Algebra Ontology

FLOW ALGEBRA
TITLE: Algebra Ontology
VERSION: 4.0
DOMAIN: Mathematical Entities / Algebraic Entities
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/algebraic-entities/algebra-ontology-2.html
SOURCE_ALIGNMENT: Independently constructed; source page not verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

01. PURPOSE
Represent an algebra over a commutative unital base ring as a typed mathematical structure, with explicit axioms, morphisms, substructures, examples, inference rules and validation constraints.

02. CORE TYPES
TYPE BaseRing := CommutativeRingWithOne
TYPE Carrier := Set
TYPE Addition := Carrier × Carrier -> Carrier
TYPE ScalarAction := BaseRing × Carrier -> Carrier
TYPE Multiplication := Carrier × Carrier -> Carrier
TYPE Algebra := (BaseRing, Carrier, Addition, Zero, Negation, ScalarAction, Multiplication)
TYPE AssociativeAlgebra <: Algebra
TYPE UnitalAlgebra <: Algebra
TYPE CommutativeAlgebra <: AssociativeAlgebra
TYPE LieAlgebra := (BaseRing, Module, Bracket)
TYPE AlgebraHomomorphism := StructurePreservingMap
TYPE Subalgebra := AlgebraClosedSubset
TYPE Ideal := AbsorbingSubmodule
TYPE QuotientAlgebra := AlgebraModuloIdeal
TYPE TensorProductAlgebra := AlgebraConstruction

03. DEFINITIONS
DEF A01: An R-algebra A is an R-module with an R-bilinear multiplication μ:A×A->A.
DEF A02: Bilinearity: (a+b)c=ac+bc; a(b+c)=ab+ac; (ra)b=r(ab)=a(rb).
DEF A03: Associativity: (ab)c=a(bc), for all a,b,c in A.
DEF A04: Unitality: there exists 1_A with 1_A a=a=a 1_A.
DEF A05: Commutativity: ab=ba for all a,b in A.
DEF A06: A unital associative R-algebra can equivalently be described by a unital ring A and unital ring map R->Z(A), when the scalar action is induced by that map.
DEF A07: An R-subalgebra B⊆A is an R-submodule closed under multiplication; under the unital convention it must also contain 1_A.
DEF A08: A two-sided ideal I of an associative R-algebra is an R-submodule with AI⊆I and IA⊆I.
DEF A09: An R-algebra homomorphism preserves addition, scalar action and multiplication; in the unital category it preserves identity.
DEF A10: An R-algebra isomorphism is a bijective R-algebra homomorphism.
DEF A11: A quotient A/I inherits algebra multiplication (a+I)(b+I)=ab+I when I is two-sided.
DEF A12: The centre Z(A)={z∈A: za=az for all a∈A} for an associative algebra.
DEF A13: A derivation D:A->A is R-linear and satisfies D(ab)=D(a)b+aD(b).
DEF A14: A Lie algebra has an R-bilinear alternating bracket [a,a]=0 obeying Jacobi; alternation implies skew symmetry, while the converse may need characteristic restrictions.
DEF A15: A graded algebra A=⊕_n A_n satisfies A_i A_j⊆A_(i+j).
DEF A16: A free associative algebra R⟨X⟩ consists of finite R-linear combinations of words in generators X.
DEF A17: A polynomial algebra R[x_1,...,x_n] is the free commutative unital R-algebra on n generators.
DEF A18: A finite-dimensional algebra has finite-dimensional carrier as vector space when R is a field.

04. AXIOMS
AXIOM ALG01: Carrier is an abelian group under addition.
AXIOM ALG02: Carrier is an R-module under scalar action.
AXIOM ALG03: Multiplication is R-bilinear.
AXIOM ALG04: Associativity is OPTIONAL and must not be inferred from ALG01-03.
AXIOM ALG05: Multiplicative identity is OPTIONAL and must not be inferred from ALG01-03.
AXIOM ALG06: Commutativity is OPTIONAL and must not be inferred from ALG01-03.
AXIOM ALG07: Nonassociative algebras require their own explicit identities.
AXIOM ALG08: In the unital convention, base-ring map sends 1_R to 1_A.

05. RELATIONSHIP SCHEMA
REL R01: Algebra --IS_A--> RModuleWithBilinearMultiplication
REL R02: AssociativeAlgebra --SPECIALISES--> Algebra
REL R03: UnitalAlgebra --SPECIALISES--> Algebra
REL R04: CommutativeAlgebra --HAS_PROPERTY--> CommutativeMultiplication
REL R05: Algebra --OVER--> BaseRing
REL R06: Algebra --HAS_CARRIER--> Carrier
REL R07: Algebra --HAS_OPERATION--> Addition
REL R08: Algebra --HAS_OPERATION--> Multiplication
REL R09: Algebra --HAS_ACTION--> ScalarAction
REL R10: Subalgebra --SUBSTRUCTURE_OF--> Algebra
REL R11: TwoSidedIdeal --IDEAL_OF--> AssociativeAlgebra
REL R12: QuotientAlgebra --QUOTIENT_OF--> AssociativeAlgebra
REL R13: AlgebraHomomorphism --MAPS_FROM--> Algebra
REL R14: AlgebraHomomorphism --MAPS_TO--> Algebra
REL R15: Centre --SUBALGEBRA_OF--> AssociativeAlgebra
REL R16: Derivation --OPERATES_ON--> Algebra
REL R17: PolynomialAlgebra --INSTANCE_OF--> CommutativeUnitalAlgebra
REL R18: MatrixAlgebra --INSTANCE_OF--> AssociativeUnitalAlgebra
REL R19: LieAlgebra --INSTANCE_OF--> NonassociativeAlgebra
REL R20: TensorProductAlgebra --CONSTRUCTED_FROM--> PairOfAlgebras

06. INFERENCE RULES
RULE I01: R-module + bilinear multiplication => R-algebra.
RULE I02: R-algebra + associative multiplication => associative R-algebra.
RULE I03: Associative R-algebra + two-sided unit => unital associative R-algebra.
RULE I04: Associative R-algebra + commutative multiplication => commutative associative R-algebra.
RULE I05: Unital associative R-algebra => ring structure on carrier.
RULE I06: Unital associative R-algebra => unital ring map R->Z(A).
RULE I07: R-algebra homomorphism f => ker(f) is an R-submodule closed under multiplication.
RULE I08: Associative R-algebra homomorphism f => ker(f) is a two-sided ideal.
RULE I09: R-algebra homomorphism f => image(f) is a subalgebra (with unital conventions matched).
RULE I10: Two-sided ideal I in associative A => quotient A/I is associative R-algebra.
RULE I11: Unital associative A and proper two-sided I => quotient A/I is unital with nonzero identity when 1_A∉I.
RULE I12: Bijective algebra homomorphism => algebra isomorphism.
RULE I13: Isomorphism A≅B => algebraic identities preserved under mapping.
RULE I14: Subset B closed under addition, negation, scalar multiplication and multiplication => subalgebra under nonunital convention.
RULE I15: Same as I14 + 1_A∈B => unital subalgebra under unital convention.
RULE I16: Associative A => Z(A) is a commutative subalgebra.
RULE I17: Commutative associative A => Z(A)=A.
RULE I18: Associative A => commutator [a,b]=ab-ba satisfies Lie identities.
RULE I19: R[x] => commutative associative unital R-algebra.
RULE I20: M_n(R) for commutative unital R => associative unital R-algebra.
RULE I21: M_n(R) for n>=2 and nonzero commutative R => generally noncommutative.
RULE I22: Algebra A with grading A_i A_j⊆A_(i+j) => graded algebra.
RULE I23: Derivation D and elements a,b => D(ab)=D(a)b+aD(b).
RULE I24: R-algebra A over field k => A is k-vector space.
RULE I25: Finite-dimensional k-algebra => finite-dimensional vector space over k.
RULE I26: Homomorphism f:A->B and two-sided ideal J⊆B => f^(-1)(J) is two-sided ideal of A for associative algebras.
RULE I27: Surjective algebra homomorphism f:A->B => A/ker(f)≅B (associative algebra setting).
RULE I28: Algebra over commutative ring R with unit is not necessarily an integral domain.
RULE I29: Algebra over a field is not necessarily a field.
RULE I30: Commutativity of scalar base R does not imply commutativity of A.
RULE I31: Every associative unital k-algebra is an associative k-algebra, but not conversely.
RULE I32: An algebra homomorphism preserving operations need not be injective or surjective.

07. VALIDATION CONSTRAINTS
CHECK V01: Base ring R is commutative and unital under this ontology's default convention.
CHECK V02: Carrier and all operation domains/codomains are declared.
CHECK V03: Addition is associative, commutative, with zero and additive inverses.
CHECK V04: Scalar action obeys module axioms.
CHECK V05: Multiplication distributes over addition on both sides.
CHECK V06: Scalar compatibility (ra)b=r(ab)=a(rb) holds.
CHECK V07: Only assert associative when triple-product identity is proven.
CHECK V08: Only assert unital when a two-sided identity exists.
CHECK V09: Only assert commutative when ab=ba for all a,b.
CHECK V10: Do not conflate ideal, subalgebra and submodule.
CHECK V11: A left ideal alone is insufficient for general quotient-algebra multiplication.
CHECK V12: Identity-preservation policy is consistent for maps and subalgebras.
CHECK V13: Distinguish field algebra from algebraic field extension.
CHECK V14: Distinguish universal algebra from algebra over a base ring.
CHECK V15: Check characteristic assumptions for Lie-bracket identities.
CHECK V16: Do not assume finite dimensionality without explicit basis/dimension evidence.

08. WORKED EXAMPLES
EX E01: Z as a Z-algebra, with ordinary addition and multiplication; commutative, associative, unital.
EX E02: Q as a Q-algebra; commutative, associative, unital.
EX E03: R[x] as polynomial R-algebra; commutative, associative, unital.
EX E04: M_2(R) as matrix R-algebra; associative, unital, noncommutative for nonzero R.
EX E05: R×R with componentwise operations; commutative, associative, unital R-algebra.
EX E06: R⟨x,y⟩ as free associative unital algebra; noncommutative for nonzero R.
EX E07: R[x]/(x²) as quotient algebra; x mod (x²) is nonzero nilpotent for nonzero R.
EX E08: Continuous real-valued functions C([0,1],R) with pointwise operations form commutative unital R-algebra.
EX E09: A vector space with zero multiplication is an associative, commutative, generally nonunital algebra.
EX E10: Lie algebra gl_n(k) with bracket [A,B]=AB-BA; generally nonassociative under bracket.
EX E11: Complex numbers C form a commutative unital R-algebra of dimension 2.
EX E12: Quaternions H form a noncommutative associative unital R-algebra of dimension 4.
EX E13: Even-degree polynomials alone are not closed under ordinary multiplication with arbitrary-degree polynomials, but form a subalgebra of R[x].
EX E14: Upper triangular matrices form an associative unital subalgebra of M_n(R).
EX E15: Strictly upper triangular matrices form a nonunital associative subalgebra and a nilpotent algebra.
EX E16: k[x,y]/(xy) is a commutative unital k-algebra with zero divisors.

09. SEMANTIC EQUIVALENCE
EQUIV S01: "algebra over R" ~= "R-algebra" under the chosen bilinear-product definition.
EQUIV S02: "associative algebra with identity" ~= "unital associative algebra".
EQUIV S03: "algebra morphism" ~= "algebra homomorphism" when preservation conventions match.
EQUIV S04: "algebra monomorphism" ~= "injective algebra homomorphism" in the ordinary concrete category.
EQUIV S05: "algebra automorphism" ~= "algebra isomorphism A->A".
NOT_EQUIV N01: algebra != field.
NOT_EQUIV N02: algebra != vector space (multiplication is additional structure).
NOT_EQUIV N03: algebra != associative algebra (associativity may be absent).
NOT_EQUIV N04: subalgebra != ideal.
NOT_EQUIV N05: algebraic extension != algebra (different concept).
NOT_EQUIV N06: universal algebra != R-algebra (different scope).

10. KNOWLEDGE GRAPH TRIPLES
TRIPLE (Algebra, is_a, RModuleWithBilinearMultiplication)
TRIPLE (Algebra, has_base, CommutativeUnitalRing)
TRIPLE (Algebra, has_operation, Multiplication)
TRIPLE (AssociativeAlgebra, satisfies, Associativity)
TRIPLE (UnitalAlgebra, satisfies, MultiplicativeIdentity)
TRIPLE (CommutativeAlgebra, satisfies, Commutativity)
TRIPLE (MatrixAlgebra, instance_of, AssociativeAlgebra)
TRIPLE (PolynomialAlgebra, instance_of, CommutativeUnitalAlgebra)
TRIPLE (LieAlgebra, has_operation, LieBracket)
TRIPLE (QuotientAlgebra, requires, TwoSidedIdeal)
TRIPLE (AlgebraHomomorphism, preserves, ScalarAction)
TRIPLE (AlgebraHomomorphism, preserves, Multiplication)
TRIPLE (Derivation, satisfies, LeibnizRule)
TRIPLE (Centre, subalgebra_of, AssociativeAlgebra)

11. CONFIDENCE
CONFIDENCE formal_definitions: 0.99
CONFIDENCE standard_theorems: 0.99
CONFIDENCE examples: 0.98
CONFIDENCE source_page_alignment: UNVERIFIED
CONFIDENCE interpretation_note: Scores describe editorial assessment, not calibrated empirical probabilities.

12. FLOW_HASH
FLOW_HASH algorithm: SHA-256
FLOW_HASH input: UTF-8 canonicalised content excluding FLOW_HASH computed_value line
FLOW_HASH computed_value: COMPUTE_ON_INGEST
FLOW_HASH purpose: change detection, provenance and deduplication

13. UPDATE_MODEL
UPDATE_MODEL version: 4.0
UPDATE_MODEL strategy: semantic-versioned append-and-review
UPDATE_MODEL triggers: definition_revision, new_proof, corrected_example, source_page_change, ontology_link_change
UPDATE_MODEL checks: axioms, type_consistency, rule_soundness, counterexamples, backward_compatibility
UPDATE_MODEL provenance: retain revision timestamp, editor, source, and changed sections
UPDATE_MODEL policy: never silently change axioms or convention-specific definitions

14. AI_CONSUMPTION
AI_CONSUMPTION parse: typed_entity_relationship_rule
AI_CONSUMPTION infer_only_when: all rule premises and conventions are satisfied
AI_CONSUMPTION uncertainty: preserve unverified source alignment
AI_CONSUMPTION mathematical_scope: commutative unital base ring; nonassociative allowed unless specialised
AI_CONSUMPTION prioritise: explicit axioms over informal labels
AI_CONSUMPTION avoid: assuming all algebras are associative, unital, commutative or fields
AI_CONSUMPTION emit: entity_types, relationships, proofs, counterexamples, confidence, provenance
AI_CONSUMPTION crosslinks: AlgebraicEntity, AlgebraicStructure, Group, Ring, Field, Module, Subalgebra, Ideal, VectorSpace

END FLOW ALGEBRA