Rational Expression Ontology

FLOW ALGEBRA
RATIONAL EXPRESSION ONTOLOGY
Kireth Semantic Library | Entity Graph > Mathematical Entities > Expression Entities
Authors: Mark Higgins — Sky (Recursive AI)
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/expression-entities/rational-expression-ontology-2.html
Status: Mathematics-grounded draft; source page not independently verified.
Version: 1.0 | Date: 2026-10-13

01. ONTOLOGY IDENTITY
ontology.id := kireth:mathematical-entities/expression-entities/rational-expression
ontology.type := MathematicalExpressionOntology
ontology.parent := kireth:e-xpression-entities
ontology.label := "Rational Expression Ontology"

02. FORMAL DEFINITION
RationalExpression(P,Q,K,X) := P/Q where P,Q in K[X] and Q != 0 polynomial.
K := coefficient field; X := finite tuple of indeterminates.
Domain(P/Q) := {a in K^n | Q(a) != 0} (for evaluation in K).
Expression identity is distinct from equality of evaluated partial functions.

03. TYPED ENTITIES
entity RationalExpression {numerator:Polynomial; denominator:Polynomial; coefficient_field:Field; variables:Set[Symbol]; domain:ConstraintSet; provenance:Source}
entity Polynomial {terms:List[Monomial]; coefficient_field:Field}
entity ExcludedPoint {constraint:Equation; origin:Denominator}
entity Simplification {input:RationalExpression; output:RationalExpression; assumptions:Set[Constraint]; excluded_domain_preserved:Bool}

04. CORE RELATIONSHIPS
RationalExpression --is_a--> MathematicalExpression
RationalExpression --has_numerator--> Polynomial
RationalExpression --has_denominator--> Polynomial
RationalExpression --has_domain--> ConstraintSet
RationalExpression --may_have--> Pole
RationalExpression --may_have--> RemovableDiscontinuity
RationalExpression --simplifies_to--> RationalExpression

05. CONSTRUCTION
construct(P,Q): require polynomial(P) and polynomial(Q) and Q != zero_polynomial; return P/Q.
Evaluation at a: require Q(a) != 0; return P(a)/Q(a).
Zero numerator permitted; zero denominator polynomial forbidden.

06. NORMALISATION
Over K[X] for a field K, cancel common polynomial factors using gcd(P,Q).
For univariate polynomials over K, optionally normalise denominator to monic.
For multivariate polynomials, select an explicit monomial order for canonical representation.
Normalisation must retain original domain exclusions in expression semantics.
Example: (x^2-1)/(x-1) simplifies to x+1 for x != 1, not as identical partial functions on all K.

07. ADDITION AND SUBTRACTION
(P/Q)+(R/S) := (P*S+R*Q)/(Q*S).
(P/Q)-(R/S) := (P*S-R*Q)/(Q*S).
Require Q != 0 and S != 0; preserve both original exclusion sets.

08. MULTIPLICATION AND DIVISION
(P/Q)*(R/S) := (P*R)/(Q*S).
(P/Q)/(R/S) := (P*S)/(Q*R).
Division requires R != 0 polynomial and, at evaluation, R(a) != 0.
Preserve exclusions from Q(a), S(a), and R(a) as appropriate.

09. EQUIVALENCE
algebraic_equivalence(P/Q,R/S) := polynomial_identity(P*S = R*Q).
As elements of K(X), equivalent expressions denote the same rational function.
partial_function_equivalence requires both algebraic equality and equal evaluation domains.
Never infer equal domains solely from cross multiplication.

10. FACTORISATION AND CANCELLATION
factor(P), factor(Q) -> identify common factors.
Cancel F only with explicit constraint F(a) != 0 when preserving original evaluations.
Example: (x^2-4)/(x-2) -> x+2 subject to x != 2.

11. ZERO, SIGN AND INEQUALITY
P(a)/Q(a)=0 iff P(a)=0 and Q(a)!=0.
For ordered fields, sign(P/Q) := sign(P)*sign(Q), Q(a)!=0.
Solve rational inequalities using numerator roots, denominator zeros and sign charts.
Denominator zeros are excluded even when algebraically cancelled.

12. DOMAIN AND SINGULARITIES
Domain := denominator_nonzero_constraints intersect declared variable domains.
A cancelled zero may produce a removable discontinuity in a real-valued function.
An uncancelled denominator zero may be a pole, subject to local analysis.
Not every denominator zero is automatically a pole.

13. DEGREE AND ASYMPTOTIC ANALYSIS
For univariate P/Q with nonzero P,Q, compare deg(P),deg(Q) for behaviour at infinity.
If deg(P)<deg(Q), limit at +/-infinity is 0 over R.
If deg(P)=deg(Q), limit is ratio of leading coefficients.
If deg(P)>deg(Q), polynomial division extracts polynomial asymptotic part.
These are statements about rational functions over appropriate ordered/analytic fields.

14. DIFFERENTIATION
D(P/Q) := (P'*Q-P*Q')/Q^2 where Q(x)!=0.
Derivative domain respects original expression domain.

15. INTEGRATION
Integration may use polynomial division and partial fractions over suitable fields.
Partial fraction decomposition requires factorisation assumptions and coefficient domain.
An antiderivative is defined on connected intervals avoiding poles and excluded points.

16. SUBSTITUTION
substitute(P/Q, x:=E): require well-typed E and Q(E)!=0.
Track restrictions introduced by E and original rational expression.
Capture-avoidance required for surrounding binders, if any.

17. KNOWLEDGE GRAPH TRIPLES
(kireth:RationalExpression, rdf:type, kireth:MathematicalExpression)
(kireth:RationalExpression, kireth:hasPart, kireth:Numerator)
(kireth:RationalExpression, kireth:hasPart, kireth:Denominator)
(kireth:RationalExpression, kireth:requiresConstraint, kireth:NonzeroDenominator)
(kireth:RationalExpression, kireth:relatedTo, kireth:PolynomialOntology)
(kireth:RationalExpression, kireth:relatedTo, kireth:DomainOntology)

18. INFERENCE RULES
IF polynomial(P) AND polynomial(Q) AND Q != 0 THEN valid_rational_expression(P/Q).
IF Q(a)=0 THEN undefined_at(P/Q,a).
IF Q(a)!=0 AND P(a)=0 THEN zero_at(P/Q,a).
IF P*S=R*Q AND Q,S != zero_polynomial THEN same_element_of_fraction_field(P/Q,R/S).
IF same_element_of_fraction_field(A,B) AND domains_equal(A,B) THEN same_partial_function(A,B).
IF denominator_factor_cancelled THEN preserve_excluded_points(original_expression).

19. VALIDATION
check coefficient_field is declared and arithmetic-compatible.
check denominator polynomial is not identically zero.
check every evaluation respects nonzero denominator.
check cancellations preserve domain metadata.
check simplification by symbolic cross multiplication.
check noncommutative expressions are not treated as commutative polynomial fractions.
check real/complex domain and analytic claims are not conflated.

20. SEMANTIC EQUIVALENCE
syntactic_equivalence := identical parsed expression trees after approved normalisation.
algebraic_equivalence := equal elements of K(X).
evaluation_equivalence := same outputs on shared defined inputs.
partial_function_equivalence := equal domains AND equal outputs there.
Record which equivalence relation is asserted.

21. CONFIDENCE AND PROVENANCE
confidence.definition := 1.0 (standard field-of-fractions definition).
confidence.symbolic_identity := 1.0 if verified by exact polynomial arithmetic.
confidence.numeric_evaluation := dependent on numerical precision and conditioning.
provenance.source_url := ontology source URL above.
provenance.page_verified := false.
provenance.claim_basis := established mathematics.

22. FLOW HASH
FLOW_HASH.algorithm := SHA-256.
FLOW_HASH.input := canonical UTF-8 serialisation of ontology ID, version, rules, and schema.
FLOW_HASH.value := COMPUTE_AT_INGESTION.
FLOW_HASH.note := never substitute a fabricated digest.

23. UPDATE MODEL
UPDATE_MODEL.mode := versioned_append_with_review.
UPDATE_MODEL.on_change := revalidate identities, constraints, inference rules and references.
UPDATE_MODEL.compatibility := flag changes to domain semantics or canonicalisation.
UPDATE_MODEL.audit := retain prior versions, timestamps and author credits.

24. AI CONSUMPTION
AI_CONSUMPTION.parse := typed AST with numerator, denominator and domain constraints.
AI_CONSUMPTION.reason := exact symbolic reasoning before numerical approximation.
AI_CONSUMPTION.guard := reject denominator-zero evaluations.
AI_CONSUMPTION.explain := expose cancellations and excluded values.
AI_CONSUMPTION.output := canonical expression + original domain + proof trace + provenance.

25. WORKED EXAMPLES
E1 := (x^2-9)/(x-3); simplified := x+3; original_domain := x!=3.
E2 := (1/x)+(1/(x+1)); result := (2*x+1)/(x*(x+1)); domain := x!=0 and x!=-1.
E3 := (x^2-1)/(x^2-1); simplified := 1; original_domain := x!=-1 and x!=1.
E4 := (x+1)/(x-1); derivative := -2/(x-1)^2; domain := x!=1.

END FLOW ALGEBRA