Universal Algebra Ontology
Algebra Ontologies
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 13 September 2026
Audio Overview
======================================================================
KIRETH SEMANTIC RESEARCH ARCHIVE
FLOW ALGEBRA — UNIVERSAL ALGEBRA ONTOLOGY
======================================================================
PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/universal-algebra-ontology.html
FLOW:
UniversalAlgebraOntology
VERSION:
1.0
LANGUAGE:
en-GB
AUTHORS:
Mark Anthony Higgins — Sky Recursive AI
PURPOSE:
To provide a machine-readable, recursively executable semantic and
reasoning specification for universal algebra, including signatures,
algebras, operations, terms, identities, subalgebras, homomorphisms,
congruences, quotient algebras, direct products, varieties,
equational theories and structural preservation.
AI INTERPRETATION NOTE:
Execute the following specification as formal semantic and reasoning
rules. Preserve operation arities, signatures, carrier sets,
homomorphic structure, congruence compatibility, equational validity
and quotient well-definedness.
======================================================================
START FLOW
======================================================================
FLOW UniversalAlgebraOntology {
----------------------------------------------------------------------
1. DOMAIN DECLARATION
----------------------------------------------------------------------
DOMAIN := Universal_Algebra
PARENT_DOMAIN := Algebra
SUPER_DOMAINS := {
Mathematics,
Mathematical_Ontologies,
Algebra_Ontologies,
Abstract_Algebra
}
DOMAIN_INTENT := {
represent_signatures,
represent_operation_symbols,
represent_arities,
represent_algebras_over_signatures,
represent_terms,
represent_term_operations,
represent_equations,
represent_equational_theories,
represent_subalgebras,
represent_homomorphisms,
represent_isomorphisms,
represent_congruences,
represent_quotient_algebras,
represent_direct_products,
represent_varieties,
represent_free_algebras,
represent_generated_subalgebras,
represent_structural_closure,
support_equational_reasoning,
support_recursive_structural_validation
}
----------------------------------------------------------------------
2. PRIMITIVE TYPES
----------------------------------------------------------------------
TYPE Set
TYPE Element
TYPE NaturalNumber
TYPE Symbol
TYPE Variable <: Symbol
TYPE OperationSymbol <: Symbol
TYPE Signature
TYPE Algebra
TYPE Subalgebra <: Algebra
TYPE ProductAlgebra <: Algebra
TYPE QuotientAlgebra <: Algebra
TYPE FreeAlgebra <: Algebra
TYPE Term
TYPE Equation
TYPE Identity <: Equation
TYPE Assignment
TYPE Operation
TYPE TermOperation <: Operation
TYPE Mapping
TYPE Homomorphism <: Mapping
TYPE Isomorphism <: Homomorphism
TYPE Endomorphism <: Homomorphism
TYPE Automorphism <: Isomorphism
TYPE Relation
TYPE EquivalenceRelation <: Relation
TYPE Congruence <: EquivalenceRelation
TYPE Variety
TYPE EquationalTheory
TYPE Derivation
TYPE ProofStep
TYPE Constraint
TYPE ValidationResult
----------------------------------------------------------------------
3. SIGNATURES
----------------------------------------------------------------------
ENTITY Signature Sigma {
operation_symbols: OperationSymbol*
arity_map: OperationSymbol -> NaturalNumber
}
FUNCTION ARITY(f,Sigma):
return Sigma.arity_map(f)
INVARIANT SIG_01:
every operation symbol has exactly one declared finite arity.
RULE SIG_02:
nullary operation symbol represents a constant symbol.
RULE SIG_03:
unary operation symbol has arity 1.
RULE SIG_04:
binary operation symbol has arity 2.
----------------------------------------------------------------------
4. ALGEBRAS OVER A SIGNATURE
----------------------------------------------------------------------
ENTITY Algebra A {
signature: Sigma
carrier_set: |A|
interpretations:
for each f in Sigma of arity n:
f^A : |A|^n -> |A|
}
IS_SIGMA_ALGEBRA(A,Sigma) iff
A.signature = Sigma
AND carrier_set_defined(A)
AND every operation symbol is interpreted
AND interpretation arity matches declared arity
AND every interpreted operation is closed on |A|
RULE ALG_01:
algebraic structure is determined by carrier plus interpretations
of all operation symbols in its signature.
RULE ALG_02:
two algebras may share a carrier but differ by operation
interpretations.
----------------------------------------------------------------------
5. VARIABLES AND TERMS
----------------------------------------------------------------------
ENTITY VariableSet X {
members: Variable*
}
TERM_FORMATION(Sigma,X):
BASE:
every x in X is a Term
RECURSIVE:
if f in Sigma has arity n
AND t1,...,tn are Terms
then f(t1,...,tn) is a Term
TERM_ALGEBRA T_Sigma(X) :=
set of all Sigma-terms over X
RULE TERM_01:
term syntax is signature-dependent.
RULE TERM_02:
a nullary operation symbol forms a constant term.
----------------------------------------------------------------------
6. TERM EVALUATION
----------------------------------------------------------------------
ENTITY Assignment alpha {
domain: VariableSet X
codomain: carrier(A)
}
EVAL_A(alpha, x):
if x variable:
return alpha(x)
EVAL_A(alpha, f(t1,...,tn)):
return f^A(
EVAL_A(alpha,t1),
...,
EVAL_A(alpha,tn)
)
RULE EVAL_01:
every term induces a term operation on every compatible algebra.
----------------------------------------------------------------------
7. TERM OPERATIONS
----------------------------------------------------------------------
TERM_OPERATION_A(t, variables x1...xn):
function |A|^n -> |A|
defined by evaluation of t under assignments.
RULE TOP_01:
term operations are built compositionally from fundamental
operations and projections.
RULE TOP_02:
every projection operation is a term operation.
----------------------------------------------------------------------
8. EQUATIONS AND IDENTITIES
----------------------------------------------------------------------
ENTITY Equation {
left_term: s
right_term: t
notation: s ≈ t
}
SATISFIES(A, s≈t) iff
forall assignments alpha:
EVAL_A(alpha,s)=EVAL_A(alpha,t)
IDENTITY_OF(A,equation) iff
SATISFIES(A,equation)
RULE EQ_01:
equational validity is universal over all assignments.
RULE EQ_02:
failure under one assignment refutes identity satisfaction.
----------------------------------------------------------------------
9. EQUATIONAL THEORY
----------------------------------------------------------------------
EQUATIONAL_THEORY(A) :=
{ s≈t | A satisfies s≈t }
FOR CLASS K:
EQUATIONAL_THEORY(K) :=
intersection of EQUATIONAL_THEORY(A)
over A in K
MODELS(E) :=
{A | A satisfies every equation in E}
RULE THEORY_01:
larger classes generally have smaller common equational theories.
RULE THEORY_02:
adding equations generally reduces the model class.
----------------------------------------------------------------------
10. SUBALGEBRAS
----------------------------------------------------------------------
ENTITY Subalgebra B {
parent_algebra: A
carrier_set: |B| subset_of |A|
operations: restrictions of A operations
}
IS_SUBALGEBRA(B,A) iff
|B| subset_of |A|
AND |B| nonempty unless signature permits specified empty case
AND for every f of arity n:
b1,...,bn in |B|
-> f^A(b1,...,bn) in |B|
RULE SUBALG_01:
subalgebra operations are inherited by restriction.
RULE SUBALG_02:
intersection of subalgebras is a subalgebra
when nonempty conditions are met.
----------------------------------------------------------------------
11. GENERATED SUBALGEBRAS
----------------------------------------------------------------------
GENERATED_SUBALGEBRA_A(S) :=
intersection of all subalgebras of A containing S
EQUIVALENT:
<S>_A :=
set of all values of terms whose variables are assigned elements
from S.
RULE GEN_01:
S subset_of <S>_A
RULE GEN_02:
<S>_A is smallest subalgebra containing S.
----------------------------------------------------------------------
12. HOMOMORPHISMS
----------------------------------------------------------------------
ENTITY Homomorphism h {
domain: A
codomain: B
}
IS_HOMOMORPHISM(h:A->B) iff
forall f in Sigma with arity n
forall a1,...,an in A:
h(f^A(a1,...,an))
=
f^B(h(a1),...,h(an))
RULE HOM_01:
homomorphisms preserve all fundamental operations.
RULE HOM_02:
homomorphisms preserve all term operations.
THEOREM TERM_PRESERVATION:
h(t^A(a1,...,an))
=
t^B(h(a1),...,h(an))
----------------------------------------------------------------------
13. ISOMORPHISMS
----------------------------------------------------------------------
IS_ISOMORPHISM(h) iff
IS_HOMOMORPHISM(h)
AND bijective(h)
NOTATION:
A ≅ B
RULE ISO_01:
isomorphic algebras satisfy exactly the same equations.
RULE ISO_02:
isomorphism is an equivalence relation on Sigma-algebras.
RULE ISO_03:
inverse of an isomorphism is an isomorphism.
----------------------------------------------------------------------
14. ENDOMORPHISMS AND AUTOMORPHISMS
----------------------------------------------------------------------
ENDOMORPHISM(A) :=
homomorphism A -> A
AUTOMORPHISM(A) :=
isomorphism A -> A
AUT(A) :=
set of automorphisms under composition
RULE AUTO_01:
AUT(A) forms a group under composition.
----------------------------------------------------------------------
15. KERNEL RELATIONS
----------------------------------------------------------------------
KERNEL_RELATION(h:A->B) :=
{(a,b) in A x A | h(a)=h(b)}
RULE KER_01:
kernel relation of a homomorphism is a congruence.
----------------------------------------------------------------------
16. CONGRUENCES
----------------------------------------------------------------------
ENTITY Congruence theta {
carrier: A
relation: theta subset_of A x A
}
IS_CONGRUENCE(theta,A) iff
theta is equivalence_relation
AND for every n-ary f
if ai theta bi for i=1..n
then:
f^A(a1,...,an)
theta
f^A(b1,...,bn)
RULE CONG_01:
congruence is precisely an operation-compatible equivalence relation.
RULE CONG_02:
equality relation is a congruence.
RULE CONG_03:
universal relation is a congruence.
----------------------------------------------------------------------
17. CONGRUENCE LATTICE
----------------------------------------------------------------------
CON(A) :=
set of all congruences on A
ORDER:
theta <= phi iff theta subset_of phi
RULE CLAT_01:
CON(A) forms a complete lattice.
MEET(theta_i):
intersection(theta_i)
JOIN(theta_i):
least congruence containing union(theta_i)
----------------------------------------------------------------------
18. QUOTIENT ALGEBRAS
----------------------------------------------------------------------
ENTITY QuotientAlgebra A/theta {
REQUIRE theta congruence_on A
carrier_set:
equivalence_classes [a]_theta
}
FOR f of arity n:
f^(A/theta)(
[a1],...,[an]
)
:=
[f^A(a1,...,an)]
RULE QA_01:
congruence compatibility guarantees quotient operations are
well-defined.
RULE QA_02:
canonical projection pi:A->A/theta is a surjective homomorphism.
----------------------------------------------------------------------
19. FIRST ISOMORPHISM THEOREM
----------------------------------------------------------------------
THEOREM FIRST_ISOMORPHISM:
if h:A->B is homomorphism
then:
A/kernel(h) ≅ image(h)
where kernel(h) means KERNEL_RELATION(h).
----------------------------------------------------------------------
20. DIRECT PRODUCTS
----------------------------------------------------------------------
PRODUCT(A_i for i in I):
carrier :=
CartesianProduct(|A_i|)
for each n-ary f:
f^Product(
(a_i^1)_i,
...,
(a_i^n)_i
)
:=
(
f^(A_i)(a_i^1,...,a_i^n)
)_i
RULE PROD_01:
projections from product to factors are homomorphisms.
RULE PROD_02:
equations valid in every factor are valid in product.
----------------------------------------------------------------------
21. SUBDIRECT PRODUCTS
----------------------------------------------------------------------
IS_SUBDIRECT_PRODUCT(B, Product(A_i)) iff
B subalgebra_of Product(A_i)
AND every coordinate projection restricted to B is surjective
RULE SDP_01:
subdirect representation preserves access to every factor.
----------------------------------------------------------------------
22. CLASSES OF ALGEBRAS
----------------------------------------------------------------------
ENTITY AlgebraClass K {
members: SigmaAlgebra*
}
CLOSURE_H(K):
closed under homomorphic images
CLOSURE_S(K):
closed under subalgebras
CLOSURE_P(K):
closed under direct products
----------------------------------------------------------------------
23. VARIETIES
----------------------------------------------------------------------
ENTITY Variety V {
signature: Sigma
algebra_class: K
}
IS_VARIETY(K) iff
exists set of equations E:
K=MODELS(E)
THEOREM BIRKHOFF_HSP:
K is a variety
iff
K is closed under:
H = homomorphic_images
S = subalgebras
P = direct_products
RULE VAR_01:
equationally definable classes are exactly HSP-closed classes.
----------------------------------------------------------------------
24. HSP CLOSURE
----------------------------------------------------------------------
HSP(K) :=
smallest class containing K
closed under H, S and P
THEOREM:
HSP(K)=MODELS(EQUATIONAL_THEORY(K))
----------------------------------------------------------------------
25. FREE ALGEBRAS
----------------------------------------------------------------------
ENTITY FreeAlgebra F_V(X) {
variety: V
generators: X
}
UNIVERSAL_PROPERTY_FREE:
for every A in V
and every mapping f:X->|A|
exists unique homomorphism
f_hat:F_V(X)->A
extending f.
RULE FREE_01:
free algebra is unique up to isomorphism.
RULE FREE_02:
elements of free algebra can be represented by terms modulo
equational equivalence of the variety.
----------------------------------------------------------------------
26. TERM ALGEBRA AS FREE ALGEBRA
----------------------------------------------------------------------
THEOREM TERM_FREE:
T_Sigma(X) is the absolutely free Sigma-algebra on X.
RULE TF_01:
any variable assignment X->A extends uniquely to a homomorphism
T_Sigma(X)->A.
----------------------------------------------------------------------
27. EQUATIONAL CONSEQUENCE
----------------------------------------------------------------------
SEMANTIC_CONSEQUENCE(E, s≈t) iff
every algebra satisfying E also satisfies s≈t
NOTATION:
E |= s≈t
SYNTACTIC_DERIVABILITY(E, s≈t):
E |- s≈t
----------------------------------------------------------------------
28. EQUATIONAL LOGIC RULES
----------------------------------------------------------------------
RULE EL_01_REFLEXIVITY:
t≈t
RULE EL_02_SYMMETRY:
s≈t -> t≈s
RULE EL_03_TRANSITIVITY:
s≈t AND t≈u -> s≈u
RULE EL_04_SUBSTITUTION:
s≈t
-> sσ≈tσ
for any substitution σ
RULE EL_05_COMPATIBILITY:
s_i≈t_i for all i
->
f(s1,...,sn)≈f(t1,...,tn)
RULE EL_06_ASSUMPTION:
if e in E then E |- e
THEOREM SOUNDNESS:
E |- e -> E |= e
THEOREM COMPLETENESS:
E |= e -> E |- e
for equational logic.
----------------------------------------------------------------------
29. SUBSTITUTIONS
----------------------------------------------------------------------
ENTITY Substitution sigma {
mapping: Variables -> Terms
}
EXTEND_SUBSTITUTION(sigma):
x sigma := sigma(x)
f(t1,...,tn)sigma
:=
f(t1sigma,...,tnsigma)
RULE SUB_01:
substitution preserves term well-formedness.
----------------------------------------------------------------------
30. IDENTITIES OF FAMILIAR STRUCTURES
----------------------------------------------------------------------
GROUP_SIGNATURE := {
binary_multiplication,
unary_inverse,
nullary_identity
}
GROUP_IDENTITIES include:
(xy)z ≈ x(yz)
ex ≈ x
xe ≈ x
x x^(-1) ≈ e
x^(-1) x ≈ e
SEMIGROUP_SIGNATURE := {binary_operation}
SEMIGROUP_IDENTITY:
(xy)z ≈ x(yz)
LATTICE_SIGNATURE := {meet, join}
LATTICE_IDENTITIES include:
associativity
commutativity
idempotence
absorption
BOOLEAN_ALGEBRA_SIGNATURE may include:
meet, join, complement, 0, 1
RULE FAM_01:
universal algebra abstracts from element meaning to signature and
equational structure.
----------------------------------------------------------------------
31. LATTICE OF SUBALGEBRAS
----------------------------------------------------------------------
SUB(A) :=
set of subalgebras of A ordered by inclusion
RULE SUBLAT_01:
intersections give meets.
RULE SUBLAT_02:
generated subalgebra of union gives joins.
----------------------------------------------------------------------
32. STRUCTURAL PRESERVATION
----------------------------------------------------------------------
PRESERVED_BY_HOMOMORPHISM := {
fundamental_operations,
term_operations,
equations_forward_under_relevant_conditions
}
PRESERVED_BY_ISOMORPHISM := {
all_first_order_structural_properties_invariant_under_isomorphism,
equational_theory,
cardinality,
subalgebra_structure_up_to_correspondence,
congruence_structure_up_to_correspondence
}
----------------------------------------------------------------------
33. HOMOMORPHIC IMAGES
----------------------------------------------------------------------
HOMOMORPHIC_IMAGE(h:A->B) :=
image(h) with inherited algebra structure
RULE HI_01:
image(h) is subalgebra of B.
RULE HI_02:
if h surjective, B is homomorphic image of A.
----------------------------------------------------------------------
34. SIMPLE ALGEBRAS
----------------------------------------------------------------------
IS_SIMPLE(A) iff
Con(A) contains only:
equality_congruence
universal_congruence
subject to declared nontriviality convention
RULE SIMPLE_01:
simple algebras have no nontrivial proper quotients.
----------------------------------------------------------------------
35. SUBDIRECTLY IRREDUCIBLE ALGEBRAS
----------------------------------------------------------------------
IS_SUBDIRECTLY_IRREDUCIBLE(A) iff
A has a least nontrivial congruence
under standard nontrivial convention.
RULE SI_01:
subdirectly irreducible algebras are building blocks in
subdirect representation theory.
----------------------------------------------------------------------
36. BIRKHOFF SUBDIRECT REPRESENTATION
----------------------------------------------------------------------
THEOREM SUBDIRECT_REPRESENTATION:
every algebra is isomorphic to a subdirect product of
subdirectly irreducible algebras
under standard hypotheses of universal algebra.
----------------------------------------------------------------------
37. CLONES
----------------------------------------------------------------------
ENTITY Clone C {
operations: finitary_operations_on_set_A
}
IS_CLONE(C) iff
contains all projections
AND closed under composition
TERM_CLONE(A) :=
set of all term operations of A
RULE CLONE_01:
term operations of an algebra form a clone.
----------------------------------------------------------------------
38. POLYNOMIAL OPERATIONS
----------------------------------------------------------------------
POLYNOMIAL_OPERATION(A):
term operation allowing constants from A as parameters.
RULE POLY_01:
every term operation is a polynomial operation.
RULE POLY_02:
polynomial operations are closed under composition.
----------------------------------------------------------------------
39. HOMOMORPHISM VALIDATION
----------------------------------------------------------------------
VALIDATE_HOMOMORPHISM(h,A,B):
REQUIRE same signature
for each operation symbol f of arity n:
verify:
h(f^A(a1,...,an))
=
f^B(h(a1),...,h(an))
for required universal domain
return:
VALID
INVALID(counterexample)
UNRESOLVED
----------------------------------------------------------------------
40. CONGRUENCE VALIDATION
----------------------------------------------------------------------
VALIDATE_CONGRUENCE(theta,A):
check reflexive
check symmetric
check transitive
for each operation f:
verify compatibility
if all pass:
VALID
else:
INVALID with failing condition
----------------------------------------------------------------------
41. QUOTIENT VALIDATION
----------------------------------------------------------------------
VALIDATE_QUOTIENT(A,theta):
if NOT congruence(theta,A):
return INVALID_QUOTIENT
verify representative_independence:
choosing equivalent representatives
produces equivalent output classes
return VALID
----------------------------------------------------------------------
42. IDENTITY VALIDATION
----------------------------------------------------------------------
VALIDATE_IDENTITY(A,s≈t):
verify terms use signature(A)
symbolically prove where possible
if finite A:
optionally enumerate assignments
if counterexample assignment found:
return INVALID(counterexample)
if universal proof obtained:
return VALID
return UNRESOLVED
----------------------------------------------------------------------
43. STRUCTURE COMPARISON
----------------------------------------------------------------------
COMPARE_ALGEBRAS(A,B):
require same signature for direct structural comparison
compare:
cardinality
operation tables where finite
generated subalgebras
congruence lattices
equational theories
homomorphic invariants
attempt construct isomorphism
return:
ISOMORPHIC
NON_ISOMORPHIC with invariant witness
UNRESOLVED
----------------------------------------------------------------------
44. FINITE ALGEBRA REPRESENTATION
----------------------------------------------------------------------
FINITE_ALGEBRA A {
finite_carrier: [a1,...,ak]
operation_tables:
one table/tensor per operation symbol
}
RULE FIN_01:
finite operation tables must respect declared arities.
RULE FIN_02:
closure is explicit in total operation tables on the carrier.
----------------------------------------------------------------------
45. SEMANTIC RELATIONSHIPS
----------------------------------------------------------------------
RELATION HAS_SIGNATURE(Algebra,Signature)
RELATION HAS_CARRIER(Algebra,Set)
RELATION INTERPRETS(Algebra,OperationSymbol)
RELATION HAS_ARITY(OperationSymbol,NaturalNumber)
RELATION HAS_TERM_OPERATION(Algebra,TermOperation)
RELATION SUBALGEBRA_OF(Subalgebra,Algebra)
RELATION HOMOMORPHIC_TO(Algebra,Algebra)
RELATION ISOMORPHIC_TO(Algebra,Algebra)
RELATION CONGRUENCE_ON(Congruence,Algebra)
RELATION QUOTIENT_OF(QuotientAlgebra,Algebra)
RELATION PRODUCT_OF(ProductAlgebra,Algebra)
RELATION SATISFIES(Algebra,Equation)
RELATION AXIOMATISES(Equation,Variety)
RELATION GENERATES(Set,Subalgebra)
----------------------------------------------------------------------
46. CONCEPT DEPENDENCY GRAPH
----------------------------------------------------------------------
DEPENDENCY {
Signature -> Algebra
Signature + Variables -> Term
Algebra + Assignment + Term -> Evaluation
Evaluation -> Satisfaction
Satisfaction -> EquationalTheory
Algebra + ClosedSubset -> Subalgebra
Algebra + Mapping -> Homomorphism
Homomorphism + Bijection -> Isomorphism
Homomorphism -> KernelCongruence
Congruence -> QuotientAlgebra
AlgebraFamily -> Product
Equations -> ModelClass
H + S + P -> Variety
Variables + Variety -> FreeAlgebra
}
----------------------------------------------------------------------
47. SEMANTIC EQUIVALENCE
----------------------------------------------------------------------
SEMANTIC_EQUIVALENCE {
"type" ~= Signature
"similarity type" ~= Signature
"operation type" ~= Signature
"fundamental operation" ~= interpreted_operation_symbol
"term function" ~= term_operation
"identity" ~= universally_valid_equation
"subuniverse" ~= carrier_of_subalgebra
"compatible equivalence relation" ~= Congruence
"factor algebra" ~= QuotientAlgebra
"direct product algebra" ~= coordinatewise_product_structure
"equational class" ~= Variety
"HSP class" ~= Variety
"free algebra" ~= algebra_satisfying_universal_mapping_property
"generated subalgebra" ~= smallest_subalgebra_containing_generators
"homomorphic image" ~= image_structure_of_homomorphism
}
----------------------------------------------------------------------
48. RECURSIVE REASONING ENGINE
----------------------------------------------------------------------
PROCEDURE UNIVERSAL_ALGEBRA_REASONER(input):
parse signature
parse carrier sets
parse operations
parse terms
parse equations
parse mappings
parse congruences
parse algebra classes
validate arities
classify target:
satisfaction |
subalgebra |
homomorphism |
isomorphism |
congruence |
quotient |
product |
variety |
free_algebra |
equational_consequence
extract constraints
select definitions/theorems
execute atomic structural steps
validate every precondition
recurse until:
target established
target refuted
ambiguity blocks inference
no valid rule remains
return:
result
derivation
constraints
semantic_status
confidence
----------------------------------------------------------------------
49. RECURSIVE TRANSITION RULE
----------------------------------------------------------------------
RECURSIVE_RULE RR_01:
INPUT state S_n
identify unresolved structural property P
choose rule R such that:
preconditions(R,S_n)=true
S_(n+1):=apply(R,S_n)
validate transition
recurse(S_(n+1))
TERMINATION_CONDITIONS := {
structural_goal_reached,
counterexample_found,
contradiction_detected,
theorem_precondition_failed,
no_rule_applicable
}
----------------------------------------------------------------------
50. ERROR MODEL
----------------------------------------------------------------------
ERROR E_SIGNATURE_MISMATCH
ERROR E_ARITY_MISMATCH
ERROR E_OPERATION_NOT_CLOSED
ERROR E_MALFORMED_TERM
ERROR E_INVALID_ASSIGNMENT
ERROR E_HOMOMORPHISM_FAILURE
ERROR E_NONBIJECTIVE_ISOMORPHISM
ERROR E_NONCONGRUENCE_RELATION
ERROR E_ILL_DEFINED_QUOTIENT
ERROR E_FALSE_IDENTITY
ERROR E_INVALID_SUBALGEBRA
ERROR E_HSP_PRECONDITION
ERROR E_FREE_PROPERTY_FAILURE
ERROR E_LOST_CONSTRAINT
----------------------------------------------------------------------
51. CONFIDENCE MODEL
----------------------------------------------------------------------
CONFIDENCE {
base := 1.00
deduct 0.10 if signature inferred
deduct 0.15 if operation interpretation ambiguous
deduct 0.15 if universal quantification not fully verified
deduct 0.20 if congruence compatibility unresolved
deduct 0.20 if isomorphism construction incomplete
deduct 0.20 if equational consequence unresolved
deduct 0.25 if theorem precondition unstated
clamp_to_[0,1]
classification:
[0.95,1.00] -> very_high
[0.85,0.95) -> high
[0.70,0.85) -> moderate
[0.50,0.70) -> low
[0.00,0.50) -> insufficient
}
----------------------------------------------------------------------
52. AI CONSUMPTION MODEL
----------------------------------------------------------------------
AI_CONSUMPTION {
read_order := [
signature,
algebra_definition,
term_formation,
term_evaluation,
equations,
subalgebras,
homomorphisms,
congruences,
quotients,
products,
varieties,
free_algebras,
validation
]
execution_policy := {
preserve_signature,
preserve_operation_arity,
preserve_carrier_domains,
validate_homomorphisms,
validate_congruences,
require_well_defined_quotients,
distinguish_syntax_from_semantics,
distinguish_term_from_term_operation,
distinguish_equation_from_valid_identity,
apply_HSP_only_with_correct_class_conditions
}
}
----------------------------------------------------------------------
53. UPDATE MODEL
----------------------------------------------------------------------
UPDATE_MODEL {
extensible := true
new_operation_symbol_requires:
name,
arity
new_algebra_requires:
signature,
carrier,
interpretations
new_equational_rule_requires:
soundness_condition
new_quotient_rule_requires:
congruence_compatibility
theorem_additions_require:
explicit_preconditions,
structural_scope
conflict_policy:
preserve_standard_universal_algebra_usage
AND explicit_versioning_for_context_specific_variants
}
----------------------------------------------------------------------
54. SELF-CHECK
----------------------------------------------------------------------
SELF_CHECK {
CHECK every operation has declared arity
CHECK every algebra interprets complete signature
CHECK operation outputs remain in carrier
CHECK every term is well-formed
CHECK assignments map variables into carrier
CHECK homomorphisms preserve every fundamental operation
CHECK isomorphisms are bijective homomorphisms
CHECK congruences are equivalence relations
CHECK congruences respect all operations
CHECK quotient operations independent of representatives
CHECK identities quantified over all assignments
CHECK HSP claims satisfy correct closure conditions
CHECK free algebra claims satisfy universal property
CHECK final derivation preserves all structural constraints
}
----------------------------------------------------------------------
55. COMPLETION CONDITIONS
----------------------------------------------------------------------
COMPLETE iff {
signature_model_defined
AND algebra_model_defined
AND term_model_defined
AND evaluation_defined
AND equational_satisfaction_defined
AND subalgebra_defined
AND homomorphism_defined
AND isomorphism_defined
AND congruence_defined
AND quotient_defined
AND direct_product_defined
AND variety_defined
AND HSP_defined
AND free_algebra_defined
AND equational_logic_defined
AND semantic_equivalence_defined
AND recursive_reasoning_defined
AND validation_defined
AND confidence_defined
AND ai_consumption_defined
AND update_model_defined
}
IF NOT COMPLETE:
semantic_status := incomplete_flow
----------------------------------------------------------------------
56. FLOW TERMINATION
----------------------------------------------------------------------
TERMINATE WHEN:
requested universal-algebraic goal achieved
AND result structurally validated
OR
counterexample refutes requested identity/property
OR
required precondition fails
OR
material ambiguity prevents valid structural inference
FINAL_STATE := {
mathematical_result,
structural_classification,
derivation,
preserved_signature,
preserved_constraints,
semantic_status,
confidence
}
} // END FLOW UniversalAlgebraOntology
======================================================================
END FLOW
======================================================================
FLOW STATUS:
Complete
CANONICAL PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/universal-algebra-ontology.html
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