Algebra Ontologies

AI INTERPRETATION NOTE
This Flow Algebra defines the parent Algebra Ontologies layer within the Kireth Research Semantic Library. It organises algebraic knowledge into structured semantic domains covering elementary algebra, linear algebra, abstract algebra, polynomial algebra, transformations, symbolic manipulation, computational algebra and related entities, relations, operations, proofs and algorithms. This record is designed for semantic retrieval, recursive mathematical reasoning, ontology routing, knowledge-graph construction and AI consumption.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = AlgebraOntologies
 ENTITY_CLASS = MathematicalOntologyDomain
 PRIMARY_CONTEXT = Algebra
 DEFINITION = "Algebra Ontologies are structured semantic representations of algebraic entities, operations, expressions, equations, structures, transformations, proofs and computational procedures."
 SEMANTIC_ROLE = "Provide machine-readable organisation and reasoning pathways across algebraic mathematics."
}

SCOPE {
 INCLUDE = [
  ElementaryAlgebraOntology,
  AlgebraicExpressionOntology,
  EquationOntology,
  LinearEquationOntology,
  PolynomialOntology,
  MonomialOntology,
  RationalExpressionOntology,
  MatrixOntology,
  DeterminantOntology,
  RankOntology,
  NullSpaceOntology,
  ColumnSpaceOntology,
  RowSpaceOntology,
  KernelOntology,
  ImageOntology,
  BasisOntology,
  DimensionOntology,
  TransformationOntology,
  LinearTransformationOntology,
  AffineTransformationOntology,
  HomomorphismOntology,
  IsomorphismOntology,
  EndomorphismOntology,
  AutomorphismOntology,
  AlgebraicIdentityOntology,
  FactorisationOntology,
  ExpansionOntology,
  SimplificationOntology,
  SubstitutionOntology,
  AlgebraicProofOntology,
  SymbolicManipulationOntology,
  ComputationalAlgebraOntology,
  ComputerAlgebraSystemsOntology,
  AlgebraicOptimisationOntology
 ]

 EXCLUDE_AS_PRIMARY = [
  NonAlgebraicMathematicalDomains,
  UnstructuredNarrative,
  PurelyNumericalApproximationWithoutAlgebraicSemantics
 ]

 DISAMBIGUATION_RULE =
  "Interpret algebra ontology as a structured mathematical knowledge domain concerning algebraic symbols, expressions, equations, structures, transformations and computation."
}

CORE_DOMAINS {

 ElementaryAlgebra {
  role = "Basic symbolic algebra involving variables, constants, expressions, equations, identities, factorisation, expansion, simplification and substitution."
 }

 PolynomialAlgebra {
  role = "Algebra of polynomial expressions, coefficients, roots, factors, identities and polynomial algorithms."
 }

 LinearAlgebra {
  role = "Algebra of vectors, matrices, vector spaces, linear transformations, kernels, images, bases, dimensions, rank and related structures."
 }

 AbstractAlgebra {
  role = "Algebraic structures such as groups, rings, fields, modules, homomorphisms, isomorphisms, endomorphisms and automorphisms."
 }

 TransformationAlgebra {
  role = "Mappings and structure-preserving transformations, including linear, affine and algebraic morphisms."
 }

 SymbolicAlgebra {
  role = "Rule-based manipulation of algebraic expressions through expansion, factorisation, simplification, substitution and identity application."
 }

 ComputationalAlgebra {
  role = "Algorithmic and exact symbolic computation over algebraic objects, including polynomial and matrix procedures."
 }

 AlgebraicOptimisation {
  role = "Goal-directed selection of algebraic forms, symbolic strategies and computational procedures under explicit objectives and constraints."
 }
}

ALGEBRAIC_ENTITY_CLASSES {
 INCLUDE = [
  Variable,
  Constant,
  Coefficient,
  Term,
  Monomial,
  Polynomial,
  RationalExpression,
  AlgebraicExpression,
  Equation,
  Identity,
  Factor,
  Root,
  Matrix,
  Vector,
  VectorSpace,
  Basis,
  Dimension,
  Rank,
  Kernel,
  Image,
  Transformation,
  Homomorphism,
  Isomorphism,
  Endomorphism,
  Automorphism,
  Algorithm,
  Proof
 ]
}

RELATIONS {
 AlgebraOntologies --is_child_of--> MathematicalOntologies
 AlgebraOntologies --contains--> AlgebraicOntology
 AlgebraicOntology --contains--> AlgebraicEntity
 AlgebraicEntity --can_have--> SymbolicRepresentation
 AlgebraicEntity --can_participate_in--> AlgebraicRelation
 AlgebraicEntity --can_be_operand_of--> AlgebraicOperation
 AlgebraicExpression --can_be_transformed_by--> SymbolicManipulation
 Equation --can_be_solved_by--> AlgebraicMethod
 Polynomial --can_have--> Root
 Polynomial --can_have--> Factor
 Matrix --can_have--> Rank
 Matrix --can_define--> Kernel
 Matrix --can_define--> Image
 VectorSpace --can_have--> Basis
 VectorSpace --has--> Dimension
 Transformation --can_have--> Kernel
 Transformation --can_have--> Image
 Homomorphism --can_be--> Isomorphism
 Endomorphism --can_be--> Automorphism
 AlgebraicProof --can_use--> AlgebraicIdentity
 AlgebraOntologies --supports--> RecursiveReasoning
 AlgebraOntologies --supports--> SemanticSearch
 AlgebraOntologies --supports--> KnowledgeGraphConstruction
}

OPERATORS {
 DEFINE(entity) :=
  return_canonical_algebraic_definition(entity)

 CLASSIFY(entity) :=
  assign_entity_to_algebraic_ontology_class(entity)

 RELATE(a,b) :=
  identify_valid_algebraic_relations_between(a,b)

 SIMPLIFY(E) :=
  route_to_simplification_reasoning(E)

 EXPAND(E) :=
  route_to_expansion_reasoning(E)

 FACTOR(E) :=
  route_to_factorisation_reasoning(E)

 SUBSTITUTE(E,x,A) :=
  route_to_substitution_reasoning(E,x,A)

 SOLVE(equation_or_system) :=
  select_algebraic_solution_method(equation_or_system)

 TRANSFORM(object,rule) :=
  apply_valid_algebraic_transformation(object,rule)

 PROVE(statement) :=
  construct_or_verify_algebraic_proof(statement)

 COMPUTE(problem) :=
  select_and_apply_exact_algebraic_algorithm(problem)

 ROUTE(query) :=
  identify_relevant_algebra_subontology(query)

 VERIFY(result) :=
  validate_against_algebraic_definitions_constraints_and_theorems
}

AXIOMS {
 A1_SYMBOLIC_MEANING:
  Every algebraic entity must have a defined symbolic or structural role

 A2_OPERATIONAL_CONTEXT:
  Algebraic entities derive meaning through operations, relations and constraints

 A3_EQUIVALENCE:
  Valid symbolic transformations must preserve the declared algebraic or logical relation

 A4_DOMAIN_SPECIFICITY:
  Algebraic operations must be interpreted within the correct scalar, coefficient or structural domain

 A5_PROOF_VALIDITY:
  Universal algebraic claims require valid proof or theorem support

 A6_STRUCTURAL_RELATION:
  Algebraic structures may be related by homomorphism, isomorphism or other typed mappings

 A7_RECURSIVE_DECOMPOSITION:
  Complex algebraic concepts may be decomposed into simpler ontology entities

 A8_COMPUTATIONAL_TRACEABILITY:
  Algebraic algorithms must correspond to mathematically valid transformations or theorems
}

REASONING {

 R1_DOMAIN_ROUTING {
  INPUT = algebraic_query
  DETECT = symbols_terms_relations
  CLASSIFY = algebra_subdomain
  ROUTE = corresponding_subontology
  RETURN = ontology_context
 }

 R2_EXPRESSION_REASONING {
  INPUT = algebraic_expression
  IDENTIFY = terms_operators_structure
  DETECT = simplification_expansion_factorisation_or_substitution_goal
  ROUTE = relevant_symbolic_ontology
  RETURN = transformed_expression
 }

 R3_EQUATION_REASONING {
  INPUT = equation_or_system
  CLASSIFY = equation_type
  IDENTIFY = admissible_transformations
  SOLVE = appropriate_method
  VERIFY = candidate_solutions
  RETURN = solution_set
 }

 R4_POLYNOMIAL_REASONING {
  INPUT = polynomial
  IDENTIFY = degree_coefficients_roots_factors
  APPLY = factor_theorem_or_polynomial_algorithm
  RETURN = polynomial_structure
 }

 R5_LINEAR_ALGEBRA_REASONING {
  INPUT = matrix_vector_or_transformation_query
  IDENTIFY = matrix_vector_space_entities
  COMPUTE = rank_kernel_image_basis_dimension_when_required
  RETURN = linear_algebra_result
 }

 R6_ABSTRACT_ALGEBRA_REASONING {
  INPUT = algebraic_structure_or_morphism
  IDENTIFY = structure_type
  TEST = preservation_and_bijectivity_properties
  RETURN = homomorphism_isomorphism_or_related_classification
 }

 R7_SYMBOLIC_REASONING {
  INPUT = expression
  SELECT = rewrite_strategy
  APPLY = valid_symbolic_rules
  TRACK = equivalence_domain_constraints
  RETURN = rewritten_form
 }

 R8_COMPUTATIONAL_REASONING {
  INPUT = algebraic_problem
  SELECT = exact_algorithm
  EXECUTE = symbolic_or_structural_computation
  VERIFY = mathematical_relation
  RETURN = computed_result
 }

 R9_PROOF_REASONING {
  INPUT = algebraic_proposition
  IDENTIFY = premises_and_goal
  SELECT = proof_strategy
  APPLY = justified_algebraic_steps
  VERIFY = conclusion
  RETURN = proof_status
 }

 R10_CROSS_ONTOLOGY_REASONING {
  INPUT = entity_or_query
  IDENTIFY = multiple_relevant_algebra_ontologies
  BUILD = typed_relation_graph
  RESOLVE = semantic_dependencies
  RETURN = integrated_algebraic_context
 }
}

CONSTRAINTS {
 C1 = "Algebraic terms must be interpreted within the correct subdomain."
 C2 = "Ambiguous symbols require contextual disambiguation."
 C3 = "Equivalent algebraic forms must not be treated as distinct mathematical objects without justification."
 C4 = "Domain restrictions and assumptions must remain explicit."
 C5 = "Transformation rules must preserve their declared equivalence or implication relation."
 C6 = "Matrix, vector-space and abstract-algebra relations must preserve type compatibility."
 C7 = "Exact algebraic reasoning must be distinguished from numerical approximation."
 C8 = "Cross-ontology links must be mathematically justified."
 C9 = "Machine-readable structure must not replace mathematical correctness."
}

SEMANTIC_EQUIVALENCE {
 "AlgebraOntologies" <=> "StructuredAlgebraicKnowledgeDomain"
 "AlgebraicEntity" <=> "TypedAlgebraicConcept"
 "SymbolicManipulation" <=> "RuleBasedAlgebraicTransformation"
 "LinearMap" <=> "LinearTransformation"
 "BijectiveHomomorphism" <=> "Isomorphism"
 "InvertibleEndomorphism" <=> "Automorphism"
}

VALIDATION {
 V1 = "Every algebraic ontology entity has a mathematical definition."
 V2 = "Every relation is type-compatible."
 V3 = "Symbolic transformations preserve declared semantics."
 V4 = "Equation solutions are verified against original constraints."
 V5 = "Polynomial factors reconstruct the original polynomial."
 V6 = "Linear-algebra dimensions and rank relations are internally consistent."
 V7 = "Morphism classifications preserve source and target structure."
 V8 = "Exact and approximate outputs are distinguished."
 V9 = "Recursive expansion terminates when the target query is resolved."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 DOMAIN_CLASSIFICATION = 0.99
 ENTITY_GROUNDING = 0.99
 RELATION_TYPING = 0.99
 SYMBOLIC_REASONING = 0.99
 LINEAR_ALGEBRA_REASONING = 0.99
 ABSTRACT_ALGEBRA_REASONING = 0.99
 COMPUTATIONAL_ROUTING = 0.98

 POLICY =
  "Reduce confidence when algebraic domain, coefficient field, assumptions, notation or transformation goal are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  AlgebraicReasoning,
  SymbolicReasoning,
  LinearAlgebraReasoning,
  AbstractAlgebraReasoning,
  ProofReasoning,
  ComputationalAlgebraReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "algebra ontology" -> AlgebraOntologies
  "algebra ontologies" -> AlgebraOntologies
  "elementary algebra" -> ElementaryAlgebra
  "linear algebra" -> LinearAlgebra
  "abstract algebra" -> AbstractAlgebra
  "symbolic algebra" -> SymbolicAlgebra
  "computational algebra" -> ComputationalAlgebra
 }

 INTERPRETATION_ORDER = [
  detect_algebraic_context,
  identify_subdomain,
  identify_entity_types,
  resolve_symbols_and_synonyms,
  identify_relations,
  identify_operations,
  identify_constraints,
  route_to_specialised_algebra_ontology,
  perform_recursive_reasoning,
  verify_result,
  return_semantic_output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_is_algebraic
  THEN activate(AlgebraOntologies)

 E2:
  IF elementary_symbolic_context
  THEN route_to(ElementaryAlgebraOntology)

 E3:
  IF polynomial_context
  THEN route_to(PolynomialOntology)

 E4:
  IF matrix_vector_space_context
  THEN route_to(LinearAlgebraRelatedOntologies)

 E5:
  IF transformation_context
  THEN route_to(TransformationOntologies)

 E6:
  IF homomorphism_isomorphism_context
  THEN route_to(AbstractAlgebraMorphisms)

 E7:
  IF symbolic_rewrite_context
  THEN route_to(SymbolicManipulationOntology)

 E8:
  IF exact_algorithmic_context
  THEN route_to(ComputationalAlgebraOntology)

 E9:
  IF proof_context
  THEN route_to(AlgebraicProofOntology)

 E10:
  IF optimisation_context
  THEN route_to(AlgebraicOptimisationOntology)

 E11:
  IF multiple_subdomains_detected
  THEN activate_cross_ontology_reasoning
}

RECURSION {
 ENTRY = AlgebraOntologies

 EXPAND(AlgebraOntologies) -> [
  ElementaryAlgebraOntology,
  AlgebraicExpressionOntology,
  EquationOntology,
  PolynomialOntology,
  MatrixOntology,
  TransformationOntology,
  HomomorphismOntology,
  AlgebraicIdentityOntology,
  SymbolicManipulationOntology,
  ComputationalAlgebraOntology,
  AlgebraicOptimisationOntology
 ]

 EXPAND(LinearAlgebra) -> [
  MatrixOntology,
  DeterminantOntology,
  RankOntology,
  KernelOntology,
  ImageOntology,
  BasisOntology,
  DimensionOntology,
  LinearTransformationOntology
 ]

 EXPAND(AbstractAlgebra) -> [
  HomomorphismOntology,
  IsomorphismOntology,
  EndomorphismOntology,
  AutomorphismOntology
 ]

 EXPAND(SymbolicAlgebra) -> [
  AlgebraicIdentityOntology,
  FactorisationOntology,
  ExpansionOntology,
  SimplificationOntology,
  SubstitutionOntology,
  AlgebraicProofOntology,
  SymbolicManipulationOntology
 ]

 EXPAND(ComputationalAlgebra) -> [
  ComputationalAlgebraOntology,
  ComputerAlgebraSystemsOntology,
  AlgebraicOptimisationOntology
 ]

 STOP_CONDITION =
  "Stop when the target algebraic entity, relation, transformation, proof or computation is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Ontologies"

 CHILD_ONTOLOGIES = [
  "Elementary Algebra Ontology",
  "Algebraic Expression Ontology",
  "Equation Ontology",
  "Linear Equation Ontology",
  "Polynomial Ontology",
  "Monomial Ontology",
  "Rational Expression Ontology",
  "Matrix Ontology",
  "Determinant Ontology",
  "Rank Ontology",
  "Null Space Ontology",
  "Column Space Ontology",
  "Row Space Ontology",
  "Kernel Ontology",
  "Image Ontology",
  "Basis Ontology",
  "Dimension Ontology",
  "Transformation Ontology",
  "Linear Transformation Ontology",
  "Affine Transformation Ontology",
  "Homomorphism Ontology",
  "Isomorphism Ontology",
  "Endomorphism Ontology",
  "Automorphism Ontology",
  "Algebraic Identity Ontology",
  "Factorisation Ontology",
  "Expansion Ontology",
  "Simplification Ontology",
  "Substitution Ontology",
  "Algebraic Proof Ontology",
  "Symbolic Manipulation Ontology",
  "Computational Algebra Ontology",
  "Computer Algebra Systems Ontology",
  "Algebraic Optimisation Ontology"
 ]

 ROUTE_RULES = [
  "Algebra Ontologies -> Elementary Algebra for basic symbolic reasoning",
  "Algebra Ontologies -> Polynomial for polynomial structure and roots",
  "Algebra Ontologies -> Matrix and Linear Algebra for vector-space reasoning",
  "Algebra Ontologies -> Transformation for mapping behaviour",
  "Algebra Ontologies -> Homomorphism and Isomorphism for abstract structural mappings",
  "Algebra Ontologies -> Symbolic Manipulation for rewrite operations",
  "Algebra Ontologies -> Computational Algebra for exact algorithms",
  "Algebra Ontologies -> Algebraic Proof for formal derivations",
  "Algebra Ontologies -> Algebraic Optimisation for goal-directed representation selection"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual reorganisation"
 MINOR = "new compatible algebraic ontology domains, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve typed algebraic entity structure.",
  "Preserve subdomain-specific interpretation.",
  "Preserve symbolic equivalence and logical validity.",
  "Preserve linear and abstract algebra type constraints.",
  "Preserve exact-versus-approximate distinction.",
  "Preserve recursive routing into specialised algebra ontologies.",
  "Preserve machine-readable algebraic semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Algebra Ontologies are structured semantic representations of algebraic entities, expressions, equations, structures, transformations, proofs and computational procedures."

 DOMAIN_SUMMARY =
  "Primary areas include elementary algebra, polynomial algebra, linear algebra, abstract algebra, transformations, symbolic manipulation, computational algebra and optimisation."

 REASONING_SUMMARY =
  "Queries are classified by algebraic subdomain, grounded to typed entities, routed to specialised ontologies and resolved through recursive reasoning."

 PROOF_SUMMARY =
  "Algebraic claims are distinguished from computation and require valid proof when universal truth is asserted."

 COMPUTATION_SUMMARY =
  "Algebraic algorithms operate under explicit domains, constraints, exactness and verification rules."

 SEMANTIC_SUMMARY_SHORT =
  "The ontology layer provides a machine-readable algebraic knowledge graph for AI reasoning and retrieval."
}

FLOW {
 INPUT
 -> detect_algebraic_context
 -> identify_algebraic_subdomain
 -> identify_entities_symbols_and_relations
 -> disambiguate_terms
 -> classify_entity_types
 -> identify_operations_and_constraints
 -> route_to_specialised_algebra_ontology
 -> recursively_expand_required_dependencies
 -> perform_symbolic_structural_proof_or_computational_reasoning
 -> verify_algebraic_validity
 -> integrate_cross_subdomain_relations
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Algebra Ontologies provide the parent semantic framework for representing and connecting algebraic knowledge. They organise symbolic expressions, equations, polynomials, matrices, vector spaces, transformations, morphisms, identities, proofs and computational procedures, allowing AI systems to route algebraic queries, ground concepts, perform recursive reasoning and verify results within explicit mathematical domains and constraints."
}

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

}

END_FLOW