FramedAlgebra(R, UP)

algcat.spad line 149 [edit on github]

• R : CommutativeRing
• UP : UnivariatePolynomialCategory(R)

A FramedAlgebra is a FiniteRankAlgebra together with a fixed R-module basis.

* : (%, %) -> %

from Magma

* : (%, R) -> %

from RightModule(R)

* : (R, %) -> %

from LeftModule(R)

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associator : (%, %, %) -> %

from NonAssociativeRng

basis : () -> Vector(%)

from FramedModule(R)

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

characteristicPolynomial : % -> UP

from FiniteRankAlgebra(R, UP)

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

from CharacteristicNonZero

coerce : R -> %

from Algebra(R)

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

convert : Vector(R) -> %

from FramedModule(R)

convert : % -> InputForm if R has Finite

from ConvertibleTo(InputForm)

convert : % -> Vector(R)

from FramedModule(R)

coordinates : Vector(%) -> Matrix(R)

from FramedModule(R)

coordinates : (Vector(%), Vector(%)) -> Matrix(R)

from FiniteRankAlgebra(R, UP)

coordinates : % -> Vector(R)

from FramedModule(R)

coordinates : (%, Vector(%)) -> Vector(R)

from FiniteRankAlgebra(R, UP)

discriminant : () -> R

discriminant() = determinant(traceMatrix()).

discriminant : Vector(%) -> R

from FiniteRankAlgebra(R, UP)

enumerate : () -> List(%) if R has Finite

from Finite

hash : % -> SingleInteger if R has Hashable

from Hashable

hashUpdate! : (HashState, %) -> HashState if R has Hashable

from Hashable

index : PositiveInteger -> % if R has Finite

from Finite

latex : % -> String

from SetCategory

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

lookup : % -> PositiveInteger if R has Finite

from Finite

minimalPolynomial : % -> UP if R has Field

from FiniteRankAlgebra(R, UP)

norm : % -> R

from FiniteRankAlgebra(R, UP)

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(R)

random : () -> % if R has Finite

from Finite

rank : () -> PositiveInteger

from FiniteRankAlgebra(R, UP)

recip : % -> Union(%, "failed")

from MagmaWithUnit

regularRepresentation : % -> Matrix(R)

regularRepresentation(a) returns the matrix m of the linear map defined by left multiplication by a with respect to the fixed basis. That is for all x we have coordinates(a*x) = m*coordinates(x).

regularRepresentation : (%, Vector(%)) -> Matrix(R)

from FiniteRankAlgebra(R, UP)

represents : Vector(R) -> %

from FramedModule(R)

represents : (Vector(R), Vector(%)) -> %

from FiniteRankAlgebra(R, UP)

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

size : () -> NonNegativeInteger if R has Finite

from Finite

smaller? : (%, %) -> Boolean if R has Finite

from Comparable

subtractIfCan : (%, %) -> Union(%, "failed")

from CancellationAbelianMonoid

trace : % -> R

from FiniteRankAlgebra(R, UP)

traceMatrix : () -> Matrix(R)

traceMatrix() is the n-by-n matrix ( Tr(vi * vj) ), where v1, ..., vn are the elements of the fixed basis.

traceMatrix : Vector(%) -> Matrix(R)

from FiniteRankAlgebra(R, UP)

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

CharacteristicNonZero

Comparable

ConvertibleTo(InputForm)

Algebra(R)

RightModule(%)

Monoid

AbelianMonoid

BiModule(R, R)

NonAssociativeSemiRng

NonAssociativeAlgebra(R)

FiniteRankAlgebra(R, UP)

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

LeftModule(%)

LeftModule(R)

Finite

SetCategory

CoercibleTo(OutputForm)

Rng

FramedModule(R)

TwoSidedRecip

Magma

SemiGroup

BiModule(%, %)

unitsKnown

AbelianGroup

AbelianSemiGroup

NonAssociativeSemiRing

RightModule(R)

Module(R)

CharacteristicZero

NonAssociativeRng

Ring

SemiRng

Hashable

BasicType

SemiRing