Computational Algebra Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of computational algebra. Computational algebra concerns the representation, manipulation and algorithmic solution of algebraic objects using symbolic and exact computational methods. It links algebraic structures, algorithms, normal forms, symbolic manipulation, polynomial systems, matrix methods and computer algebra systems. 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 = "Computational Algebra Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Algebra"
 SPECIALISATION = "Computational Algebra / Computer Algebra"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/computational-algebra-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ComputationalAlgebra
 SYMBOLIC_FORMS = ["algorithm(E)", "normal_form(E)", "solve(P)", "reduce(G)"]
 ENTITY_CLASS = AlgorithmicMathematicsDomain
 PRIMARY_CONTEXT = Algebra
 SECONDARY_CONTEXTS = [ComputerAlgebra, SymbolicComputation, ComputationalMathematics]
 DEFINITION = "Computational algebra is the study and algorithmic manipulation of algebraic structures and expressions using exact symbolic methods."
 SEMANTIC_ROLE = "Connects algebraic theory with executable procedures for representation, transformation, decision and solution."
}

SCOPE {
 INCLUDE = [
  ComputationalAlgebra,
  ComputerAlgebra,
  SymbolicComputation,
  AlgebraicAlgorithm,
  ExactArithmetic,
  PolynomialArithmetic,
  PolynomialDivision,
  GreatestCommonDivisor,
  Factorisation,
  Simplification,
  NormalForm,
  GröbnerBasis,
  PolynomialSystemSolving,
  MatrixComputation,
  LinearAlgebraAlgorithms,
  Resultant,
  Elimination,
  ComputerAlgebraSystem
 ]

 EXCLUDE_AS_PRIMARY = [
  PureNumericalAnalysis,
  FloatingPointApproximationOnly,
  StatisticalComputation,
  GenericProgrammingWithoutAlgebraicSemantics
 ]

 DISAMBIGUATION_RULE =
  "When computational algebra occurs with symbolic manipulation, polynomials, Gröbner bases, factorisation, exact arithmetic, algebraic algorithms, matrices or computer algebra systems, interpret it as algorithmic symbolic algebra."
}

CORE_CONCEPTS {

 ComputationalAlgebra {
  definition = "Algorithmic treatment of algebraic expressions, equations and structures using exact or symbolic representations."
  goals = [
   ExactComputation,
   SymbolicTransformation,
   DecisionProcedures,
   EquationSolving,
   Canonicalisation,
   StructuralAnalysis
  ]
 }

 AlgebraicAlgorithm {
  definition = "A finite executable procedure operating on algebraic objects and preserving or deriving mathematically valid results."
 }

 ExactArithmetic {
  definition = "Arithmetic performed without floating-point approximation when exact representation is available."
  examples = [IntegerArithmetic, RationalArithmetic, PolynomialArithmetic]
 }

 PolynomialArithmetic {
  operations = [
   Addition,
   Subtraction,
   Multiplication,
   DivisionWithRemainder,
   GCD,
   Factorisation
  ]
 }

 NormalForm {
  definition = "A canonical or selected reduced representation used to compare or simplify algebraic objects."
 }

 GröbnerBasis {
  definition = "A specially structured generating set for a polynomial ideal relative to a monomial order, enabling algorithmic reduction and elimination."
  role = "Supports ideal membership, polynomial-system solving and elimination."
 }

 Elimination {
  definition = "Algorithmic removal of variables from systems of polynomial equations."
 }

 Resultant {
  definition = "An algebraic construction used to eliminate a variable and detect common roots."
 }

 ComputerAlgebraSystem {
  definition = "Software implementing symbolic algebraic algorithms."
  role = "Executes algebraic representations and transformations according to formal rules."
 }
}

RELATIONS {
 ComputationalAlgebra --uses--> AlgebraicAlgorithm
 ComputationalAlgebra --uses--> SymbolicManipulation
 ComputationalAlgebra --uses--> ExactArithmetic
 ComputationalAlgebra --operates_on--> AlgebraicExpression
 ComputationalAlgebra --operates_on--> Polynomial
 ComputationalAlgebra --operates_on--> Matrix
 ComputationalAlgebra --can_compute--> NormalForm
 ComputationalAlgebra --can_compute--> Factorisation
 ComputationalAlgebra --can_compute--> GreatestCommonDivisor
 ComputationalAlgebra --can_compute--> GröbnerBasis
 ComputationalAlgebra --can_solve--> PolynomialSystem
 GröbnerBasis --supports--> IdealMembership
 GröbnerBasis --supports--> Elimination
 Resultant --supports--> Elimination
 ComputerAlgebraSystem --implements--> AlgebraicAlgorithm
 SymbolicManipulation --is_operational_component_of--> ComputationalAlgebra
}

OPERATORS {
 COMPUTE(E,ALGORITHM) :=
  execute_algorithm_on_algebraic_object(E)

 EXACT_ADD(a,b) :=
  algebraically_exact_sum(a,b)

 POLYNOMIAL_DIVIDE(P,Q) :=
  return(quotient,remainder)

 POLYNOMIAL_GCD(P,Q) :=
  EuclideanAlgorithm(P,Q)

 FACTOR_POLYNOMIAL(P,D) :=
  factor_over_domain_D(P)

 NORMAL_FORM(E,RULESET) :=
  reduce_expression_under_ruleset(E)

 GROEBNER_BASIS(F,ORDER) :=
  Buchberger_or_equivalent_algorithm(F,ORDER)

 REDUCE_POLYNOMIAL(P,G) :=
  multivariate_division_remainder(P,G)

 IDEAL_MEMBERSHIP(P,G) :=
  REDUCE_POLYNOMIAL(P,G)==0 WHEN G_is_GroebnerBasis

 ELIMINATE(F,VARIABLES) :=
  compute_elimination_representation(F,VARIABLES)

 SOLVE_POLYNOMIAL_SYSTEM(F) :=
  derive_solution_structure_using_exact_algebraic_methods
}

DEFINITIONS {
 D1 = "Computational algebra applies algorithms to algebraic objects."
 D2 = "Symbolic computation manipulates expressions according to exact algebraic rules."
 D3 = "Exact arithmetic avoids approximation when exact representations are retained."
 D4 = "Polynomial division produces quotient and remainder satisfying P=Q*quotient+remainder."
 D5 = "Polynomial GCD generalises integer greatest-common-divisor reasoning."
 D6 = "Normal forms support comparison and algorithmic decision."
 D7 = "A Gröbner basis depends on a chosen monomial order."
 D8 = "Reduction by a Gröbner basis can decide polynomial ideal membership."
 D9 = "Elimination algorithms reduce variables in polynomial systems."
 D10 = "Computer algebra systems operationalise computational algebra algorithms."
}

AXIOMS {
 A1_EXACTNESS:
  IF an exact representation is used
  THEN algebraic operations should preserve exactness unless approximation is explicitly requested

 A2_ALGORITHMIC_VALIDITY:
  Every computational transformation must correspond to a mathematically valid algebraic rule or theorem

 A3_POLYNOMIAL_DIVISION:
  FOR suitable polynomial division setting
  P = Q*S + R

 A4_GCD:
  GCD(P,Q) divides both P and Q and is maximal up to multiplication by units

 A5_NORMAL_FORM:
  A chosen reduction system maps algebraic objects to a reduced representation

 A6_GROEBNER_REDUCTION:
  IF G is a Gröbner basis for ideal I
  THEN P belongs to I iff the remainder of P on reduction by G is zero

 A7_DOMAIN_DEPENDENCE:
  Computational results such as factorisation depend on the coefficient domain
}

REASONING {

 R1_ALGORITHM_SELECTION {
  INPUT = algebraic_task
  CLASSIFY = object_type_and_goal
  SELECT = appropriate_exact_algorithm
  RETURN = algorithm
 }

 R2_POLYNOMIAL_GCD {
  INPUT = [P,Q]
  APPLY = polynomial_Euclidean_algorithm_or_subresultant_method
  NORMALISE = gcd_by_unit_convention
  RETURN = gcd
 }

 R3_FACTORISATION {
  INPUT = polynomial P and coefficient domain D
  DETECT = square_free_structure
  APPLY = factorisation_algorithm_for_D
  VERIFY = multiply_factors
  RETURN = irreducible_factorisation_over_D
 }

 R4_GROEBNER {
  INPUT = generating_polynomials F and monomial_order O
  INITIALISE = basis F
  COMPUTE = S_polynomials
  REDUCE = S_polynomials_against_basis
  ADD = nonzero_remainders
  REPEAT = until_completion_criterion
  RETURN = GroebnerBasis
 }

 R5_IDEAL_MEMBERSHIP {
  INPUT = [P,G]
  REQUIRE = G_is_GroebnerBasis
  REDUCE = P modulo G
  IF remainder==0
   RETURN Member
  ELSE
   RETURN NotMember
 }

 R6_ELIMINATION {
  INPUT = polynomial_system F
  SELECT = elimination_order_or_resultant_method
  COMPUTE = elimination_objects
  RETURN = reduced_variable_system
 }

 R7_MATRIX_COMPUTATION {
  INPUT = matrix_problem
  SELECT = exact_linear_algebra_algorithm
  POSSIBLE = [RowReduction, Determinant, Rank, NullSpace, SmithNormalForm]
  RETURN = exact_algebraic_result
 }

 R8_NORMALISATION {
  INPUT = expression E
  APPLY = rewrite_and_reduction_strategy
  VERIFY = equivalence
  RETURN = normal_form
 }
}

CONSTRAINTS {
 C1 = "Algorithm selection must respect the algebraic structure and coefficient domain."
 C2 = "Exact symbolic methods should not silently replace exact values with floating-point approximations."
 C3 = "Factorisation and irreducibility are domain-dependent."
 C4 = "A Gröbner basis is defined relative to a monomial order."
 C5 = "Reduction results may depend on basis properties and ordering strategy."
 C6 = "Algorithmic termination and complexity are separate from mathematical correctness."
 C7 = "Canonical or normal forms require an explicit convention."
 C8 = "Do not confuse computational algebra with purely numerical computation."
 C9 = "Computer algebra output should remain traceable to valid mathematical transformations."
}

SEMANTIC_EQUIVALENCE {
 "ComputationalAlgebra" <=> "AlgorithmicAlgebra"
 "ComputerAlgebra" <=> "SymbolicAlgebraComputation"
 "CAS" <=> "ComputerAlgebraSystem"
 "NormalForm" <=> "SelectedReducedCanonicalRepresentation"
 "GroebnerBasis" <=> "AlgorithmicallyStructuredIdealGeneratingSet"
}

VALIDATION {
 V1 = "Polynomial division satisfies dividend=divisor*quotient+remainder."
 V2 = "Computed factors multiply back to the original polynomial."
 V3 = "Exact arithmetic retains exact values."
 V4 = "A Gröbner basis depends on monomial order."
 V5 = "Reduction by a Gröbner basis can test ideal membership."
 V6 = "Normal-form computation preserves the intended algebraic equivalence."
 V7 = "Domain selection affects factorisation results."
 V8 = "Computer algebra procedures must implement mathematically valid rules."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 COMPUTATIONAL_ALGEBRA_DEFINITION = 0.99
 EXACT_ARITHMETIC = 0.99
 POLYNOMIAL_ALGORITHMS = 0.99
 NORMAL_FORM_RELATION = 0.98
 GROEBNER_BASIS_RELATION = 0.99
 IDEAL_MEMBERSHIP_RELATION = 0.99
 DOMAIN_DEPENDENCE = 0.99

 POLICY =
  "Reduce confidence when coefficient domain, monomial order, algebraic structure or algorithmic assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SymbolicComputation,
  AutomatedAlgebra,
  PolynomialReasoning,
  ExactLinearAlgebra,
  EquationSolving,
  IdealReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "computational algebra" -> ComputationalAlgebra
  "computer algebra" -> ComputationalAlgebra
  "symbolic computation" -> SymbolicComputation
  "computer algebra system" -> ComputerAlgebraSystem
  "CAS" -> ComputerAlgebraSystem
  "Groebner basis" -> GröbnerBasis
  "normal form" -> NormalForm
  "polynomial algorithm" -> AlgebraicAlgorithm
 }

 INTERPRETATION_ORDER = [
  identify_algebraic_object,
  identify_coefficient_domain,
  identify_computational_goal,
  select_exact_algorithm,
  identify_required_ordering_or_normal_form,
  execute_symbolic_transformation,
  preserve_exactness,
  verify_result,
  expose_complexity_or_termination_constraints_when_relevant,
  return_semantic_result
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("computational algebra")
  OR query_mentions("computer algebra")
  THEN activate(ComputationalAlgebraOntology)

 E2:
  IF query_mentions("Groebner basis")
  THEN activate(ComputationalAlgebraOntology)
  AND require(monomial_order_context)

 E3:
  IF polynomial_factorisation_requested
  THEN identify_coefficient_domain_before_factorisation

 E4:
  IF exact_result_requested
  THEN prohibit_unrequested_floating_point_approximation

 E5:
  IF ideal_membership_requested
  THEN require_or_compute_GroebnerBasis

 E6:
  IF normal_form_requested
  THEN identify_reduction_rules_or_canonicalisation_policy

 E7:
  IF matrix_computation_context
  THEN route_to_exact_linear_algebra_algorithms

 E8:
  IF request=="verify computational result"
  THEN reconstruct_and_validate_mathematical_relation
}

RECURSION {
 ENTRY = ComputationalAlgebra

 EXPAND(ComputationalAlgebra) -> [
  SymbolicManipulation,
  AlgebraicAlgorithm,
  Polynomial,
  Matrix,
  ExactArithmetic,
  NormalForm,
  GröbnerBasis,
  Factorisation,
  EquationSystem,
  ComputerAlgebraSystem
 ]

 EXPAND(PolynomialAlgorithms) -> [
  PolynomialDivision,
  GCD,
  Factorisation,
  Resultant,
  Root,
  Irreducibility
 ]

 EXPAND(GröbnerBasis) -> [
  PolynomialIdeal,
  MonomialOrder,
  SPolynomial,
  Reduction,
  Elimination,
  IdealMembership
 ]

 EXPAND(MatrixComputation) -> [
  RowReduction,
  Rank,
  Determinant,
  NullSpace,
  Kernel,
  Image,
  CanonicalMatrixForm
 ]

 STOP_CONDITION =
  "Stop when the requested algebraic computation is completed, exactness and assumptions are resolved, and the result has been mathematically verified."
}

ROUTING {
 PARENT_ONTOLOGY = "Algebra Ontology"

 IMMEDIATE_CONTEXTS = [
  "Symbolic Manipulation Ontology",
  "Polynomial Ontology",
  "Matrix Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Symbolic Manipulation Ontology",
  "Simplification Ontology",
  "Expansion Ontology",
  "Factorisation Ontology",
  "Substitution Ontology",
  "Algebraic Proof Ontology",
  "Polynomial Ontology",
  "Matrix Ontology",
  "Rank Ontology",
  "Kernel Ontology",
  "Image Ontology",
  "Basis Ontology",
  "Dimension Ontology"
 ]

 ROUTE_RULES = [
  "Computational Algebra -> Symbolic Manipulation for expression rewriting",
  "Computational Algebra -> Polynomial for exact polynomial algorithms",
  "Computational Algebra -> Factorisation for decomposition algorithms",
  "Computational Algebra -> Matrix for exact linear algebra",
  "Computational Algebra -> Algebraic Proof for verification and derivation",
  "Computational Algebra -> Kernel, Image and Rank for matrix-space computation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible algorithms, concepts, operators or relations"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve exact symbolic computation semantics.",
  "Preserve coefficient-domain dependence.",
  "Preserve mathematical validity of every algorithmic transformation.",
  "Preserve monomial-order dependence of Gröbner bases.",
  "Preserve exactness unless approximation is explicitly requested.",
  "Preserve traceability between computation and algebraic theorem.",
  "Explicitly distinguish computational algebra from purely numerical computation."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Computational algebra is the algorithmic manipulation and solution of algebraic objects using exact symbolic methods."

 SYMBOLIC_SUMMARY =
  "algebraic object + exact algorithm -> verified algebraic result"

 POLYNOMIAL_SUMMARY =
  "Core operations include polynomial arithmetic, division, GCD, factorisation and elimination."

 GROEBNER_SUMMARY =
  "A Gröbner basis supports reduction, ideal membership and elimination relative to a chosen monomial order."

 EXACTNESS_SUMMARY =
  "Exact arithmetic preserves exact algebraic values unless approximation is explicitly requested."

 CAS_SUMMARY =
  "Computer algebra systems implement computational algebra algorithms for symbolic mathematics."

 VERIFICATION_SUMMARY =
  "Every computational result should be checked against the defining mathematical relation or theorem."
}

FLOW {
 INPUT
 -> detect_computational_algebra_context
 -> identify_algebraic_object
 -> identify_coefficient_domain
 -> identify_computational_goal
 -> identify_required_order_or_normal_form
 -> select_exact_algorithm
 -> execute_symbolic_computation
 -> preserve_exactness_and_constraints
 -> derive_result
 -> verify_result_mathematically
 -> connect_related_algebraic_structures
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Computational algebra turns algebraic theory into executable exact procedures. It covers symbolic manipulation, polynomial arithmetic, factorisation, normal forms, Gröbner bases, elimination, equation solving and exact matrix computation. Its correctness depends on valid algebraic algorithms, explicit domains and ordering conventions, preservation of exactness, and verification of computed results."
}

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

}

END_FLOW