomorphism_established
  THEN classify(Automorphism)

 E7:
  IF quotient_kernel_image_context
  THEN consider(FirstIsomorphismTheorem)
}

RECURSION {
 ENTRY = Isomorphism

 EXPAND(Isomorphism) -> [
  Homomorphism, Injectivity, Surjectivity,
  Bijectivity, InverseHomomorphism,
  StructuralEquivalence, Automorphism,
  AlgebraicInvariant
 ]

 EXPAND(Homomorphism) -> [
  AlgebraicStructure, Operation, Kernel, Image
 ]

 EXPAND(VectorSpaceIsomorphism) -> [
  LinearTransformation, Basis, Dimension,
  Matrix, Rank, Kernel
 ]

 EXPAND(StructuralEquivalence) -> [
  Invariant, Representation, EquivalenceRelation
 ]

 EXPAND(FirstIsomorphismTheorem) -> [
  Homomorphism, Kernel, QuotientStructure,
  Image, Isomorphism
 ]

 STOP_CONDITION =
  "Stop when isomorphism status or the requested structural-equivalence consequence is resolved with sufficient confidence."
}

ROUTING {
 PARENT_ONTOLOGY = "Homomorphism Ontology"

 IMMEDIATE_CONTEXTS = [
  "Algebra Ontology",
  "Abstract Algebra Ontology",
  "Linear Transformation Ontology"
 ]

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

 ROUTE_RULES = [
  "Isomorphism -> Homomorphism for structure-preservation reasoning",
  "Isomorphism -> Kernel for injectivity reasoning",
  "Isomorphism -> Image for surjectivity reasoning",
  "Isomorphism -> Dimension for finite vector-space classification",
  "Isomorphism -> Linear Transformation for vector-space maps",
  "Isomorphism -> Matrix for coordinate representations",
  "Isomorphism -> 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 isomorphism as bijective structure-preserving mapping.",
  "Preserve inverse-isomorphism property.",
  "Preserve reflexive, symmetric and transitive structural-equivalence behaviour.",
  "Preserve distinction between equality and isomorphism.",
  "Preserve invariant-based rejection reasoning.",
  "Preserve vector-space dimension criterion under compatible scalar fields.",
  "Explicitly distinguish isomorphism from homeomorphism and arbitrary bijection."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "An isomorphism is a bijective homomorphism that establishes structural equivalence between mathematical objects."

 SYMBOLIC_SUMMARY =
  "A ≅ B"

 TEST_SUMMARY =
  "isomorphism = homomorphism + injectivity + surjectivity"

 INVERSE_SUMMARY =
  "f is an isomorphism => f^-1 is an isomorphism"

 VECTOR_SPACE_SUMMARY =
  "Finite-dimensional V and W over the same field: V ≅ W iff dim(V)=dim(W)"

 AUTOMORPHISM_SUMMARY =
  "automorphism = isomorphism from a structure to itself"

 FIRST_ISOMORPHISM_SUMMARY =
  "Domain(f)/ker(f) ≅ im(f)"
}

FLOW {
 INPUT
 -> detect_term("isomorphism")
 -> identify_source_structure
 -> identify_target_structure
 -> identify_algebraic_signature
 -> identify_candidate_mapping
 -> verify_structure_preservation
 -> verify_injectivity
 -> verify_surjectivity
 -> establish_bijective_homomorphism
 -> classify_isomorphism
 -> construct_inverse_when_required
 -> compare_structural_invariants
 -> classify_automorphism_when_self_map
 -> apply_isomorphism_theorems_when_applicable
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "An isomorphism is a bijective structure-preserving map and expresses that two mathematical structures are equivalent in their algebraic organisation even when their elements or representations differ. Isomorphisms possess structure-preserving inverses, compose to form further isomorphisms, preserve relevant invariants and define an equivalence relation among structures of the same type."
}

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

}

END_FLOW