OctonionCategory RΒΆ

oct.spad line 1

OctonionCategory gives the categorial frame for the octonions, and eight-dimensional non-associative algebra, doubling the the quaternions in the same way as doubling the Complex numbers to get the quaternions.

0: %
from AbelianMonoid
1: % if R has CharacteristicNonZero or R has CharacteristicZero
from MagmaWithUnit
*: (%, %) -> %
from Magma
*: (%, R) -> %
from RightModule R
*: (Integer, %) -> %
from AbelianGroup
*: (NonNegativeInteger, %) -> %
from AbelianMonoid
*: (PositiveInteger, %) -> %
from AbelianSemiGroup
*: (R, %) -> %
from LeftModule R
+: (%, %) -> %
from AbelianSemiGroup
-: % -> %
from AbelianGroup
-: (%, %) -> %
from AbelianGroup
<: (%, %) -> Boolean if R has OrderedSet
from PartialOrder
<=: (%, %) -> Boolean if R has OrderedSet
from PartialOrder
=: (%, %) -> Boolean
from BasicType
>: (%, %) -> Boolean if R has OrderedSet
from PartialOrder
>=: (%, %) -> Boolean if R has OrderedSet
from PartialOrder
^: (%, NonNegativeInteger) -> % if R has CharacteristicNonZero or R has CharacteristicZero
from MagmaWithUnit
^: (%, PositiveInteger) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
abs: % -> R if R has RealNumberSystem
abs(o) computes the absolute value of an octonion, equal to the square root of the norm.
alternative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
annihilate?: (%, %) -> Boolean if R has CharacteristicNonZero or R has CharacteristicZero
from Rng
antiAssociative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
antiCommutative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
antiCommutator: (%, %) -> %
from NonAssociativeSemiRng
apply: (Matrix R, %) -> %
from FramedNonAssociativeAlgebra R
associative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
associator: (%, %, %) -> %
from NonAssociativeRng
associatorDependence: () -> List Vector R if R has IntegralDomain
from FiniteRankNonAssociativeAlgebra R
basis: () -> Vector %
from FramedModule R
characteristic: () -> NonNegativeInteger if R has CharacteristicNonZero or R has CharacteristicZero
from NonAssociativeRing
charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
from CharacteristicNonZero
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
coerce: Integer -> % if R has CharacteristicZero or R has CharacteristicNonZero or R has RetractableTo Integer
from NonAssociativeRing
coerce: R -> %
from RetractableTo R
commutative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
commutator: (%, %) -> %
from NonAssociativeRng
conditionsForIdempotents: () -> List Polynomial R
from FramedNonAssociativeAlgebra R
conditionsForIdempotents: Vector % -> List Polynomial R
from FiniteRankNonAssociativeAlgebra R
conjugate: % -> %
conjugate(o) negates the imaginary parts i, j, k, E, I, J, K of octonian o.
convert: % -> InputForm if R has ConvertibleTo InputForm
from ConvertibleTo InputForm
convert: % -> Vector R
from FramedModule R
convert: Vector R -> %
from FramedModule R
coordinates: % -> Vector R
from FramedModule R
coordinates: (%, Vector %) -> Vector R
from FiniteRankNonAssociativeAlgebra R
coordinates: (Vector %, Vector %) -> Matrix R
from FiniteRankNonAssociativeAlgebra R
coordinates: Vector % -> Matrix R
from FramedModule R
elt: (%, Integer) -> R
from FramedNonAssociativeAlgebra R
elt: (%, R) -> % if R has Eltable(R, R)
from Eltable(R, %)
enumerate: () -> List % if R has Finite
from Finite
eval: (%, Equation R) -> % if R has Evalable R
from Evalable R
eval: (%, List Equation R) -> % if R has Evalable R
from Evalable R
eval: (%, List R, List R) -> % if R has Evalable R
from InnerEvalable(R, R)
eval: (%, List Symbol, List R) -> % if R has InnerEvalable(Symbol, R)
from InnerEvalable(Symbol, R)
eval: (%, R, R) -> % if R has Evalable R
from InnerEvalable(R, R)
eval: (%, Symbol, R) -> % if R has InnerEvalable(Symbol, R)
from InnerEvalable(Symbol, R)
flexible?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
imagE: % -> R
imagE(o) extracts the imaginary E part of octonion o.
imagi: % -> R
imagi(o) extracts the i part of octonion o.
imagI: % -> R
imagI(o) extracts the imaginary I part of octonion o.
imagj: % -> R
imagj(o) extracts the j part of octonion o.
imagJ: % -> R
imagJ(o) extracts the imaginary J part of octonion o.
imagk: % -> R
imagk(o) extracts the k part of octonion o.
imagK: % -> R
imagK(o) extracts the imaginary K part of octonion o.
index: PositiveInteger -> % if R has Finite
from Finite
inv: % -> % if R has Field
inv(o) returns the inverse of o if it exists.
jacobiIdentity?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
jordanAdmissible?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
jordanAlgebra?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
latex: % -> String
from SetCategory
leftAlternative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
leftCharacteristicPolynomial: % -> SparseUnivariatePolynomial R
from FiniteRankNonAssociativeAlgebra R
leftDiscriminant: () -> R
from FramedNonAssociativeAlgebra R
leftDiscriminant: Vector % -> R
from FiniteRankNonAssociativeAlgebra R
leftMinimalPolynomial: % -> SparseUnivariatePolynomial R if R has IntegralDomain
from FiniteRankNonAssociativeAlgebra R
leftNorm: % -> R
from FiniteRankNonAssociativeAlgebra R
leftPower: (%, NonNegativeInteger) -> % if R has CharacteristicNonZero or R has CharacteristicZero
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> %
from Magma
leftRankPolynomial: () -> SparseUnivariatePolynomial Polynomial R if R has Field
from FramedNonAssociativeAlgebra R
leftRecip: % -> Union(%, failed) if R has IntegralDomain or R has CharacteristicNonZero or R has CharacteristicZero
from FiniteRankNonAssociativeAlgebra R
leftRegularRepresentation: % -> Matrix R
from FramedNonAssociativeAlgebra R
leftRegularRepresentation: (%, Vector %) -> Matrix R
from FiniteRankNonAssociativeAlgebra R
leftTrace: % -> R
from FiniteRankNonAssociativeAlgebra R
leftTraceMatrix: () -> Matrix R
from FramedNonAssociativeAlgebra R
leftTraceMatrix: Vector % -> Matrix R
from FiniteRankNonAssociativeAlgebra R
leftUnit: () -> Union(%, failed) if R has IntegralDomain
from FiniteRankNonAssociativeAlgebra R
leftUnits: () -> Union(Record(particular: %, basis: List %), failed) if R has IntegralDomain
from FiniteRankNonAssociativeAlgebra R
lieAdmissible?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
lieAlgebra?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
lookup: % -> PositiveInteger if R has Finite
from Finite
map: (R -> R, %) -> %
from FullyEvalableOver R
max: (%, %) -> % if R has OrderedSet
from OrderedSet
min: (%, %) -> % if R has OrderedSet
from OrderedSet
noncommutativeJordanAlgebra?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
norm: % -> R
norm(o) returns the norm of an octonion, equal to the sum of the squares of its coefficients.
octon: (R, R, R, R, R, R, R, R) -> %
octon(re, ri, rj, rk, rE, rI, rJ, rK) constructs an octonion from scalars.
one?: % -> Boolean if R has CharacteristicNonZero or R has CharacteristicZero
from MagmaWithUnit
opposite?: (%, %) -> Boolean
from AbelianMonoid
plenaryPower: (%, PositiveInteger) -> %
from NonAssociativeAlgebra R
powerAssociative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
random: () -> % if R has Finite
from Finite
rank: () -> PositiveInteger
from FramedModule R
rational: % -> Fraction Integer if R has IntegerNumberSystem
rational(o) returns the real part if all seven imaginary parts are 0. Error: if o is not rational.
rational?: % -> Boolean if R has IntegerNumberSystem
rational?(o) tests if o is rational, i.e. that all seven imaginary parts are 0.
rationalIfCan: % -> Union(Fraction Integer, failed) if R has IntegerNumberSystem
rationalIfCan(o) returns the real part if all seven imaginary parts are 0, and “failed” otherwise.
real: % -> R
real(o) extracts real part of octonion o.
recip: % -> Union(%, failed) if R has IntegralDomain or R has CharacteristicNonZero or R has CharacteristicZero
from FiniteRankNonAssociativeAlgebra R
represents: (Vector R, Vector %) -> %
from FiniteRankNonAssociativeAlgebra R
represents: Vector R -> %
from FramedModule R
retract: % -> Fraction Integer if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retract: % -> Integer if R has RetractableTo Integer
from RetractableTo Integer
retract: % -> R
from RetractableTo R
retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer
from RetractableTo Integer
retractIfCan: % -> Union(R, failed)
from RetractableTo R
rightAlternative?: () -> Boolean
from FiniteRankNonAssociativeAlgebra R
rightCharacteristicPolynomial: % -> SparseUnivariatePolynomial R
from FiniteRankNonAssociativeAlgebra R
rightDiscriminant: () -> R
from FramedNonAssociativeAlgebra R
rightDiscriminant: Vector % -> R
from FiniteRankNonAssociativeAlgebra R
rightMinimalPolynomial: % -> SparseUnivariatePolynomial R if R has IntegralDomain
from FiniteRankNonAssociativeAlgebra R
rightNorm: % -> R
from FiniteRankNonAssociativeAlgebra R
rightPower: (%, NonNegativeInteger) -> % if R has CharacteristicNonZero or R has CharacteristicZero
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> %
from Magma
rightRankPolynomial: () -> SparseUnivariatePolynomial Polynomial R if R has Field
from FramedNonAssociativeAlgebra R
rightRecip: % -> Union(%, failed) if R has IntegralDomain or R has CharacteristicNonZero or R has CharacteristicZero
from FiniteRankNonAssociativeAlgebra R
rightRegularRepresentation: % -> Matrix R
from FramedNonAssociativeAlgebra R
rightRegularRepresentation: (%, Vector %) -> Matrix R
from FiniteRankNonAssociativeAlgebra R
rightTrace: % -> R
from FiniteRankNonAssociativeAlgebra R
rightTraceMatrix: () -> Matrix R
from FramedNonAssociativeAlgebra R
rightTraceMatrix: Vector % -> Matrix R
from FiniteRankNonAssociativeAlgebra R
rightUnit: () -> Union(%, failed) if R has IntegralDomain
from FiniteRankNonAssociativeAlgebra R
rightUnits: () -> Union(Record(particular: %, basis: List %), failed) if R has IntegralDomain
from FiniteRankNonAssociativeAlgebra R
sample: %
from AbelianMonoid
size: () -> NonNegativeInteger if R has Finite
from Finite
smaller?: (%, %) -> Boolean if R has OrderedSet or R has Finite
from Comparable
someBasis: () -> Vector %
from FiniteRankNonAssociativeAlgebra R
structuralConstants: () -> Vector Matrix R
from FramedNonAssociativeAlgebra R
structuralConstants: Vector % -> Vector Matrix R
from FiniteRankNonAssociativeAlgebra R
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
unit: () -> Union(%, failed) if R has IntegralDomain
from FiniteRankNonAssociativeAlgebra R
zero?: % -> Boolean
from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %) if R has CharacteristicNonZero or R has CharacteristicZero

BiModule(R, R)

CancellationAbelianMonoid

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleTo OutputForm

Comparable if R has OrderedSet or R has Finite

ConvertibleTo InputForm if R has ConvertibleTo InputForm

Eltable(R, %) if R has Eltable(R, R)

Evalable R if R has Evalable R

Finite if R has Finite

FiniteRankNonAssociativeAlgebra R

FramedModule R

FramedNonAssociativeAlgebra R

FullyEvalableOver R

FullyRetractableTo R

InnerEvalable(R, R) if R has Evalable R

InnerEvalable(Symbol, R) if R has InnerEvalable(Symbol, R)

LeftModule % if R has CharacteristicNonZero or R has CharacteristicZero

LeftModule R

Magma

MagmaWithUnit if R has CharacteristicNonZero or R has CharacteristicZero

Module R

Monoid if R has CharacteristicNonZero or R has CharacteristicZero

NonAssociativeAlgebra R

NonAssociativeRing if R has CharacteristicNonZero or R has CharacteristicZero

NonAssociativeRng

NonAssociativeSemiRing if R has CharacteristicNonZero or R has CharacteristicZero

NonAssociativeSemiRng

OrderedSet if R has OrderedSet

PartialOrder if R has OrderedSet

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer

RetractableTo Integer if R has RetractableTo Integer

RetractableTo R

RightModule % if R has CharacteristicNonZero or R has CharacteristicZero

RightModule R

Ring if R has CharacteristicNonZero or R has CharacteristicZero

Rng if R has CharacteristicNonZero or R has CharacteristicZero

SemiGroup if R has CharacteristicNonZero or R has CharacteristicZero

SemiRing if R has CharacteristicNonZero or R has CharacteristicZero

SemiRng if R has CharacteristicNonZero or R has CharacteristicZero

SetCategory

unitsKnown if R has IntegralDomain or R has CharacteristicNonZero or R has CharacteristicZero