Algebraic Structure Ontology

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

1. ONTOLOGY
ENTITY AlgebraicStructure := (CarrierSet, Operations, Constants, Axioms)
ENTITY CarrierSet := nonempty set (unless a signature explicitly permits empty carriers)
ENTITY Operation := total function CarrierSet^n -> CarrierSet, n >= 0
ENTITY Signature := operation symbols with arities and optional relation symbols
ENTITY IdentityElement := e in CarrierSet satisfying a specified neutral-element law
ENTITY InverseElement := inverse relative to a specified operation and identity
ENTITY Substructure := subset closed under operations and containing named constants
ENTITY Homomorphism := mapping preserving all operations and constants in the signature
ENTITY Isomorphism := bijective homomorphism whose inverse preserves the structure
ENTITY Congruence := equivalence relation compatible with all operations
ENTITY QuotientStructure := carrier of congruence classes with induced operations

2. TYPED RELATIONSHIPS
HAS_CARRIER(AlgebraicStructure, CarrierSet)
HAS_SIGNATURE(AlgebraicStructure, Signature)
HAS_OPERATION(AlgebraicStructure, Operation)
SATISFIES(AlgebraicStructure, Axiom)
HAS_SUBSTRUCTURE(AlgebraicStructure, Substructure)
HOMOMORPHIC_TO(AlgebraicStructure, AlgebraicStructure)
ISOMORPHIC_TO(AlgebraicStructure, AlgebraicStructure)
HAS_CONGRUENCE(AlgebraicStructure, Congruence)
HAS_QUOTIENT(AlgebraicStructure, QuotientStructure)

3. STRUCTURE HIERARCHY
Magma: one closed binary operation.
Semigroup: magma with associative operation.
Monoid: semigroup with two-sided identity.
Group: monoid where every element has a two-sided inverse.
AbelianGroup: group with commutative operation.
Ring: additive abelian group with associative multiplication and distributivity; unity optional by convention.
Field: commutative ring with 1 != 0 in which each nonzero element has a multiplicative inverse.
Lattice: partially ordered set with binary meets and joins, equivalently a suitable equational algebra.
BooleanAlgebra: bounded distributive lattice with complements.

4. FORMAL LAWS
CLOSURE: for f of arity n and a_1,...,a_n in A, f(a_1,...,a_n) in A.
ASSOCIATIVITY: (a*b)*c = a*(b*c).
COMMUTATIVITY: a*b = b*a.
IDENTITY: e*a = a = a*e.
INVERSE: a*a^-1 = e = a^-1*a.
DISTRIBUTIVITY: a*(b+c) = a*b+a*c and (a+b)*c = a*c+b*c.
IDEMPOTENCE: a*a = a (only for operations explicitly satisfying this axiom).

5. OPERATIONS
BUILD_STRUCTURE(carrier, signature, interpretation, axioms) -> candidate structure
CHECK_CLOSURE(structure) -> Boolean / unknown
CHECK_AXIOM(structure, axiom) -> Boolean / unknown
FIND_SUBSTRUCTURE(structure, generators) -> generated substructure when defined
CHECK_HOMOMORPHISM(f, source, target) -> Boolean / unknown
CHECK_ISOMORPHISM(f, source, target) -> Boolean / unknown
BUILD_QUOTIENT(structure, congruence) -> quotient structure

6. INFERENCE RULES
IF associative(A,*) AND identity(A,*,e) THEN monoid(A,*) provided closure holds.
IF monoid(A,*) AND every_element_invertible(A,*) THEN group(A,*).
IF group(A,*) AND commutative(A,*) THEN abelian_group(A,*).
IF isomorphism(f,A,B) THEN all signature-expressible equations true in A are true in B.
IF congruence(theta,A) THEN operations on A/theta are well-defined.
IF homomorphism(f,A,B) THEN kernel_equivalence(f) is a congruence on A.

7. VALIDATION
REQUIRE every operation has declared arity and carrier-compatible inputs and outputs.
REQUIRE all named constants belong to the carrier.
REQUIRE axiom checks match the declared signature.
REQUIRE inverse claims specify an operation and identity.
REQUIRE homomorphisms preserve each named operation and constant.
REQUIRE quotient construction uses a compatible congruence, not an arbitrary equivalence.
REQUIRE finite exhaustive checks are distinguished from proofs for infinite carriers.

8. SEMANTIC_EQUIVALENCE
"algebraic structure" ~= "set equipped with specified operations and axioms" [context: universal algebra]
"structure-preserving map" ~= "homomorphism" [context: fixed algebraic signature]
"bijective structure-preserving equivalence" ~= "isomorphism" [context: same signature]
NOT_EQUIVALENT: group != monoid; ring != field; homomorphism != isomorphism.

9. CONFIDENCE
CONFIDENCE(formal_definitions) = 0.99 [established mathematical conventions]
CONFIDENCE(structure_classification) = 0.98 [subject to explicit ring and empty-carrier conventions]
CONFIDENCE(live_page_alignment) = unverified
Confidence values describe editorial assessment, not measured probabilities.

10. FLOW_HASH
FLOW_HASH_METHOD: SHA-256 over canonical UTF-8 content, excluding any stored hash field.
FLOW_HASH_VALUE: compute upon ingestion; no uncomputed digest asserted.

11. UPDATE_MODEL
UPDATE_TRIGGER: changes in source content, signature, axiom conventions, or mathematical corrections.
UPDATE_ACTION: revise affected definitions, revalidate dependent laws, increment version, recompute FLOW_HASH.
PROVENANCE: source URL, version, update date, authors, validation results.

12. AI_CONSUMPTION
INPUT: mathematical concept, carrier, operations, axioms, optional example.
PROCESS: normalise signature -> type-check -> validate axioms -> classify structure -> derive justified consequences -> record provenance.
OUTPUT: typed entities, verified relationships, proofs or counterexamples, uncertainty markers, canonical metadata.
GUARDRAIL: never infer an algebraic law solely from the structure's informal name without checking the adopted definition.

END FLOW ALGEBRA