Field Ontology

FLOW ALGEBRA
TITLE: Field Ontology
DOMAIN: Discrete Mathematics / Algebraic Structures
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/algebraic-structures/field-ontology-2.html
VERSION: 1.0
DATE: 2026-10-11
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
SOURCE_STATUS: Mathematical synthesis; live source page not independently verified.

1. PURPOSE
Represent fields as typed algebraic structures with machine-checkable axioms, operations, relationships, inference rules and validation conditions. A field is a commutative ring with identity 1 != 0 in which every nonzero element has a multiplicative inverse.

2. TYPED ENTITIES
entity Field { carrier: Set; add: BinaryOperation; multiply: BinaryOperation; zero: Element; one: Element; additive_inverse: UnaryOperation; multiplicative_inverse: PartialUnaryOperation }
entity Element { value: Any; belongs_to: Field }
entity Subfield { parent: Field; carrier: Subset }
entity FieldHomomorphism { source: Field; target: Field; map: Function }
entity FieldExtension { base: Field; extension: Field; inclusion: InjectiveHomomorphism }
entity Characteristic { field: Field; value: 0 | PrimeNumber }
entity PolynomialRing { coefficients: Field; variable: Symbol }
entity FiniteField { field: Field; order: PrimePower }

3. CORE AXIOMS
forall a,b,c in F:
  closure_add: a+b in F
  closure_multiply: a*b in F
  add_associativity: (a+b)+c = a+(b+c)
  add_commutativity: a+b = b+a
  add_identity: a+0 = a
  add_inverse: a+(-a) = 0
  multiply_associativity: (a*b)*c = a*(b*c)
  multiply_commutativity: a*b = b*a
  multiply_identity: a*1 = a
  distributivity: a*(b+c) = a*b+a*c
  nontriviality: 0 != 1
forall a in F where a != 0:
  inverse_exists: exists b in F such that a*b = 1
  inverse_unique: b = a^(-1)

4. DERIVED OPERATIONS
subtract(a,b) := a+(-b)
divide(a,b) := a*b^(-1) provided b != 0
power(a,n) := repeated multiplication for n >= 0; negative exponents require a != 0
solve_linear(a,b) := a^(-1)*b provided a != 0

5. ALGEBRAIC LAWS
zero_absorption: a*0 = 0
cancellation_add: a+c=b+c => a=b
cancellation_multiply: a*c=b*c and c!=0 => a=b
no_zero_divisors: a*b=0 => a=0 or b=0
inverse_product: (a*b)^(-1)=a^(-1)*b^(-1) for a,b!=0
inverse_inverse: (a^(-1))^(-1)=a for a!=0
negative_product: (-a)*b=-(a*b)

6. SEMANTIC RELATIONSHIPS
Field IS_A CommutativeRingWithIdentity
Field HAS_STRUCTURE AdditiveAbelianGroup
Field HAS_STRUCTURE MultiplicativeAbelianGroupOnNonzeroElements
Field SATISFIES DistributiveLaw
Field HAS_PROPERTY NoZeroDivisors
Field MAY_HAVE Subfield
Field MAY_EXTEND Field
FiniteField HAS_CARDINALITY PrimePower
FieldHomomorphism PRESERVES Addition
FieldHomomorphism PRESERVES Multiplication
FieldHomomorphism PRESERVES ZeroAndOne

7. EXAMPLES AND NONEXAMPLES
EXAMPLE: RationalNumbers Q with ordinary addition and multiplication
EXAMPLE: RealNumbers R with ordinary addition and multiplication
EXAMPLE: ComplexNumbers C with ordinary addition and multiplication
EXAMPLE: IntegersModuloPrime Z/pZ where p is prime
EXAMPLE: Finite field GF(p^n), p prime and n>=1
NONEXAMPLE: Integers Z (2 has no multiplicative inverse in Z)
NONEXAMPLE: Z/6Z (nonzero zero divisors)
NOTE: A field's characteristic is 0 or prime; finite fields have positive prime characteristic.

8. INFERENCE RULES
RULE F01: Field(F) => CommutativeRing(F) and HasIdentity(F)
RULE F02: Field(F) and a in F and a!=0 => ExistsUniqueMultiplicativeInverse(a)
RULE F03: Field(F) and a*b=0 => a=0 or b=0
RULE F04: Field(F) and a!=0 and a*x=b => x=a^(-1)*b
RULE F05: Prime(p) => Field(Z/pZ)
RULE F06: FiniteField(F) => exists prime p and integer n>=1 such that |F|=p^n
RULE F07: FieldHomomorphism(h:F->K) => Injective(h), assuming unital field homomorphism and nontrivial fields
RULE F08: Field(F) => Characteristic(F)=0 or Prime(Characteristic(F))

9. VALIDATION
validate carrier_nonempty and zero_in_carrier and one_in_carrier
validate 0 != 1
validate binary_operations_total_on_carrier
validate all_field_axioms
validate multiplicative_inverse_defined_exactly_for_nonzero_elements
validate finite_field_order_is_prime_power_if_finite
validate homomorphism_preserves_0_and_1_and_both_operations
counterexample Z/6Z: 2*3=0 mod 6 although 2!=0 and 3!=0; therefore not a field

10. SEMANTIC_EQUIVALENCE
"field" == "commutative division ring" (under standard unital conventions)
"multiplicative unit of F" == "nonzero element of F" when F is a field
"GF(p)" == "Z/pZ" up to field isomorphism for prime p
CAUTION: A division ring need not be commutative; it is not necessarily a field.

11. CONFIDENCE
formal_axioms: 1.00
standard_theorems: 0.99
source_page_alignment: unverified
confidence_policy: Confidence labels describe verification status, not probabilistic proofs.

12. FLOW_HASH
algorithm: SHA-256
input: canonical UTF-8 representation of the flow after excluding FLOW_HASH output
value: compute at publication time; not precomputed

13. UPDATE_MODEL
version: 1.0
update_triggers: mathematical correction | terminology refinement | ontology extension | source-page revision
procedure: validate axioms -> check examples -> revise version -> recompute hash -> publish

14. AI_CONSUMPTION
parse_mode: typed mathematical knowledge graph
reasoning_mode: deductive with explicit preconditions
truth_policy: distinguish theorem, definition, example and unverified page alignment
constraints: prohibit division by zero; require prime modulus for Z/pZ field assertions
output: entity links | validated algebraic deductions | provenance | confidence metadata

END FLOW ALGEBRA