# FunctionSpace R¶

fspace.spad line 388 [edit on github]

R: Comparable

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 LinearlyExplicitOver Integer and R has Ring
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, Kernel %), SparseMultivariatePolynomial(R, Kernel %)) -> % if R has IntegralDomain
`p1/p2`

returns the quotient of`p1`

and`p2`

as an element of %.

- ^: (%, 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

- algtower: % -> List Kernel % if R has IntegralDomain
`algtower(f)`

is algtower([`f`

])

- algtower: List % -> List Kernel % 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

- 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)`

.

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

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

- belong?: BasicOperator -> Boolean
from ExpressionSpace

- box: % -> %
from ExpressionSpace

- characteristic: () -> NonNegativeInteger if R has Ring
from NonAssociativeRing

- charthRoot: % -> Union(%, failed) if R has 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 Ring or R has RetractableTo Integer
from NonAssociativeRing

- coerce: Kernel % -> %
from CoercibleFrom Kernel %

- 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, Kernel %) -> % 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

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

- definingPolynomial: % -> % if % has Ring
from ExpressionSpace

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

- distribute: % -> %
from ExpressionSpace

- distribute: (%, %) -> %
from ExpressionSpace

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

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

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

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

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

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

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

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

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

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

- elt: (BasicOperator, List %) -> %
from ExpressionSpace

- euclideanSize: % -> NonNegativeInteger if R has IntegralDomain
from EuclideanDomain

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

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

- 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 ExpressionSpace

- eval: (%, Equation %) -> %
from Evalable %

- eval: (%, Kernel %, %) -> %
from InnerEvalable(Kernel %, %)

- 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 ExpressionSpace

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

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

- eval: (%, List Kernel %, List %) -> %
from InnerEvalable(Kernel %, %)

- 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 ExpressionSpace

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

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

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

- 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 ExpressionSpace

- 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

- freeOf?: (%, %) -> Boolean
from ExpressionSpace

- freeOf?: (%, Symbol) -> Boolean
from ExpressionSpace

- 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 ExpressionSpace

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

- is?: (%, BasicOperator) -> Boolean
from ExpressionSpace

- is?: (%, Symbol) -> Boolean
from ExpressionSpace

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

returns`[x, n]`

if`p = x^n`

and`n ~= 0`

.

- isExpt: (%, BasicOperator) -> Union(Record(var: Kernel %, 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: Kernel %, 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: Kernel %), 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 ExpressionSpace

- kernel: (BasicOperator, List %) -> %
from ExpressionSpace

- kernels: % -> List Kernel %
from ExpressionSpace

- kernels: List % -> List Kernel %
from ExpressionSpace

- 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(Kernel %, failed)
from ExpressionSpace

- map: (% -> %, Kernel %) -> %
from ExpressionSpace

- minPoly: Kernel % -> SparseUnivariatePolynomial % if % has Ring
from ExpressionSpace

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

- numer: % -> SparseMultivariatePolynomial(R, Kernel %) 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 ExpressionSpace

- one?: % -> Boolean if R has SemiGroup
from MagmaWithUnit

- operators: % -> List BasicOperator
from ExpressionSpace

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

- paren: % -> %
from ExpressionSpace

- 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

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

- reducedSystem: Matrix % -> Matrix Integer if R has LinearlyExplicitOver Integer and R has Ring
- 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 Fraction Integer or R has RetractableTo Integer and R has IntegralDomain
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: % -> Kernel %
from RetractableTo Kernel %

- 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 Fraction Integer or R has RetractableTo Integer and R has IntegralDomain
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(Kernel %, failed)
from RetractableTo Kernel %

- 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

- squareFreePart: % -> % if R has IntegralDomain

- subst: (%, Equation %) -> %
from ExpressionSpace

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

- subst: (%, List Kernel %, List %) -> %
from ExpressionSpace

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

- tower: % -> List Kernel %
from ExpressionSpace

- tower: List % -> List Kernel %
from ExpressionSpace

- 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: (%, Kernel %) -> Fraction SparseUnivariatePolynomial % if R has IntegralDomain
`univariate(f, k)`

returns`f`

viewed as a univariate fraction in`k`

.

- 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

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 Fraction Integer or R has RetractableTo Integer and R has IntegralDomain

CoercibleFrom Fraction Polynomial R if R has IntegralDomain

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom Polynomial R if R has Ring

CommutativeRing if R has IntegralDomain

CommutativeStar if R has IntegralDomain

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 %

Field if R has IntegralDomain

FullyLinearlyExplicitOver R if R has Ring

GcdDomain if R has IntegralDomain

InnerEvalable(%, %)

InnerEvalable(Kernel %, %)

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 LinearlyExplicitOver Integer and R has Ring

LinearlyExplicitOver R if R has Ring

MagmaWithUnit if R has SemiGroup

Module % if R has IntegralDomain

Module Fraction Integer if R has IntegralDomain

Module R if R has CommutativeRing

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

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 Polynomial R if R has Ring

RightModule % if R has Ring

RightModule Fraction Integer if R has IntegralDomain

RightModule Integer if R has LinearlyExplicitOver Integer and R has Ring

RightModule R if R has Ring

TwoSidedRecip if R has IntegralDomain or R has Group

UniqueFactorizationDomain if R has IntegralDomain

unitsKnown if R has Ring or R has Group