Monomial Ontology
Algebra Ontologies
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 11 September 2026
Audio Overview
======================================================================
KIRETH SEMANTIC RESEARCH ARCHIVE
FLOW ALGEBRA — MONOMIAL ONTOLOGY
======================================================================
PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/monomial-ontology.html
FLOW:
MonomialOntology
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 monomials, coefficients, variables,
exponents, degree, like terms, multiplication, division, powers,
factor relationships, monomial ordering and monomial roles within
polynomial algebra.
AI INTERPRETATION NOTE:
Execute the following specification as formal semantic and reasoning
rules. Preserve coefficient domains, variable sets, nonnegative integer
exponents, monomial structure, divisibility conditions and degree
invariants.
======================================================================
START FLOW
======================================================================
FLOW MonomialOntology {
----------------------------------------------------------------------
1. DOMAIN DECLARATION
----------------------------------------------------------------------
DOMAIN := Monomial_Theory
PARENT_DOMAIN := Polynomial_Theory
SUPER_DOMAINS := {
Mathematics,
Mathematical_Ontologies,
Algebra_Ontologies,
Elementary_Algebra,
Polynomial_Theory
}
DOMAIN_INTENT := {
represent_monomials,
represent_coefficients,
represent_variables,
represent_exponents,
represent_monomial_degree,
represent_like_terms,
represent_monomial_multiplication,
represent_monomial_division,
represent_monomial_powers,
represent_monomial_divisibility,
represent_common_factors,
represent_greatest_common_monomials,
represent_least_common_monomials,
represent_monomial_orders,
support_recursive_monomial_reasoning,
support_domain_aware_validation
}
DEFAULT_COEFFICIENT_DOMAIN := Integer | Rational | Real | Complex | Field
----------------------------------------------------------------------
2. PRIMITIVE TYPES
----------------------------------------------------------------------
TYPE Scalar
TYPE Coefficient <: Scalar
TYPE Variable
TYPE Exponent
TYPE ExponentVector
TYPE Monomial
TYPE ConstantMonomial <: Monomial
TYPE PurePowerMonomial <: Monomial
TYPE SquareFreeMonomial <: Monomial
TYPE UnitMonomial <: Monomial
TYPE ZeroTerm
TYPE Degree
TYPE MonomialOrder
TYPE Factor
TYPE Constraint
TYPE Derivation
TYPE ValidationResult
----------------------------------------------------------------------
3. MONOMIAL DEFINITION
----------------------------------------------------------------------
ENTITY Monomial M {
coefficient: a
variables: [x1,...,xn]
exponents: [e1,...,en]
}
FORM:
M = a*x1^e1*x2^e2*...*xn^en
CONSTRAINT EXP_01:
each ei is a nonnegative integer.
CONSTRAINT COEF_01:
coefficient a belongs to declared coefficient domain.
RULE MONO_01:
a monomial contains exactly one multiplicative term.
RULE MONO_02:
variables with exponent 0 may be omitted from written form.
RULE MONO_03:
variable ordering is representational and must be normalised
consistently.
----------------------------------------------------------------------
4. ZERO COEFFICIENT CASE
----------------------------------------------------------------------
RULE ZERO_01:
coefficient 0 produces the zero term.
RULE ZERO_02:
zero term is not treated as an ordinary nonzero monomial for degree
and divisibility unless an explicit convention is declared.
----------------------------------------------------------------------
5. CONSTANT MONOMIALS
----------------------------------------------------------------------
ENTITY ConstantMonomial C {
form: a
}
RULE CONST_01:
nonzero constant monomial has degree 0.
RULE CONST_02:
1 is the multiplicative identity monomial.
----------------------------------------------------------------------
6. PURE POWER MONOMIALS
----------------------------------------------------------------------
ENTITY PurePowerMonomial P {
form:
a*x^n
single_variable_support: true
}
RULE PP_01:
x^n has degree n.
----------------------------------------------------------------------
7. SQUARE-FREE MONOMIALS
----------------------------------------------------------------------
IS_SQUARE_FREE(M) iff
every exponent ei in {0,1}
RULE SQF_01:
square-free monomial contains no repeated variable factor.
----------------------------------------------------------------------
8. SUPPORT
----------------------------------------------------------------------
SUPPORT(M) :=
{xi | exponent ei > 0}
RULE SUP_01:
support ignores variables with exponent 0.
RULE SUP_02:
constant nonzero monomial has empty support.
----------------------------------------------------------------------
9. EXPONENT VECTOR
----------------------------------------------------------------------
EXPONENT_VECTOR(M):
alpha=(e1,...,en)
NOTATION:
x^alpha
:=
x1^e1 ... xn^en
RULE EV_01:
monomial structure is uniquely determined by coefficient and
exponent vector once variable order is fixed.
----------------------------------------------------------------------
10. DEGREE
----------------------------------------------------------------------
TOTAL_DEGREE(M):
sum_i ei
RULE DEG_01:
degree(a*x1^e1...xn^en)=sum ei
for nonzero coefficient a.
RULE DEG_02:
degree(nonzero_constant)=0.
RULE DEG_03:
zero-term degree requires separate convention.
----------------------------------------------------------------------
11. DEGREE IN A VARIABLE
----------------------------------------------------------------------
DEGREE_IN(M,x_i):
exponent e_i
RULE DVAR_01:
degree in absent variable is 0.
----------------------------------------------------------------------
12. LIKE MONOMIAL TERMS
----------------------------------------------------------------------
LIKE_MONOMIALS(M1,M2) iff
EXPONENT_VECTOR(M1)=EXPONENT_VECTOR(M2)
RULE LIKE_01:
coefficients may differ while terms remain like.
RULE LIKE_02:
like monomial terms may be combined additively.
EXAMPLE:
3x^2y + 5x^2y = 8x^2y
----------------------------------------------------------------------
13. UNLIKE MONOMIALS
----------------------------------------------------------------------
UNLIKE_MONOMIALS(M1,M2) iff
exponent_vectors differ
RULE UNLIKE_01:
unlike monomial terms cannot be combined by coefficient addition.
----------------------------------------------------------------------
14. NORMAL FORM
----------------------------------------------------------------------
NORMALISE_MONOMIAL(M):
reorder variables according to declared variable order
combine repeated factors by adding exponents
simplify coefficient
remove exponent-0 variables
return:
a*x1^e1...xn^en
RULE NF_01:
normalised monomial representation is unique for fixed variable
order and coefficient representation.
----------------------------------------------------------------------
15. MULTIPLICATION
----------------------------------------------------------------------
MULTIPLY(M1,M2):
coefficients:
c = a*b
exponents:
gamma_i = alpha_i + beta_i
RETURN:
c*x^gamma
RULE MUL_01:
monomial multiplication is commutative
when coefficient ring and variables commute.
RULE MUL_02:
degree(M1*M2)=degree(M1)+degree(M2)
for nonzero monomials over suitable domain.
----------------------------------------------------------------------
16. DIVISION
----------------------------------------------------------------------
DIVIDE_MONOMIALS(M1,M2):
REQUIRE:
coefficient division valid
AND for all i:
alpha_i >= beta_i
RETURN:
(a/b)*x^(alpha-beta)
RULE DIV_01:
quotient is monomial only if exponent differences are nonnegative
integers.
RULE DIV_02:
denominator coefficient must be nonzero.
----------------------------------------------------------------------
17. MONOMIAL DIVISIBILITY
----------------------------------------------------------------------
DIVIDES(M1,M2) iff
coefficient condition satisfied
AND for every variable:
exponent(M1)<=exponent(M2)
OVER FIELD_COEFFICIENTS:
nonzero coefficient divisibility automatically satisfied.
RULE MDIV_01:
x^alpha divides x^beta iff alpha_i<=beta_i for all i.
----------------------------------------------------------------------
18. POWERS OF MONOMIALS
----------------------------------------------------------------------
POWER(M,k):
REQUIRE:
k nonnegative_integer
if:
M=a*x^alpha
then:
M^k=a^k*x^(k alpha)
RULE POW_01:
degree(M^k)=k*degree(M)
for nonzero M.
RULE POW_02:
M^0=1
for nonzero monomial under standard convention.
----------------------------------------------------------------------
19. NEGATION
----------------------------------------------------------------------
NEGATE(M):
multiply coefficient by -1
RULE NEG_01:
exponent vector unchanged.
----------------------------------------------------------------------
20. RECIPROCAL
----------------------------------------------------------------------
RECIPROCAL(M):
formal result:
a^(-1) x^(-alpha)
NOTE:
generally not a monomial in ordinary polynomial ring unless
alpha=0.
RULE RECIP_01:
reciprocal belongs naturally to Laurent monomial context when
negative exponents are allowed.
----------------------------------------------------------------------
21. MONOMIAL FACTORS
----------------------------------------------------------------------
FACTOR_MONOMIAL(M):
represent as:
coefficient factors
variable prime-like factors
EXAMPLE:
12x^3y^2
=
2^2 * 3 * x^3 * y^2
RULE FAC_01:
variable factors encoded by exponents.
----------------------------------------------------------------------
22. GREATEST COMMON MONOMIAL FACTOR
----------------------------------------------------------------------
GCM(M1,...,Mk):
coefficient_part:
gcd of coefficients
when coefficient domain supports gcd
variable_part:
each exponent :=
minimum exponent among inputs
RULE GCM_01:
GCM divides every input monomial.
----------------------------------------------------------------------
23. LEAST COMMON MONOMIAL
----------------------------------------------------------------------
LCM_MONOMIAL(M1,...,Mk):
coefficient_part:
lcm under declared coefficient convention
variable_part:
each exponent :=
maximum exponent among inputs
RULE LCM_01:
every input monomial divides monomial LCM.
----------------------------------------------------------------------
24. CONTENT EXTRACTION
----------------------------------------------------------------------
FOR polynomial terms M1,...,Mk:
COMMON_MONOMIAL_FACTOR :=
gcd/coefficient common factor
times
variable exponents using minima
RULE CONT_01:
factoring common monomial reduces every term by exact monomial division.
----------------------------------------------------------------------
25. MONOMIAL ORDER
----------------------------------------------------------------------
ENTITY MonomialOrder <_m
IS_MONOMIAL_ORDER(<_m) iff
total_order
well_order
multiplicative_compatibility:
if A<_m B
then AC<_m BC
RULE MO_01:
monomial orders compare exponent vectors.
----------------------------------------------------------------------
26. LEXICOGRAPHIC ORDER
----------------------------------------------------------------------
LEX(alpha,beta):
compare first position where alpha_i != beta_i
alpha > beta iff alpha_i > beta_i there
RULE LEX_01:
variable priority determined by declared variable order.
----------------------------------------------------------------------
27. GRADED LEXICOGRAPHIC ORDER
----------------------------------------------------------------------
GRLEX(alpha,beta):
first compare total degrees
if equal:
compare lexicographically
----------------------------------------------------------------------
28. GRADED REVERSE LEXICOGRAPHIC ORDER
----------------------------------------------------------------------
GREVLEX(alpha,beta):
first compare total degree
if equal:
inspect last differing exponent
declared standard convention required.
----------------------------------------------------------------------
29. LEADING MONOMIAL
----------------------------------------------------------------------
LEADING_MONOMIAL(P,order):
greatest monomial appearing in P under order
RULE LM_01:
leading monomial depends on monomial order in multivariate setting.
----------------------------------------------------------------------
30. LEADING TERM
----------------------------------------------------------------------
LEADING_TERM(P):
leading_coefficient(P)*leading_monomial(P)
RULE LT_01:
leading term includes coefficient;
leading monomial excludes coefficient.
----------------------------------------------------------------------
31. MONOMIAL IDEALS
----------------------------------------------------------------------
ENTITY MonomialIdeal I {
generators:
monomials M1,...,Mk
}
GENERATED_IDEAL:
<M1,...,Mk>
MEMBERSHIP:
monomial M belongs to I
iff some generator Mi divides M.
RULE MI_01:
monomial ideal membership reduces to divisibility testing.
----------------------------------------------------------------------
32. MINIMAL MONOMIAL GENERATORS
----------------------------------------------------------------------
MINIMAL_GENERATING_SET(I):
remove generator Mj if another generator Mi divides Mj.
RULE MMG_01:
minimal monomial generating set is unique up to ordering.
----------------------------------------------------------------------
33. MONOMIAL LCM IN GROEBNER THEORY
----------------------------------------------------------------------
LCM(x^alpha,x^beta):
x^gamma
where:
gamma_i=max(alpha_i,beta_i)
RULE GL_01:
monomial LCM supports S-polynomial construction.
----------------------------------------------------------------------
34. S-POLYNOMIAL LINK
----------------------------------------------------------------------
FOR polynomials f,g:
S(f,g)
=
LCM(LM(f),LM(g))/LT(f) * f
-
LCM(LM(f),LM(g))/LT(g) * g
NOTE:
detailed Groebner basis reasoning belongs to specialised ontology.
----------------------------------------------------------------------
35. MONOMIAL SUBSTITUTION
----------------------------------------------------------------------
SUBSTITUTE(M, x_i -> expression):
preserve multiplicative exponent structure
expand only when requested
RULE SUB_01:
substituting scalars may reduce monomial to scalar.
RULE SUB_02:
substituting monomials yields monomial if mappings are monomial and
coefficient operations remain valid.
----------------------------------------------------------------------
36. MONOMIAL EVALUATION
----------------------------------------------------------------------
EVALUATE(M,assignment):
M=a*x1^e1...xn^en
return:
a * product_i assignment(x_i)^ei
RULE EVAL_01:
missing assignment for positive-exponent variable prevents numeric
evaluation.
----------------------------------------------------------------------
37. ZERO EVALUATION
----------------------------------------------------------------------
RULE ZEVAL_01:
if any variable with positive exponent evaluates to 0,
monomial evaluates to 0.
RULE ZEVAL_02:
nonzero constant monomial never evaluates to zero.
----------------------------------------------------------------------
38. SIGN OF REAL MONOMIAL
----------------------------------------------------------------------
SIGN(M,value_assignment):
determine coefficient sign
multiply by signs induced by variable powers
RULE SIGN_01:
even exponents remove sign of nonzero real variable.
RULE SIGN_02:
odd exponents preserve sign of real variable.
----------------------------------------------------------------------
39. HOMOGENEITY
----------------------------------------------------------------------
ENTITY HomogeneousMonomial
RULE HOM_01:
every monomial is homogeneous with total degree equal to its degree.
RULE HOM_02:
polynomial is homogeneous if all nonzero monomials have same
total degree.
----------------------------------------------------------------------
40. MULTIDEGREE
----------------------------------------------------------------------
MULTIDEGREE(M):
exponent_vector alpha
RULE MDEG_01:
multidegree contains more information than total degree.
----------------------------------------------------------------------
41. WEIGHTED DEGREE
----------------------------------------------------------------------
WEIGHTED_DEGREE(M,w):
w=(w1,...,wn)
return:
sum_i wi*ei
RULE WDEG_01:
weighted degree depends on declared weight vector.
----------------------------------------------------------------------
42. SQUARE AND CUBE MONOMIALS
----------------------------------------------------------------------
IS_PERFECT_SQUARE_MONOMIAL(M) iff
coefficient is square in declared domain
AND every exponent even
IS_PERFECT_CUBE_MONOMIAL(M) iff
coefficient is cube in declared domain
AND every exponent divisible by 3
----------------------------------------------------------------------
43. RADICAL OF A MONOMIAL
----------------------------------------------------------------------
RADICAL_MONOMIAL(M):
variable part:
product of all variables with positive exponent, exponent 1
coefficient handling:
domain-dependent
EXAMPLE:
rad(x^3 y^2)=xy
----------------------------------------------------------------------
44. SUPPORT SIZE
----------------------------------------------------------------------
SUPPORT_SIZE(M):
cardinality(SUPPORT(M))
RULE SS_01:
support_size=0 for nonzero constant monomial.
----------------------------------------------------------------------
45. UNIVARIATE MONOMIALS
----------------------------------------------------------------------
UNIVARIATE_MONOMIAL:
a*x^n
RULE UNI_01:
degree=n for nonzero coefficient.
RULE UNI_02:
divisibility:
x^m divides x^n iff m<=n.
----------------------------------------------------------------------
46. MULTIVARIATE MONOMIALS
----------------------------------------------------------------------
MULTIVARIATE_MONOMIAL:
a*x1^e1...xn^en
RULE MULTI_01:
total degree=sum exponents.
RULE MULTI_02:
divisibility is componentwise on exponent vectors.
----------------------------------------------------------------------
47. COEFFICIENT NORMALISATION
----------------------------------------------------------------------
NORMALISE_COEFFICIENT(a):
apply domain-specific canonical form.
EXAMPLES:
Rational -> lowest terms
Integer -> integer normal form
Complex -> a+bi canonical representation
RULE CN_01:
coefficient normalisation must not alter monomial algebraic value.
----------------------------------------------------------------------
48. UNITS
----------------------------------------------------------------------
UNIT_MONOMIAL(M) iff
M invertible in declared monomial/polynomial ring.
OVER field polynomial ring:
units are nonzero constants.
RULE UNIT_01:
any monomial containing positive variable exponent is nonunit in F[x1,...,xn].
----------------------------------------------------------------------
49. ASSOCIATES
----------------------------------------------------------------------
ASSOCIATES(M1,M2) iff
M1=u*M2
for unit u
RULE ASSOC_01:
over field coefficients,
monomials with same exponent vector are associates if both coefficients nonzero.
----------------------------------------------------------------------
50. SEMANTIC RELATIONSHIPS
----------------------------------------------------------------------
RELATION HAS_COEFFICIENT(Monomial,Coefficient)
RELATION HAS_VARIABLE(Monomial,Variable)
RELATION HAS_EXPONENT(Monomial,Exponent)
RELATION HAS_DEGREE(Monomial,Degree)
RELATION DIVIDES(Monomial,Monomial)
RELATION FACTOR_OF(Monomial,Polynomial)
RELATION LIKE_TO(Monomial,Monomial)
RELATION LEADING_MONOMIAL_OF(Monomial,Polynomial)
RELATION MEMBER_OF(Monomial,MonomialIdeal)
----------------------------------------------------------------------
51. SEMANTIC EQUIVALENCE
----------------------------------------------------------------------
SEMANTIC_EQUIVALENCE {
"single term" ~= Monomial
"coefficient" ~= scalar_multiplier
"power" ~= variable_with_exponent
"degree of monomial" ~= sum_of_variable_exponents
"like term" ~= same_exponent_vector
"monomial factor" ~= divisor_monomial
"common monomial factor" ~= monomial_dividing_all_terms
"leading monomial" ~= greatest_monomial_under_declared_order
"support" ~= variables_with_positive_exponents
"multidegree" ~= exponent_vector
}
----------------------------------------------------------------------
52. QUERY INTERPRETATION
----------------------------------------------------------------------
QUERY Q1:
"degree of monomial"
-> TOTAL_DEGREE(M)
QUERY Q2:
"multiply monomials"
-> MULTIPLY(M1,M2)
QUERY Q3:
"divide monomials"
-> DIVIDE_MONOMIALS(M1,M2)
QUERY Q4:
"does M1 divide M2"
-> DIVIDES(M1,M2)
QUERY Q5:
"greatest common monomial"
-> GCM(inputs)
QUERY Q6:
"least common monomial"
-> LCM_MONOMIAL(inputs)
QUERY Q7:
"are these like terms"
-> compare exponent vectors
QUERY Q8:
"leading monomial"
-> require polynomial and monomial order
----------------------------------------------------------------------
53. VALIDATION ENGINE
----------------------------------------------------------------------
PROCEDURE VALIDATE_MONOMIAL(M):
check coefficient domain
check coefficient canonicality if required
check all exponents nonnegative integers
check variable order
verify zero-coefficient handling
return:
VALID
INVALID(reason)
UNRESOLVED
PROCEDURE VALIDATE_OPERATION(operation,inputs,result):
validate each input
check coefficient operation
check exponent arithmetic
check divisibility preconditions
recompute expected exponent vector
compare with result
return validation status
----------------------------------------------------------------------
54. RECURSIVE REASONING ENGINE
----------------------------------------------------------------------
PROCEDURE MONOMIAL_REASONER(input):
parse coefficient
parse variables
parse exponents
infer coefficient domain
normalise variable order
classify requested operation
extract constraints
select valid monomial rules
execute atomic transformations
preserve exponent legality
validate intermediate states
recurse until goal achieved
validate final result
RETURN:
result,
derivation,
coefficient_domain,
exponent_vector,
semantic_status,
confidence
----------------------------------------------------------------------
55. RECURSIVE TRANSITION
----------------------------------------------------------------------
RECURSIVE_RULE RR_01:
INPUT state S_n
identify unresolved monomial property
choose rule R such that:
coefficient_preconditions=true
exponent_preconditions=true
divisibility_preconditions=true
S_(n+1):=apply(R,S_n)
validate transition
recurse(S_(n+1))
TERMINATION_CONDITIONS := {
monomial_goal_achieved,
normal_form_reached,
divisibility_refuted,
invalid_exponent_detected,
coefficient_domain_failure,
no_valid_rule
}
----------------------------------------------------------------------
56. ERROR MODEL
----------------------------------------------------------------------
ERROR E_NEGATIVE_EXPONENT
ERROR E_NONINTEGER_EXPONENT
ERROR E_COEFFICIENT_DOMAIN_MISMATCH
ERROR E_DIVISION_BY_ZERO_COEFFICIENT
ERROR E_NONDIVISIBLE_MONOMIAL
ERROR E_VARIABLE_ORDER_AMBIGUOUS
ERROR E_ZERO_TERM_DEGREE
ERROR E_MONOMIAL_ORDER_UNDECLARED
ERROR E_LIKE_TERM_MISCLASSIFICATION
ERROR E_LOST_EXPONENT
----------------------------------------------------------------------
57. CONFIDENCE MODEL
----------------------------------------------------------------------
CONFIDENCE {
base:=1.00
deduct 0.10 if coefficient domain inferred
deduct 0.10 if variable order inferred
deduct 0.15 if monomial order unspecified but material
deduct 0.15 if zero-term convention unresolved
deduct 0.20 if coefficient divisibility ambiguous
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
}
----------------------------------------------------------------------
58. AI CONSUMPTION MODEL
----------------------------------------------------------------------
AI_CONSUMPTION {
read_order := [
monomial_definition,
coefficient_domain,
exponent_vector,
degree,
like_terms,
multiplication,
division,
divisibility,
common_factors,
monomial_orders,
validation
]
execution_policy := {
preserve_nonnegative_integer_exponents,
preserve_coefficient_domain,
normalise_variable_order,
distinguish_coefficient_from_monomial_part,
validate_divisibility_componentwise,
distinguish_total_degree_from_multidegree,
require_monomial_order_for_leading_monomial
}
}
----------------------------------------------------------------------
59. UPDATE MODEL
----------------------------------------------------------------------
UPDATE_MODEL {
extensible:=true
new_monomial_class_requires:
defining_exponent_or_coefficient_predicate
new_order_requires:
totality,
well_ordering,
multiplicative_compatibility
new_divisibility_rule_requires:
coefficient_domain_conditions,
exponent_conditions
conflicts_require:
explicit_versioning
standard_polynomial_algebra_meaning_has_priority:=true
}
----------------------------------------------------------------------
60. SELF-CHECK
----------------------------------------------------------------------
SELF_CHECK {
CHECK coefficient belongs to declared domain
CHECK all exponents nonnegative integers
CHECK repeated variables combined correctly
CHECK total degree equals exponent sum
CHECK like-term classification uses exponent vectors
CHECK multiplication adds exponents
CHECK division subtracts exponents only when nonnegative
CHECK GCM uses minimum exponents
CHECK LCM uses maximum exponents
CHECK monomial order declared when required
CHECK final result validated
}
----------------------------------------------------------------------
61. COMPLETION CONDITIONS
----------------------------------------------------------------------
COMPLETE iff {
monomial_definition_defined
AND coefficient_defined
AND exponent_vector_defined
AND degree_defined
AND like_terms_defined
AND normalisation_defined
AND multiplication_defined
AND division_defined
AND divisibility_defined
AND powers_defined
AND common_factor_rules_defined
AND monomial_orders_defined
AND monomial_ideals_link_defined
AND semantic_equivalence_defined
AND validation_defined
AND recursive_reasoning_defined
AND confidence_defined
AND ai_consumption_defined
AND update_model_defined
}
IF NOT COMPLETE:
semantic_status:=incomplete_flow
----------------------------------------------------------------------
62. FLOW TERMINATION
----------------------------------------------------------------------
TERMINATE WHEN:
requested monomial goal achieved
AND result validated
OR
exponent constraints make object non-monomial
OR
divisibility conditions fail
OR
coefficient-domain conditions prevent operation
OR
no valid monomial transformation remains
FINAL_STATE := {
monomial_result,
derivation,
coefficient_domain,
exponent_vector,
degree,
semantic_status,
confidence
}
} // END FLOW MonomialOntology
======================================================================
END FLOW
======================================================================
FLOW STATUS:
Complete
CANONICAL PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/monomial-ontology.html
FLOW_HASH_SHA256: 234ef90a27db2bdeb59d4e42d4e58377660d75a6f1b29920ef98175eaaea0598