FunctionSpace2(R, K)

fspace.spad line 389 [edit on github]

• R: Comparable

• K: KernelCategory %

A space of formal functions with arguments in an arbitrary ordered set.

0: % if R has AbelianSemiGroup

from AbelianMonoid

1: % if R has SemiGroup

from MagmaWithUnit

*: (%, %) -> % if R has SemiGroup

from Magma

*: (%, Fraction Integer) -> % if R has IntegralDomain

from RightModule Fraction Integer

*: (%, Integer) -> % if R has Ring and R has LinearlyExplicitOver Integer

from RightModule Integer

*: (%, R) -> % if R has Ring

from RightModule R

*: (Fraction Integer, %) -> % if R has IntegralDomain

from LeftModule Fraction Integer

*: (Integer, %) -> % if R has AbelianGroup

from AbelianGroup

*: (NonNegativeInteger, %) -> % if R has AbelianSemiGroup

from AbelianMonoid

*: (PositiveInteger, %) -> % if R has AbelianSemiGroup

from AbelianSemiGroup

*: (R, %) -> % if R has CommutativeRing

from LeftModule R

+: (%, %) -> % if R has AbelianSemiGroup

from AbelianSemiGroup

-: % -> % if R has AbelianGroup

from AbelianGroup

-: (%, %) -> % if R has AbelianGroup

from AbelianGroup

/: (%, %) -> % if R has IntegralDomain or R has Group

from Group

/: (SparseMultivariatePolynomial(R, K), SparseMultivariatePolynomial(R, K)) -> % if R has IntegralDomain

p1/p2 returns the quotient of p1 and p2 as an element of %.

=: (%, %) -> Boolean

from BasicType

^: (%, Integer) -> % if R has IntegralDomain or R has Group

from Group

^: (%, NonNegativeInteger) -> % if R has SemiGroup

from MagmaWithUnit

^: (%, PositiveInteger) -> % if R has SemiGroup

from Magma

~=: (%, %) -> Boolean

from BasicType

algtower: % -> List K if R has IntegralDomain

algtower(f) is algtower([f])

algtower: List % -> List K if R has IntegralDomain

algtower([f1, ..., fn]) returns list of kernels [ak1, ..., akl] such that each toplevel algebraic kernel in one of f1, …, fn or in arguments of ak1, …, akl is one of ak1, …, akl.

annihilate?: (%, %) -> Boolean if R has Ring

from Rng

antiCommutator: (%, %) -> % if R has Ring

from NonAssociativeSemiRng

applyQuote: (Symbol, %) -> %

applyQuote(foo, x) returns 'foo(x).

applyQuote: (Symbol, %, %) -> %

applyQuote(foo, x, y) returns 'foo(x, y).

applyQuote: (Symbol, %, %, %) -> %

applyQuote(foo, x, y, z) returns 'foo(x, y, z).

applyQuote: (Symbol, %, %, %, %) -> %

applyQuote(foo, x, y, z, t) returns 'foo(x, y, z, t).

applyQuote: (Symbol, List %) -> %

applyQuote(foo, [x1, ..., xn]) returns 'foo(x1, ..., xn).

associates?: (%, %) -> Boolean if R has IntegralDomain

from EntireRing

associator: (%, %, %) -> % if R has Ring

from NonAssociativeRng

belong?: BasicOperator -> Boolean

from ExpressionSpace2 K

box: % -> %

from ExpressionSpace2 K

characteristic: () -> NonNegativeInteger if R has Ring

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero

from CharacteristicNonZero

coerce: % -> % if R has IntegralDomain

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> % if R has IntegralDomain or R has RetractableTo Fraction Integer

from CoercibleFrom Fraction Integer

coerce: Fraction Polynomial Fraction R -> % if R has IntegralDomain

coerce(f) returns f as an element of %.

coerce: Fraction Polynomial R -> % if R has IntegralDomain

from CoercibleFrom Fraction Polynomial R

coerce: Fraction R -> % if R has IntegralDomain

coerce(q) returns q as an element of %.

coerce: Integer -> % if R has RetractableTo Integer or R has Ring

from NonAssociativeRing

coerce: K -> %

from CoercibleFrom K

coerce: Polynomial Fraction R -> % if R has IntegralDomain

coerce(p) returns p as an element of %.

coerce: Polynomial R -> % if R has Ring

from CoercibleFrom Polynomial R

coerce: R -> %

from Algebra R

coerce: SparseMultivariatePolynomial(R, K) -> % if R has Ring

coerce(p) returns p as an element of %.

coerce: Symbol -> %

from CoercibleFrom Symbol

commutator: (%, %) -> % if R has Ring or R has Group

from NonAssociativeRng

conjugate: (%, %) -> % if R has Group

from Group

convert: % -> InputForm if R has ConvertibleTo InputForm

from ConvertibleTo InputForm

convert: % -> Pattern Float if R has ConvertibleTo Pattern Float

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer

from ConvertibleTo Pattern Integer

convert: Factored % -> % if R has IntegralDomain

convert(f1\^e1 ... fm\^em) returns (f1)\^e1 ... (fm)\^em as an element of %, using formal kernels created using a paren.

D: (%, List Symbol) -> % if R has Ring

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if R has Ring

from PartialDifferentialRing Symbol

D: (%, Symbol) -> % if R has Ring

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if R has Ring

from PartialDifferentialRing Symbol

definingPolynomial: % -> % if % has Ring

from ExpressionSpace2 K

denom: % -> SparseMultivariatePolynomial(R, K) if R has IntegralDomain

denom(f) returns the denominator of f viewed as a polynomial in the kernels over R.

denominator: % -> % if R has IntegralDomain

denominator(f) returns the denominator of f converted to %.

differentiate: (%, List Symbol) -> % if R has Ring

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has Ring

from PartialDifferentialRing Symbol

differentiate: (%, Symbol) -> % if R has Ring

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if R has Ring

from PartialDifferentialRing Symbol

distribute: % -> %

from ExpressionSpace2 K

distribute: (%, %) -> %

from ExpressionSpace2 K

divide: (%, %) -> Record(quotient: %, remainder: %) if R has IntegralDomain

from EuclideanDomain

elt: (BasicOperator, %) -> %

from ExpressionSpace2 K

elt: (BasicOperator, %, %) -> %

from ExpressionSpace2 K

elt: (BasicOperator, %, %, %) -> %

from ExpressionSpace2 K

elt: (BasicOperator, %, %, %, %) -> %

from ExpressionSpace2 K

elt: (BasicOperator, %, %, %, %, %) -> %

from ExpressionSpace2 K

elt: (BasicOperator, %, %, %, %, %, %) -> %

from ExpressionSpace2 K

elt: (BasicOperator, %, %, %, %, %, %, %) -> %

from ExpressionSpace2 K

elt: (BasicOperator, %, %, %, %, %, %, %, %) -> %

from ExpressionSpace2 K

elt: (BasicOperator, %, %, %, %, %, %, %, %, %) -> %

from ExpressionSpace2 K

elt: (BasicOperator, List %) -> %

from ExpressionSpace2 K

euclideanSize: % -> NonNegativeInteger if R has IntegralDomain

from EuclideanDomain

eval: (%, %, %) -> %

from InnerEvalable(%, %)

eval: (%, BasicOperator, % -> %) -> %

from ExpressionSpace2 K

eval: (%, BasicOperator, %, Symbol) -> % if R has ConvertibleTo InputForm

eval(x, s, f, y) replaces every s(a) in x by f(y) with y replaced by a for any a.

eval: (%, BasicOperator, List % -> %) -> %

from ExpressionSpace2 K

eval: (%, Equation %) -> %

from Evalable %

eval: (%, K, %) -> %

from InnerEvalable(K, %)

eval: (%, List %, List %) -> %

from InnerEvalable(%, %)

eval: (%, List BasicOperator, List %, Symbol) -> % if R has ConvertibleTo InputForm

eval(x, [s1, ..., sm], [f1, ..., fm], y) replaces every si(a) in x by fi(y) with y replaced by a for any a.

eval: (%, List BasicOperator, List(% -> %)) -> %

from ExpressionSpace2 K

eval: (%, List BasicOperator, List(List % -> %)) -> %

from ExpressionSpace2 K

eval: (%, List Equation %) -> %

from Evalable %

eval: (%, List K, List %) -> %

from InnerEvalable(K, %)

eval: (%, List Symbol, List NonNegativeInteger, List(% -> %)) -> % if R has Ring

eval(x, [s1, ..., sm], [n1, ..., nm], [f1, ..., fm]) replaces every si(a)^ni in x by fi(a) for any a.

eval: (%, List Symbol, List NonNegativeInteger, List(List % -> %)) -> % if R has Ring

eval(x, [s1, ..., sm], [n1, ..., nm], [f1, ..., fm]) replaces every si(a1, ..., an)^ni in x by fi(a1, ..., an) for any a1, …, am.

eval: (%, List Symbol, List(% -> %)) -> %

from ExpressionSpace2 K

eval: (%, List Symbol, List(List % -> %)) -> %

from ExpressionSpace2 K

eval: (%, Symbol, % -> %) -> %

from ExpressionSpace2 K

eval: (%, Symbol, List % -> %) -> %

from ExpressionSpace2 K

eval: (%, Symbol, NonNegativeInteger, % -> %) -> % if R has Ring

eval(x, s, n, f) replaces every s(a)^n in x by f(a) for any a.

eval: (%, Symbol, NonNegativeInteger, List % -> %) -> % if R has Ring

eval(x, s, n, f) replaces every s(a1, ..., am)^n in x by f(a1, ..., am) for any a1, …, am.

even?: % -> Boolean if % has RetractableTo Integer

from ExpressionSpace2 K

expressIdealMember: (List %, %) -> Union(List %, failed) if R has IntegralDomain

from PrincipalIdealDomain

exquo: (%, %) -> Union(%, failed) if R has IntegralDomain

from EntireRing

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has IntegralDomain

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if R has IntegralDomain

from EuclideanDomain

factor: % -> Factored % if R has IntegralDomain

from UniqueFactorizationDomain

freeOf?: (%, %) -> Boolean

from ExpressionSpace2 K

freeOf?: (%, Symbol) -> Boolean

from ExpressionSpace2 K

gcd: (%, %) -> % if R has IntegralDomain

from GcdDomain

gcd: List % -> % if R has IntegralDomain

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has IntegralDomain

from GcdDomain

ground?: % -> Boolean

ground?(f) tests if f is an element of R.

ground: % -> R

ground(f) returns f as an element of R. An error occurs if f is not an element of R.

height: % -> NonNegativeInteger

from ExpressionSpace2 K

inv: % -> % if R has IntegralDomain or R has Group

from Group

is?: (%, BasicOperator) -> Boolean

from ExpressionSpace2 K

is?: (%, Symbol) -> Boolean

from ExpressionSpace2 K

isExpt: % -> Union(Record(var: K, exponent: Integer), failed) if R has SemiGroup

isExpt(p) returns [x, n] if p = x^n and n ~= 0.

isExpt: (%, BasicOperator) -> Union(Record(var: K, exponent: Integer), failed) if R has Ring

isExpt(p, op) returns [x, n] if p = x^n and n ~= 0 and x = op(a).

isExpt: (%, Symbol) -> Union(Record(var: K, exponent: Integer), failed) if R has Ring

isExpt(p, f) returns [x, n] if p = x^n and n ~= 0 and x = f(a).

isMult: % -> Union(Record(coef: Integer, var: K), failed) if R has AbelianSemiGroup

isMult(p) returns [n, x] if p = n * x and n ~= 0.

isPlus: % -> Union(List %, failed) if R has AbelianSemiGroup

isPlus(p) returns [m1, ..., mn] if p = m1 +...+ mn and n > 1.

isPower: % -> Union(Record(val: %, exponent: Integer), failed) if R has Ring

isPower(p) returns [x, n] if p = x^n and n ~= 0.

isTimes: % -> Union(List %, failed) if R has SemiGroup

isTimes(p) returns [a1, ..., an] if p = a1*...*an and n > 1.

kernel: (BasicOperator, %) -> %

from ExpressionSpace2 K

kernel: (BasicOperator, List %) -> %

from ExpressionSpace2 K

kernels: % -> List K

from ExpressionSpace2 K

kernels: List % -> List K

from ExpressionSpace2 K

latex: % -> String

from SetCategory

lcm: (%, %) -> % if R has IntegralDomain

from GcdDomain

lcm: List % -> % if R has IntegralDomain

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has IntegralDomain

from LeftOreRing

leftPower: (%, NonNegativeInteger) -> % if R has SemiGroup

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> % if R has SemiGroup

from Magma

leftRecip: % -> Union(%, failed) if R has SemiGroup

from MagmaWithUnit

mainKernel: % -> Union(K, failed)

from ExpressionSpace2 K

map: (% -> %, K) -> %

from ExpressionSpace2 K

minPoly: K -> SparseUnivariatePolynomial % if % has Ring

from ExpressionSpace2 K

multiEuclidean: (List %, %) -> Union(List %, failed) if R has IntegralDomain

from EuclideanDomain

numer: % -> SparseMultivariatePolynomial(R, K) if R has Ring

numer(f) returns the numerator of f viewed as a polynomial in the kernels over R if R is an integral domain. If not, then numer(f) = f viewed as a polynomial in the kernels over R.

numerator: % -> % if R has Ring

numerator(f) returns the numerator of f converted to %.

odd?: % -> Boolean if % has RetractableTo Integer

from ExpressionSpace2 K

one?: % -> Boolean if R has SemiGroup

from MagmaWithUnit

operator: BasicOperator -> BasicOperator

from ExpressionSpace2 K

operators: % -> List BasicOperator

from ExpressionSpace2 K

opposite?: (%, %) -> Boolean if R has AbelianSemiGroup

from AbelianMonoid

paren: % -> %

from ExpressionSpace2 K

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if R has PatternMatchable Float

from PatternMatchable Float

patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has PatternMatchable Integer

from PatternMatchable Integer

plenaryPower: (%, PositiveInteger) -> % if R has CommutativeRing

from NonAssociativeAlgebra R

prime?: % -> Boolean if R has IntegralDomain

from UniqueFactorizationDomain

principalIdeal: List % -> Record(coef: List %, generator: %) if R has IntegralDomain

from PrincipalIdealDomain

quo: (%, %) -> % if R has IntegralDomain

from EuclideanDomain

recip: % -> Union(%, failed) if R has SemiGroup

from MagmaWithUnit

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has Ring and R has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R) if R has Ring

from LinearlyExplicitOver R

reducedSystem: Matrix % -> Matrix Integer if R has Ring and R has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix R if R has Ring

from LinearlyExplicitOver R

rem: (%, %) -> % if R has IntegralDomain

from EuclideanDomain

retract: % -> Fraction Integer if R has RetractableTo Integer and R has IntegralDomain or R has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retract: % -> Fraction Polynomial R if R has IntegralDomain

from RetractableTo Fraction Polynomial R

retract: % -> Integer if R has RetractableTo Integer

from RetractableTo Integer

retract: % -> K

from RetractableTo K

retract: % -> Polynomial R if R has Ring

from RetractableTo Polynomial R

retract: % -> R

from RetractableTo R

retract: % -> Symbol

from RetractableTo Symbol

retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Integer and R has IntegralDomain or R has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Fraction Polynomial R, failed) if R has IntegralDomain

from RetractableTo Fraction Polynomial R

retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer

from RetractableTo Integer

retractIfCan: % -> Union(K, failed)

from RetractableTo K

retractIfCan: % -> Union(Polynomial R, failed) if R has Ring

from RetractableTo Polynomial R

retractIfCan: % -> Union(R, failed)

from RetractableTo R

retractIfCan: % -> Union(Symbol, failed)

from RetractableTo Symbol

rightPower: (%, NonNegativeInteger) -> % if R has SemiGroup

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> % if R has SemiGroup

from Magma

rightRecip: % -> Union(%, failed) if R has SemiGroup

from MagmaWithUnit

sample: % if R has SemiGroup or R has AbelianSemiGroup

from AbelianMonoid

sizeLess?: (%, %) -> Boolean if R has IntegralDomain

from EuclideanDomain

smaller?: (%, %) -> Boolean

from Comparable

squareFree: % -> Factored % if R has IntegralDomain

from UniqueFactorizationDomain

squareFreePart: % -> % if R has IntegralDomain

from UniqueFactorizationDomain

subst: (%, Equation %) -> %

from ExpressionSpace2 K

subst: (%, List Equation %) -> %

from ExpressionSpace2 K

subst: (%, List K, List %) -> %

from ExpressionSpace2 K

subtractIfCan: (%, %) -> Union(%, failed) if R has AbelianGroup

from CancellationAbelianMonoid

tower: % -> List K

from ExpressionSpace2 K

tower: List % -> List K

from ExpressionSpace2 K

unit?: % -> Boolean if R has IntegralDomain

from EntireRing

unitCanonical: % -> % if R has IntegralDomain

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has IntegralDomain

from EntireRing

univariate: (%, K) -> Fraction SparseUnivariatePolynomial % if R has IntegralDomain

univariate(f, k) returns f viewed as a univariate fraction in k.

variables: % -> List Symbol

variables(f) returns the list of all the variables of f.

variables: List % -> List Symbol

variables([f1, ..., fn]) returns the list of all the variables of f1, …, fn.

zero?: % -> Boolean if R has AbelianSemiGroup

from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoid if R has AbelianSemiGroup

AbelianSemiGroup if R has AbelianSemiGroup

Algebra % if R has IntegralDomain

Algebra Fraction Integer if R has IntegralDomain

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %) if R has Ring

BiModule(Fraction Integer, Fraction Integer) if R has IntegralDomain

BiModule(R, R) if R has CommutativeRing

CancellationAbelianMonoid if R has AbelianGroup

canonicalsClosed if R has IntegralDomain

canonicalUnitNormal if R has IntegralDomain

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleFrom Fraction Integer if R has RetractableTo Integer and R has IntegralDomain or R has RetractableTo Fraction Integer

CoercibleFrom Fraction Polynomial R if R has IntegralDomain

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom K

CoercibleFrom Polynomial R if R has Ring

CoercibleFrom R

CoercibleFrom Symbol

CoercibleTo OutputForm

CommutativeRing if R has IntegralDomain

CommutativeStar if R has IntegralDomain

Comparable

ConvertibleTo InputForm if R has ConvertibleTo InputForm

ConvertibleTo Pattern Float if R has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has ConvertibleTo Pattern Integer

DivisionRing if R has IntegralDomain

EntireRing if R has IntegralDomain

EuclideanDomain if R has IntegralDomain

Evalable %

ExpressionSpace2 K

Field if R has IntegralDomain

FullyLinearlyExplicitOver R if R has Ring

FullyPatternMatchable R

FullyRetractableTo R

GcdDomain if R has IntegralDomain

Group if R has Group

InnerEvalable(%, %)

InnerEvalable(K, %)

IntegralDomain if R has IntegralDomain

LeftModule % if R has Ring

LeftModule Fraction Integer if R has IntegralDomain

LeftModule R if R has CommutativeRing

LeftOreRing if R has IntegralDomain

LinearlyExplicitOver Integer if R has Ring and R has LinearlyExplicitOver Integer

LinearlyExplicitOver R if R has Ring

Magma if R has SemiGroup

MagmaWithUnit if R has SemiGroup

Module % if R has IntegralDomain

Module Fraction Integer if R has IntegralDomain

Module R if R has CommutativeRing

Monoid if R has SemiGroup

NonAssociativeAlgebra % if R has IntegralDomain

NonAssociativeAlgebra Fraction Integer if R has IntegralDomain

NonAssociativeAlgebra R if R has CommutativeRing

NonAssociativeRing if R has Ring

NonAssociativeRng if R has Ring

NonAssociativeSemiRing if R has Ring

NonAssociativeSemiRng if R has Ring

noZeroDivisors if R has IntegralDomain

PartialDifferentialRing Symbol if R has Ring

Patternable R

PatternMatchable Float if R has PatternMatchable Float

PatternMatchable Integer if R has PatternMatchable Integer

PrincipalIdealDomain if R has IntegralDomain

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer or R has RetractableTo Integer and R has IntegralDomain

RetractableTo Fraction Polynomial R if R has IntegralDomain

RetractableTo Integer if R has RetractableTo Integer

RetractableTo K

RetractableTo Polynomial R if R has Ring

RetractableTo R

RetractableTo Symbol

RightModule % if R has Ring

RightModule Fraction Integer if R has IntegralDomain

RightModule Integer if R has Ring and R has LinearlyExplicitOver Integer

RightModule R if R has Ring

Ring if R has Ring

Rng if R has Ring

SemiGroup if R has SemiGroup

SemiRing if R has Ring

SemiRng if R has Ring

SetCategory

TwoSidedRecip if R has IntegralDomain or R has Group

UniqueFactorizationDomain if R has IntegralDomain

unitsKnown if R has Ring or R has Group