Dimension Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of dimension, principally within linear algebra and vector-space theory. Dimension measures the number of independent directions in a vector space and is equal to the cardinality of any basis in the finite-dimensional case. This record is intended for semantic retrieval, recursive mathematical reasoning, ontology linking, and AI consumption within the Kireth Research Semantic Library.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
    PRIMARY_ENTITY = Dimension
    SYMBOLIC_FORMS = ["dim(V)", "dim(W)", "dim(ker(T))", "dim(im(T))"]
    ENTITY_CLASS = VectorSpaceInvariant
    PRIMARY_CONTEXT = LinearAlgebra
    DEFINITION = "The dimension of a vector space is the cardinality of any basis of that space."
    FINITE_FORM = "If B is a finite basis of V, then dim(V)=|B|."
}

SCOPE {
    INCLUDE = [
        Dimension,
        FiniteDimension,
        InfiniteDimension,
        DimensionOfVectorSpace,
        DimensionOfSubspace,
        Rank,
        Nullity,
        DimensionOfKernel,
        DimensionOfImage,
        DimensionOfColumnSpace,
        DimensionOfRowSpace,
        BasisCardinality,
        RankNullityRelationship
    ]

    EXCLUDE_AS_PRIMARY = [
        PhysicalDimension,
        SpatialDimension,
        DimensionalAnalysis,
        DatabaseDimension,
        DataWarehouseDimension
    ]

    DISAMBIGUATION_RULE =
        "When dimension occurs with vector spaces, bases, matrices, rank, nullity, kernel, image, row space, column space, span or linear independence, interpret it as linear-algebraic dimension."
}

CORE_CONCEPTS {

    Dimension {
        definition =
            "A structural invariant measuring the number of vectors in any basis of a vector space."

        finite_form =
            "dim(V)=|B| for any finite basis B of V"

        semantic_role =
            "Measures the number of independent directions available in a vector space."
    }

    BasisCardinality {
        definition =
            "The number of vectors in a basis."

        equivalence =
            "For finite-dimensional V, basis cardinality equals dim(V)."
    }

    FiniteDimensionalSpace {
        definition =
            "A vector space possessing a finite basis."
    }

    InfiniteDimensionalSpace {
        definition =
            "A vector space whose bases have infinite cardinality."
    }

    SubspaceDimension {
        definition =
            "The dimension of a subspace W is the cardinality of any basis of W."

        inequality =
            "If W is a subspace of finite-dimensional V, then dim(W) <= dim(V)."
    }

    Rank {
        definition =
            "The dimension of the image of a linear transformation or equivalently the column space of a matrix."

        symbolic_form =
            "rank(T)=dim(im(T))"
    }

    Nullity {
        definition =
            "The dimension of the kernel of a linear transformation."

        symbolic_form =
            "nullity(T)=dim(ker(T))"
    }

    RankNullityTheorem {
        symbolic_form =
            "dim(V)=rank(T)+nullity(T)"

        condition =
            "T: V -> W with finite-dimensional domain V."
    }
}

RELATIONS {
    Dimension --is_invariant_of--> VectorSpace
    Dimension --is_determined_by--> Basis
    Basis --has_cardinality--> Dimension
    Subspace --has_dimension--> Dimension
    Kernel --has_dimension--> Nullity
    Image --has_dimension--> Rank
    ColumnSpace --has_dimension--> Rank
    RowSpace --has_dimension--> Rank
    Dimension --participates_in--> RankNullityTheorem
    Dimension --constrains--> LinearIndependence
    Dimension --constrains--> SpanningSets
    Dimension --is_preserved_by--> VectorSpaceIsomorphism
}

OPERATORS {
    DIMENSION(V) :=
        CARDINALITY(any_basis_of(V))

    DIMENSION_FROM_BASIS(B) :=
        CARDINALITY(B)

    NULLITY(T) :=
        DIMENSION(KERNEL(T))

    RANK(T) :=
        DIMENSION(IMAGE(T))

    COLUMN_RANK(A) :=
        DIMENSION(COLUMN_SPACE(A))

    ROW_RANK(A) :=
        DIMENSION(ROW_SPACE(A))

    IS_FINITE_DIMENSIONAL(V) :=
        EXISTS finite basis B of V

    SUBSPACE_DIMENSION(W) :=
        CARDINALITY(any_basis_of(W))
}

DEFINITIONS {
    D1: "dim(V)=|B| for any basis B of finite-dimensional V."
    D2: "All bases of a finite-dimensional vector space have the same cardinality."
    D3: "If W is a subspace of finite-dimensional V, then dim(W)<=dim(V)."
    D4: "If W is a subspace of V and dim(W)=dim(V), then W=V."
    D5: "rank(T)=dim(im(T))."
    D6: "nullity(T)=dim(ker(T))."
    D7: "For finite-dimensional V, dim(V)=rank(T)+nullity(T)."
    D8: "Isomorphic vector spaces have equal dimension."
    D9: "The zero vector space has dimension 0."
    D10: "F^n has dimension n."
}

AXIOMS {
    A1_BASIS_CARDINALITY:
        IF B is a basis of finite-dimensional V
        THEN dim(V)=|B|

    A2_EQUAL_BASIS_SIZE:
        IF B1 and B2 are bases of finite-dimensional V
        THEN |B1|=|B2|

    A3_SUBSPACE_BOUND:
        IF W is a subspace of finite-dimensional V
        THEN dim(W)<=dim(V)

    A4_EQUAL_DIMENSION_SUBSPACE:
        IF W is a subspace of finite-dimensional V
        AND dim(W)=dim(V)
        THEN W=V

    A5_ZERO_SPACE:
        IF V={0}
        THEN dim(V)=0

    A6_STANDARD_SPACE:
        IF V=F^n
        THEN dim(V)=n

    A7_ISOMORPHISM:
        IF V is isomorphic to W
        THEN dim(V)=dim(W)

    A8_RANK_NULLITY:
        IF T: V -> W
        AND V is finite-dimensional
        THEN dim(V)=rank(T)+nullity(T)
}

REASONING {

    R1_DIMENSION_FROM_BASIS {
        INPUT = basis B
        RETURN = cardinality(B)
    }

    R2_DIMENSION_FROM_SPANNING_SET {
        INPUT = finite spanning set S
        REDUCE = remove_linearly_dependent_vectors
        BASIS = resulting_basis
        RETURN = cardinality(BASIS)
    }

    R3_DIMENSION_FROM_MATRIX_COLUMN_SPACE {
        INPUT = matrix A
        REDUCE = RREF(A)
        COUNT = number_of_pivot_columns
        RETURN = rank(A)
    }

    R4_DIMENSION_FROM_ROW_SPACE {
        INPUT = matrix A
        REDUCE = RREF(A)
        COUNT = number_of_nonzero_rows
        RETURN = row_rank(A)
    }

    R5_KERNEL_DIMENSION {
        INPUT = matrix A
        SOLVE = "Ax=0"
        COUNT = number_of_free_variables
        RETURN = nullity(A)
    }

    R6_IMAGE_DIMENSION {
        INPUT = linear transformation T
        COMPUTE = basis(im(T))
        RETURN = cardinality(basis(im(T)))
    }

    R7_RANK_NULLITY {
        REQUIRE = finite_dimensional_domain
        GIVEN = dim(V)
        GIVEN_OR_COMPUTE = rank(T)
        GIVEN_OR_COMPUTE = nullity(T)
        VERIFY = "dim(V)=rank(T)+nullity(T)"
    }

    R8_SUBSPACE_EQUALITY {
        INPUT = [W,V]
        REQUIRE = W subspace_of V
        IF dim(W)==dim(V)
            RETURN "W=V"
        ELSE
            RETURN "dimension alone does not establish equality"
    }

    R9_LINEAR_INDEPENDENCE_BOUND {
        INPUT = finite-dimensional V with dim(V)=n
        RULE = "Any set containing more than n vectors is linearly dependent."
    }

    R10_SPANNING_BOUND {
        INPUT = finite-dimensional V with dim(V)=n
        RULE = "Any spanning set contains at least n vectors."
    }
}

CONSTRAINTS {
    C1: "Dimension is well-defined because all bases of a vector space have the same cardinality."
    C2: "For finite-dimensional vector spaces, dimension is a non-negative integer."
    C3: "The dimension of a subspace cannot exceed that of its finite-dimensional ambient space."
    C4: "Rank is the dimension of the image."
    C5: "Nullity is the dimension of the kernel."
    C6: "Do not infer vector-space equality from equal dimensions unless an inclusion or isomorphism relationship is also established."
    C7: "A set of more than dim(V) vectors in finite-dimensional V must be linearly dependent."
    C8: "A spanning set of finite-dimensional V must contain at least dim(V) vectors."
    C9: "Do not confuse linear-algebraic dimension with physical, spatial or dimensional-analysis meanings."
}

SEMANTIC_EQUIVALENCE {
    "dim(V)" <=> "cardinality of any basis of V"
    "rank(T)" <=> "dim(im(T))"
    "nullity(T)" <=> "dim(ker(T))"
    "column rank(A)" <=> "dim(Col(A))"
    "row rank(A)" <=> "dim(Row(A))"
}

VALIDATION {
    V1 = "All finite bases of the same vector space have equal cardinality."
    V2 = "dim({0})=0."
    V3 = "dim(F^n)=n."
    V4 = "If W is a subspace of V, then dim(W)<=dim(V) for finite-dimensional V."
    V5 = "If W is a subspace of V and dimensions are equal, then W=V."
    V6 = "rank(T)=dim(im(T))."
    V7 = "nullity(T)=dim(ker(T))."
    V8 = "dim(V)=rank(T)+nullity(T) for finite-dimensional domain V."
    V9 = "Isomorphic vector spaces have equal dimension."
}

CONFIDENCE {
    BASE_CONFIDENCE = 0.99
    DIMENSION_DEFINITION = 0.99
    BASIS_CARDINALITY = 0.99
    SUBSPACE_BOUND = 0.99
    RANK_RELATION = 0.99
    NULLITY_RELATION = 0.99
    RANK_NULLITY_RELATION = 0.99
    ISOMORPHISM_INVARIANCE = 0.99

    POLICY =
        "Reduce confidence when dimension appears without sufficient mathematical context to distinguish vector-space dimension from other meanings."
}

AI_CONSUMPTION {
    PURPOSES = [
        SemanticSearch,
        OntologyGrounding,
        MathematicalReasoning,
        EntityResolution,
        DimensionReasoning,
        MatrixReasoning,
        KnowledgeGraphConstruction,
        RetrievalAugmentation,
        RecursiveInference
    ]

    NORMALISE {
        "dimension of V" -> Dimension
        "dim V" -> Dimension
        "dim(V)" -> Dimension
        "dimension of kernel" -> Nullity
        "dimension of image" -> Rank
        "dimension of column space" -> Rank
        "dimension of row space" -> Rank
        "number of basis vectors" -> Dimension
    }

    INTERPRETATION_ORDER = [
        identify_vector_space_or_subspace,
        identify_basis_if_available,
        establish_finite_or_infinite_dimension,
        compute_basis_cardinality,
        resolve_rank_or_nullity_context,
        apply_subspace_dimension_rules,
        apply_rank_nullity_when_applicable,
        evaluate_isomorphism_invariance,
        return_semantic_result
    ]
}

EXECUTION_RULES {
    E1:
        IF query_mentions("dimension")
        AND linear_algebra_context
        THEN activate(DimensionOntology)

    E2:
        IF basis_known
        THEN infer(dim(V)=cardinality(basis))

    E3:
        IF query_asks("dimension of column space")
        THEN compute(number_of_pivots)

    E4:
        IF query_asks("dimension of row space")
        THEN compute(number_of_nonzero_rows_in_RREF)

    E5:
        IF query_asks("dimension of kernel")
        THEN solve_homogeneous_system
        THEN count_free_variables

    E6:
        IF query_asks("dimension of image")
        THEN compute(rank)

    E7:
        IF rank_and_nullity_known
        THEN infer(domain_dimension=rank+nullity)

    E8:
        IF domain_dimension_and_rank_known
        THEN infer(nullity=domain_dimension-rank)

    E9:
        IF domain_dimension_and_nullity_known
        THEN infer(rank=domain_dimension-nullity)

    E10:
        IF W subspace_of V
        AND dim(W)==dim(V)
        THEN infer(W=V)
}

RECURSION {
    ENTRY = Dimension

    EXPAND(Dimension) -> [
        Basis,
        VectorSpace,
        Subspace,
        Rank,
        Nullity,
        Kernel,
        Image,
        ColumnSpace,
        RowSpace,
        RankNullityTheorem,
        Isomorphism
    ]

    EXPAND(Basis) -> [
        LinearIndependence,
        Span,
        BasisCardinality
    ]

    EXPAND(Rank) -> [
        Image,
        ColumnSpace,
        RowSpace,
        Pivot,
        Dimension
    ]

    EXPAND(Nullity) -> [
        Kernel,
        NullSpace,
        FreeVariable,
        Dimension
    ]

    EXPAND(RankNullityTheorem) -> [
        DomainDimension,
        Rank,
        Nullity,
        Kernel,
        Image
    ]

    STOP_CONDITION =
        "Stop when the requested dimension or dimension-derived relation is established with sufficient confidence and no required dependencies remain unresolved."
}

ROUTING {
    PARENT_ONTOLOGY = "Algebra Ontology"

    IMMEDIATE_CONTEXTS = [
        "Linear Algebra Ontology",
        "Vector Space Ontology"
    ]

    RELATED_ONTOLOGIES = [
        "Basis Ontology",
        "Vector Space Ontology",
        "Subspace Ontology",
        "Span Ontology",
        "Linear Independence Ontology",
        "Matrix Ontology",
        "Rank Ontology",
        "Kernel Ontology",
        "Null Space Ontology",
        "Image Ontology",
        "Column Space Ontology",
        "Row Space Ontology",
        "Linear Transformation Ontology"
    ]

    ROUTE_RULES = [
        "Dimension -> Basis when basis cardinality is required",
        "Dimension -> Rank when image dimension is involved",
        "Dimension -> Nullity when kernel dimension is involved",
        "Dimension -> Column Space for matrix column-rank reasoning",
        "Dimension -> Row Space for row-rank reasoning",
        "Dimension -> Rank Nullity for domain decomposition reasoning",
        "Dimension -> Subspace for dimension-bound 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 dimension as basis cardinality.",
        "Preserve equal-cardinality theorem for bases.",
        "Preserve rank as image dimension.",
        "Preserve nullity as kernel dimension.",
        "Preserve rank-nullity for finite-dimensional domains.",
        "Preserve subspace dimension inequalities.",
        "Explicitly disambiguate unrelated meanings of dimension."
    ]
}

OUTPUT {
    CANONICAL_DEFINITION =
        "The dimension of a vector space is the cardinality of any basis of that space."

    BASIS_SUMMARY =
        "If B is a finite basis of V, then dim(V)=|B|."

    SUBSPACE_SUMMARY =
        "W subspace of V => dim(W)<=dim(V) for finite-dimensional V."

    RANK_SUMMARY =
        "rank(T)=dim(im(T))"

    NULLITY_SUMMARY =
        "nullity(T)=dim(ker(T))"

    RANK_NULLITY_SUMMARY =
        "dim(V)=rank(T)+nullity(T)"
}

FLOW {
    INPUT
    -> detect_term("dimension")
    -> disambiguate_linear_algebra_context
    -> identify_vector_space_or_subspace
    -> identify_basis_or_construct_basis
    -> determine_finite_or_infinite_dimension
    -> derive_basis_cardinality
    -> derive_dimension
    -> apply_subspace_constraints
    -> resolve_rank_context
    -> resolve_nullity_context
    -> apply_rank_nullity_when_applicable
    -> evaluate_isomorphism_invariance
    -> link_related_ontologies
    -> assign_confidence
    -> produce_semantic_output
    -> END
}

SEMANTIC_SUMMARY {
    "Dimension is the cardinality of a basis and measures the number of independent directions in a vector space. In finite-dimensional linear algebra it governs limits on independent and spanning sets, determines the size of bases, and connects directly to rank and nullity through the rank-nullity theorem. It is invariant under vector-space isomorphism."
}

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

}

END_FLOW