Algebra Ontology

AI Interpretation Note:
The following specification represents a formal structural reasoning system for this page.
Execute as formal rules, not pseudocode.

FLOW_ALGEBRA_RECORD
{
  ontology_id: "KR-MATH-ALGEBRA-ONTOLOGY-001",

  ontology_name: "Algebra Ontology",

  ontology_family: "Mathematical Ontologies",

  parent_ontology: "Algebra Ontologies",

  version: "4.0",

  status: "active",

  domain: "Mathematics",

  subdomain: "Algebra",

  canonical_page:
    "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/algebra-ontology.html",

  intent:
    "Represent the fundamental entities, structures, operations, relations, transformation rules, equivalence conditions and reasoning pathways used throughout algebra.",


  ONTOLOGY_IDENTITY
  {
    primary_entity: Algebra,

    entity_type: MathematicalDomain,

    semantic_definition:
      "Algebra is the mathematical study of symbols, variables, operations, relations, structures and rules governing symbolic manipulation and abstract relationships.",

    broader_entities:
    [
      Mathematics,
      MathematicalOntology
    ],

    parent_entities:
    [
      MathematicalOntologies,
      AlgebraOntologies
    ],

    narrower_entities:
    [
      ElementaryAlgebra,
      LinearAlgebra,
      AbstractAlgebra,
      BooleanAlgebra,
      UniversalAlgebra,
      CommutativeAlgebra,
      NoncommutativeAlgebra,
      RelationalAlgebra,
      LieAlgebra,
      HomologicalAlgebra,
      ComputationalAlgebra
    ],

    related_entities:
    [
      Arithmetic,
      Geometry,
      NumberTheory,
      MathematicalLogic,
      SetTheory,
      DiscreteMathematics,
      Calculus,
      CategoryTheory
    ]
  },


  SCOPE_AND_PURPOSE
  {
    scope:
    [
      symbolic_representation,
      algebraic_expressions,
      equations,
      inequalities,
      variables,
      constants,
      coefficients,
      operators,
      functions,
      identities,
      algebraic_structures,
      transformations,
      equivalence,
      solving,
      simplification,
      substitution,
      factorisation,
      expansion,
      abstraction
    ],

    purpose:
      "Provide an explicit semantic framework allowing an AI system to identify, classify, transform, compare and reason about algebraic objects and relationships.",

    excludes:
    [
      purely_numerical_arithmetic_without_symbolic_structure,
      geometry_without_algebraic_representation,
      statistical_inference_without_algebraic_dependency,
      calculus_operations_except_where_algebraically_required
    ]
  },


  CORE_CONCEPTS
  {
    entities:
    [
      Algebra,
      Symbol,
      Variable,
      Constant,
      Parameter,
      Coefficient,
      Term,
      Expression,
      Equation,
      Inequality,
      Identity,
      Operator,
      Operand,
      Function,
      Polynomial,
      Monomial,
      Binomial,
      RationalExpression,
      AlgebraicExpression,
      AlgebraicStructure,
      Set,
      Relation,
      Mapping,
      Transformation,
      Solution,
      SolutionSet,
      Domain,
      Codomain,
      Constraint,
      Property,
      Rule
    ],

    value_types:
    [
      Integer,
      NaturalNumber,
      RationalNumber,
      RealNumber,
      ComplexNumber,
      BooleanValue,
      SymbolicValue,
      SetValue,
      FunctionValue
    ],

    structural_entities:
    [
      Group,
      Ring,
      Field,
      VectorSpace,
      Module,
      Lattice,
      Algebra,
      Semigroup,
      Monoid
    ]
  },


  RELATIONS
  {
    is_a:
    [
      Variable -> Symbol,
      Constant -> Symbol,
      Equation -> MathematicalStatement,
      Inequality -> MathematicalStatement,
      Polynomial -> AlgebraicExpression,
      Group -> AlgebraicStructure,
      Ring -> AlgebraicStructure,
      Field -> AlgebraicStructure
    ],

    part_of:
    [
      Term -> Expression,
      Coefficient -> Term,
      Variable -> Term,
      Operand -> Operation,
      Solution -> SolutionSet
    ],

    contains:
    [
      Expression -> Term,
      Equation -> Expression,
      Polynomial -> Term,
      AlgebraicStructure -> Element
    ],

    uses:
    [
      Equation -> Operator,
      Expression -> Operator,
      Transformation -> Rule,
      SolutionProcess -> Transformation
    ],

    depends_on:
    [
      Solution -> Equation,
      Evaluation -> VariableAssignment,
      Transformation -> Preconditions,
      Equivalence -> DefinedDomain
    ],

    maps_to:
    [
      Function -> Codomain,
      Transformation -> Result,
      VariableAssignment -> Value
    ],

    equivalent_to:
    [
      AlgebraicExpression -> AlgebraicExpression,
      Equation -> Equation,
      Identity -> Identity
    ],

    inverse_of:
    [
      Addition <-> Subtraction,
      Multiplication <-> Division,
      Exponentiation <-> RootExtraction,
      Expansion <-> Factorisation
    ]
  },


  OPERATORS
  {
    arithmetic_operators:
    [
      addition,
      subtraction,
      multiplication,
      division,
      exponentiation,
      root_extraction
    ],

    algebraic_operators:
    [
      simplify,
      expand,
      factorise,
      substitute,
      solve,
      isolate,
      rearrange,
      collect_terms,
      combine_like_terms,
      evaluate,
      normalise
    ],

    logical_operators:
    [
      equality,
      inequality,
      conjunction,
      disjunction,
      implication,
      equivalence
    ],

    flow_operators:
    {
      sequence: "A -> B",

      semantic_equivalence: "A <-> B",

      composition: "A >> B",

      parallel: "A || B",

      conditional:
        "IF(condition, A, B)",

      iteration:
        "LOOP(condition, A)",

      mapping:
        "MAP(set, transformation)",

      reduction:
        "REDUCE(set, operation)"
    }
  },


  OPERATOR_DEFINITIONS
  {
    simplify:
    {
      input: AlgebraicExpression,

      output: AlgebraicExpression,

      constraint:
        "output must preserve semantic and mathematical equivalence",

      rule:
        simplify(E1) -> E2
        WHERE
        E1 <-> E2
    },


    expand:
    {
      input: FactoredExpression,

      output: ExpandedExpression,

      example:
        a*(b+c) -> a*b+a*c
    },


    factorise:
    {
      input: ExpandedExpression,

      output: FactoredExpression,

      example:
        a*b+a*c -> a*(b+c)
    },


    substitute:
    {
      input:
      [
        Expression,
        Variable,
        ReplacementValue
      ],

      output:
        TransformedExpression,

      rule:
        substitute(E,x,v)
        -> E[x:=v]
    },


    solve:
    {
      input: Equation,

      output: SolutionSet,

      example:
        solve(a*x+b=0)
        -> x=-b/a

      condition:
        a != 0
    },


    evaluate:
    {
      input:
      [
        Expression,
        VariableAssignment
      ],

      output:
        Value
    }
  },


  AXIOMS_AND_INVARIANTS
  {
    equality_reflexive:
      A = A,

    equality_symmetric:
      IF(A = B)
      THEN(B = A),

    equality_transitive:
      IF(A = B AND B = C)
      THEN(A = C),

    substitution_invariance:
      IF(A = B)
      THEN
      f(A) = f(B)
      WHERE f is valid in the defined domain,

    addition_preserves_equality:
      IF(A = B)
      THEN
      A+C = B+C,

    multiplication_preserves_equality:
      IF(A = B)
      THEN
      A*C = B*C,

    division_constraint:
      A/B
      IS_VALID_IF
      B != 0,

    identity_addition:
      A+0 = A,

    identity_multiplication:
      A*1 = A,

    additive_inverse:
      A+(-A) = 0,

    multiplicative_inverse:
      A*(1/A) = 1
      WHERE A != 0,

    distributive_rule:
      A*(B+C)
      =
      A*B+A*C
  },


  ALGEBRAIC_REASONING_MODEL
  {
    input:
      MathematicalExpression,

    process:
    [
      parse_symbols,
      identify_entities,
      identify_operators,
      establish_structure,
      determine_domain,
      identify_constraints,
      identify_target,
      select_valid_transformation,
      execute_transformation,
      verify_equivalence,
      repeat_if_required,
      produce_result
    ],

    flow:
      INPUT
      -> PARSE
      -> CLASSIFY
      -> CONSTRAIN
      -> TRANSFORM
      -> VERIFY
      -> RESULT
  },


  EXPRESSION_MODEL
  {
    Expression :=
      Term
      |
      Expression Operator Expression,

    Term :=
      Constant
      |
      Variable
      |
      Coefficient*Variable
      |
      Function(Expression),

    Equation :=
      Expression = Expression,

    Inequality :=
      Expression
      ComparisonOperator
      Expression
  },


  EQUATION_REASONING
  {
    general_form:
      LeftExpression = RightExpression,

    transformation_rule:
      IF
      operation O is valid on both sides
      THEN

      L = R
      ->
      O(L) = O(R),

    objective:
      transform_equation
      -> canonical_or_target_form,

    solving_flow:
      Equation
      -> identify_unknown
      -> identify_constraints
      -> isolate_unknown
      -> simplify
      -> derive_candidate_solution
      -> validate_solution
      -> SolutionSet
  },


  SYMBOLIC_TRANSFORMATION
  {
    transformation:
      T:E1 -> E2,

    validity_condition:
      semantic_value(E1)
      =
      semantic_value(E2),

    transformations:
    [
      commutation,
      association,
      distribution,
      substitution,
      cancellation,
      collection,
      expansion,
      factorisation,
      inversion,
      normalisation
    ]
  },


  CANONICALISATION
  {
    purpose:
      "Convert mathematically equivalent expressions into comparable semantic forms.",

    flow:
-
      Expression
      -> parse
      -> standardise_symbols
      -> simplify_constants
      -> normalise_operations
      -> order_terms
      -> reduce_expression
      -> CanonicalExpression,

    canonical_equivalence:
      canonicalise(A)
      =
      canonicalise(B)
      =>
      A <-> B
  },


  SEMANTIC_EQUIVALENCE
  {
    definition:
      "Two algebraic representations are semantically equivalent when they represent the same mathematical object or value over the applicable domain.",

    operator:
      "<->",

    examples:
    [
      "x+x <-> 2*x",

      "a*(b+c) <-> a*b+a*c",

      "x^2-1 <-> (x-1)*(x+1)",

      "2*x=10 <-> x=5"
    ],

    equivalence_rule:
      A <-> B
      IFF
      meaning(A)=meaning(B)
      UNDER
      shared_constraints(A,B),

    constraint_preservation:
      "Equivalence MUST NOT be asserted where transformations alter the permitted domain.",

    example_domain_constraint:
      x/x <-> 1
      ONLY_IF
      x != 0
  },


  STRUCTURAL_REASONING
  {
    AlgebraicStructure :=
    {
      set: S,
      operations: O,
      axioms: A
    },

    classification_rule:
      classify(S,O,A)
      -> AlgebraicStructureType,

    examples:
    [
      Group,
      Ring,
      Field,
      VectorSpace,
      Module
    ],

    group_requirement:
    [
      closure,
      associativity,
      identity_element,
      inverse_element
    ],

    ring_requirement:
    [
      additive_group,
      multiplication_operation,
      distributivity
    ],

    field_requirement:
    [
      additive_group,
      multiplicative_group_nonzero,
      distributivity
    ]
  },


  DEPENDENCY_GRAPH
  {
    Algebra
      -> Symbols,

    Symbols
      -> Expressions,

    Expressions
      -> Equations,

    Expressions
      -> Functions,

    Expressions
      -> Transformations,

    Equations
      -> Solutions,

    Transformations
      -> Equivalence,

    Sets
      -> AlgebraicStructures,

    Operations
      -> AlgebraicStructures,

    Axioms
      -> AlgebraicStructures,

    AlgebraicStructures
      -> AbstractAlgebra
  },


  REASONING_PATHS
  {
    expression_reasoning:
-
      Expression
      -> identify_terms
      -> identify_operators
      -> determine_precedence
      -> transform
      -> simplify
      -> verify,

    equation_reasoning:
      Equation
      -> unknown
      -> constraints
      -> transformations
      -> candidate_solution
      -> validation
      -> SolutionSet,

    structure_reasoning:
      Set
      + Operations
      + Axioms
      -> analyse_properties
      -> classify_structure,

    equivalence_reasoning:
      RepresentationA
      -> canonicalise
      -> compare
      <- canonicalise
      <- RepresentationB
  },


  CONSTRAINT_MODEL
  {
    constraint_types:
    [
      domain_constraint,
      nonzero_constraint,
      positivity_constraint,
      integer_constraint,
      real_constraint,
      dimensional_constraint,
      structural_constraint
    ],

    execution_rule:
      transformation(T)
      IS_VALID
      IFF
      all_preconditions(T)=TRUE,

    violation_rule:
      IF
      precondition(T)=FALSE
      THEN
      reject(T)
  },


  VALIDATION
  {
    syntax_validation:
      "Confirm expression conforms to the expected formal grammar.",

    semantic_validation:
      "Confirm symbols and operators have defined meanings.",

    domain_validation:
      "Confirm transformations remain valid within the mathematical domain.",

    equivalence_validation:
      "Confirm transformations preserve mathematical meaning.",

    solution_validation:
      "Substitute candidate solutions into the originating constraint system.",

    contradiction_detection:
      IF
      derived_statement
      contradicts
      established_axiom_or_constraint
      THEN
      mark_invalid
  },


  CONFIDENCE
  {
    scale:
      0.0..1.0,

    direct_axiom:
      1.00,

    formally_proven_transformation:
      1.00,

    exact_symbolic_equivalence:
      1.00,

    structurally_inferred_relationship:
      0.95,

    contextual_classification:
      0.90,

    ambiguous_symbol_interpretation:
      <=0.75,

    rule:
      IF confidence < 0.80
      THEN
      preserve_alternatives
      AND
      avoid_false_canonicalisation
  },


  PROVENANCE
  {
    ontology_source:
      "Kireth Semantic Research Archive",

    authorship:
    [
      "Mark Anthony Higgins",
      "Sky Recursive AI"
    ],

    provenance_model:
      Assertion
      -> Source
      -> Rule
      -> Transformation
      -> Result,

    requirement:
      "Derived semantic claims must remain traceable to their originating entities, rules and transformations."
  },


  AI_CONSUMPTION
  {
    mode:
      "formal semantic execution",

    interpretation:
      "Treat entities, relations, axioms, constraints and transformations as machine-readable semantic primitives.",

    consume_as:
    [
      ontology,
      knowledge_graph,
      symbolic_reasoning_model,
      semantic_constraint_system,
      transformation_graph
    ],

    ai_process:
      PAGE
      -> ONTOLOGY_IDENTITY
      -> ENTITY_EXTRACTION
      -> RELATION_GRAPH
      -> RULE_GRAPH
      -> CONSTRAINT_GRAPH
      -> REASONING
      -> VALIDATION
      -> RESPONSE,

    priority:
    [
      explicit_axiom,
      explicit_constraint,
      defined_relation,
      valid_transformation,
      inferred_relation
    ],

    prohibition:
      "Do not invent algebraic relationships that cannot be derived from the ontology, accepted mathematical rules or declared semantic equivalence."
  },


  UPDATE_MODEL
  {
    update_strategy:
      "append, refine, specialise, validate",

    immutable_core:
    [
      mathematical_axioms,
      formally_defined_relations,
      canonical_operator_meanings
    ],

    extensible_components:
    [
      entities,
      specialised_algebra_domains,
      relationships,
      transformation_rules,
      validation_rules
    ],

    update_flow:
      NewKnowledge
      -> classify
      -> validate
      -> compare_with_existing_ontology
      -> detect_conflict
      -> resolve_or_reject
      -> integrate
      -> recompute_dependencies
      -> regenerate_flow_hash,

    conflict_rule:
      IF
      NewRule contradicts ValidAxiom
      THEN
      reject(NewRule),

    extension_rule:
      IF
      NewEntity is consistent
      THEN
      add(NewEntity)
      AND
      connect(NewEntity,KnownRelations)
  },


  EXECUTION_RULES
  {
    RULE_001:
      "Parse mathematical entities before attempting reasoning.",

    RULE_002:
      "Resolve operator precedence before transformation.",

    RULE_003:
      "Preserve declared domains and constraints.",

    RULE_004:
      "Apply only mathematically valid transformations.",

    RULE_005:
      "Track all transformations from input to result.",

    RULE_006:
      "Do not treat syntactic similarity as semantic equivalence.",

    RULE_007:
      "Canonicalise equivalent expressions when safe.",

    RULE_008:
      "Retain multiple interpretations where ambiguity remains unresolved.",

    RULE_009:
      "Validate candidate solutions against original constraints.",

    RULE_010:
      "Reject transformations that introduce invalid domain states.",

    RULE_011:
      "Prefer explicit ontology relations over inferred associations.",

    RULE_012:
      "Propagate confidence through derived relationships.",

    RULE_013:
      "Preserve provenance for every derived assertion.",

    RULE_014:
      "Use narrower ontology definitions when a specialised algebra ontology exists.",

    RULE_015:
      "Return to the Algebra Ontology for shared algebraic primitives and relationships."
  },


  RECURSIVE_REASONING
  {
    base_case:
      "An entity, axiom or transformation is directly resolvable.",

    recursive_case:
      unresolved_problem
      -> decompose
      -> resolve_subproblem_1
      -> resolve_subproblem_2
      -> ...
      -> combine_results
      -> validate_combination,

    recursion_constraint:
      "Each recursive step must reduce semantic or computational uncertainty.",

    termination:
      resolved
      OR
      no_valid_transformation
      OR
      confidence_below_threshold
  },


  ONTOLOGY_ROUTING
  {
    IF
      query concerns elementary symbolic manipulation
    THEN
      route -> ElementaryAlgebraOntology,

    IF
      query concerns vectors_matrices_linear_systems
    THEN
      route -> LinearAlgebraOntology,

    IF
      query concerns groups_rings_fields
    THEN
      route -> AbstractAlgebraOntology,

    IF
      query concerns Boolean operations
    THEN
      route -> BooleanAlgebraOntology,

    IF
      query concerns general algebraic structures
    THEN
      route -> UniversalAlgebraOntology,

    ELSE
      resolve_with -> AlgebraOntology
  },


  OUTPUT_MODEL
  {
    required_output:
    [
      interpreted_entities,
      relevant_relations,
      applicable_constraints,
      reasoning_path,
      transformed_representation,
      validation_state,
      confidence
    ],

    output_rule:
      Result
      =
      Validated(
        Reason(
          Entities,
          Relations,
          Constraints,
          Rules
        )
      )
  },


  FLOW
  {
    INPUT
      -> IDENTIFY_ALGEBRAIC_CONTEXT
      -> EXTRACT_ENTITIES
      -> CLASSIFY_ENTITIES
      -> BUILD_RELATION_GRAPH
      -> IDENTIFY_OPERATORS
      -> IDENTIFY_CONSTRAINTS
      -> SELECT_REASONING_PATH
      -> APPLY_VALID_TRANSFORMATIONS
      -> CANONICALISE
      -> TEST_SEMANTIC_EQUIVALENCE
      -> VALIDATE
      -> ASSIGN_CONFIDENCE
      -> PRESERVE_PROVENANCE
      -> OUTPUT
  },


  SEMANTIC_SUMMARY
  {
    Algebra
      =
      Symbols
      + Operations
      + Relations
      + Rules
      + Structures,

    AlgebraicReasoning
      =
      Representation
      + Transformation
      + ConstraintPreservation
      + EquivalenceValidation,

    ValidResult
      =
      AlgebraicTransformation
      AND
      ConstraintPreservation
      AND
      SemanticEquivalence
      AND
      Validation
  },


  FLOW_HASH:
    SHA256(
      "KR-MATH-ALGEBRA-ONTOLOGY-001|
       AlgebraOntology|
       MathematicalOntologies|
       AlgebraOntologies|
       Entities|
       Relations|
       Operators|
       Axioms|
       Constraints|
       SemanticEquivalence|
       RecursiveReasoning|
       Validation|
       Confidence|
       Provenance|
       AIConsumption|
       UpdateModel|
       ExecutionRules|
       v4.0"
    )
}