DirectProductCategory(dim, R)ΒΆ

vector.spad line 244

attribute to indicate an aggregate of finite size

0: % if R has AbelianMonoid
from AbelianMonoid
1: % if R has Monoid
from MagmaWithUnit
*: (%, %) -> % if R has SemiGroup
from Magma
*: (%, R) -> % if R has SemiGroup
y * r multiplies each component of the vector y by the element r.
*: (Integer, %) -> % if R has AbelianGroup
from AbelianGroup
*: (NonNegativeInteger, %) -> % if R has AbelianMonoid
from AbelianMonoid
*: (PositiveInteger, %) -> % if R has AbelianMonoid or R has SemiRng
from AbelianSemiGroup
*: (R, %) -> % if R has SemiGroup
r * y multiplies the element r times each component of the vector y.
+: (%, %) -> % if R has AbelianMonoid or R has SemiRng
from AbelianSemiGroup
-: % -> % if R has AbelianGroup
from AbelianGroup
-: (%, %) -> % if R has AbelianGroup
from AbelianGroup
/: (%, R) -> % if R has Field
from VectorSpace R
<: (%, %) -> Boolean if R has OrderedSet
from PartialOrder
<=: (%, %) -> Boolean if R has OrderedSet
from PartialOrder
=: (%, %) -> Boolean if R has BasicType
from BasicType
>: (%, %) -> Boolean if R has OrderedSet
from PartialOrder
>=: (%, %) -> Boolean if R has OrderedSet
from PartialOrder
^: (%, NonNegativeInteger) -> % if R has Monoid
from MagmaWithUnit
^: (%, PositiveInteger) -> % if R has SemiGroup
from Magma
~=: (%, %) -> Boolean if R has BasicType
from BasicType
annihilate?: (%, %) -> Boolean if R has Ring
from Rng
antiCommutator: (%, %) -> % if R has SemiRng
from NonAssociativeSemiRng
associator: (%, %, %) -> % if R has Ring
from NonAssociativeRng
characteristic: () -> NonNegativeInteger if R has Ring
from NonAssociativeRing
coerce: % -> OutputForm if R has CoercibleTo OutputForm
from CoercibleTo OutputForm
coerce: % -> Vector R
from CoercibleTo Vector R
coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer and R has SetCategory
from RetractableTo Fraction Integer
coerce: Integer -> % if R has RetractableTo Integer and R has SetCategory or R has Ring
from NonAssociativeRing
coerce: R -> % if R has SetCategory
from RetractableTo R
commutator: (%, %) -> % if R has Ring
from NonAssociativeRng
convert: % -> InputForm if R has Finite
from ConvertibleTo InputForm
copy: % -> %
from Aggregate
count: (R, %) -> NonNegativeInteger if R has BasicType
from HomogeneousAggregate R
D: % -> % if R has Ring and R has DifferentialRing
from DifferentialRing
D: (%, List Symbol) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, List Symbol, List NonNegativeInteger) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, NonNegativeInteger) -> % if R has Ring and R has DifferentialRing
from DifferentialRing
D: (%, R -> R) -> % if R has Ring
from DifferentialExtension R
D: (%, R -> R, NonNegativeInteger) -> % if R has Ring
from DifferentialExtension R
D: (%, Symbol) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
D: (%, Symbol, NonNegativeInteger) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: % -> % if R has Ring and R has DifferentialRing
from DifferentialRing
differentiate: (%, List Symbol) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, NonNegativeInteger) -> % if R has Ring and R has DifferentialRing
from DifferentialRing
differentiate: (%, R -> R) -> % if R has Ring
from DifferentialExtension R
differentiate: (%, R -> R, NonNegativeInteger) -> % if R has Ring
from DifferentialExtension R
differentiate: (%, Symbol) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
differentiate: (%, Symbol, NonNegativeInteger) -> % if R has Ring and R has PartialDifferentialRing Symbol
from PartialDifferentialRing Symbol
dimension: () -> CardinalNumber if R has Field
from VectorSpace R
directProduct: Vector R -> %
directProduct(v) converts the vector v to become a direct product. Error: if the length of v is different from dim.
dot: (%, %) -> R if R has SemiRng and R has AbelianMonoid
dot(x, y) computes the inner product of the vectors x and y.
elt: (%, Integer) -> R
from Eltable(Integer, R)
elt: (%, Integer, R) -> R
from EltableAggregate(Integer, R)
empty: () -> %
from Aggregate
empty?: % -> Boolean
from Aggregate
entries: % -> List R
from IndexedAggregate(Integer, R)
entry?: (R, %) -> Boolean if R has BasicType
from IndexedAggregate(Integer, R)
enumerate: () -> List % if R has Finite
from Finite
eq?: (%, %) -> Boolean
from Aggregate
eval: (%, Equation R) -> % if R has SetCategory and R has Evalable R
from Evalable R
eval: (%, List Equation R) -> % if R has SetCategory and R has Evalable R
from Evalable R
eval: (%, List R, List R) -> % if R has SetCategory and R has Evalable R
from InnerEvalable(R, R)
eval: (%, R, R) -> % if R has SetCategory and R has Evalable R
from InnerEvalable(R, R)
first: % -> R
from IndexedAggregate(Integer, R)
hash: % -> SingleInteger if R has SetCategory
from SetCategory
hashUpdate!: (HashState, %) -> HashState if R has SetCategory
from SetCategory
index: PositiveInteger -> % if R has Finite
from Finite
index?: (Integer, %) -> Boolean
from IndexedAggregate(Integer, R)
indices: % -> List Integer
from IndexedAggregate(Integer, R)
latex: % -> String if R has SetCategory
from SetCategory
leftPower: (%, NonNegativeInteger) -> % if R has Monoid
from MagmaWithUnit
leftPower: (%, PositiveInteger) -> % if R has SemiGroup
from Magma
leftRecip: % -> Union(%, failed) if R has Monoid
from MagmaWithUnit
less?: (%, NonNegativeInteger) -> Boolean
from Aggregate
lookup: % -> PositiveInteger if R has Finite
from Finite
map: (R -> R, %) -> %
from HomogeneousAggregate R
max: (%, %) -> % if R has OrderedSet
from OrderedSet
maxIndex: % -> Integer
from IndexedAggregate(Integer, R)
member?: (R, %) -> Boolean if R has BasicType
from HomogeneousAggregate R
min: (%, %) -> % if R has OrderedSet
from OrderedSet
minIndex: % -> Integer
from IndexedAggregate(Integer, R)
more?: (%, NonNegativeInteger) -> Boolean
from Aggregate
one?: % -> Boolean if R has Monoid
from MagmaWithUnit
opposite?: (%, %) -> Boolean if R has AbelianMonoid
from AbelianMonoid
qelt: (%, Integer) -> R
from EltableAggregate(Integer, R)
random: () -> % if R has Finite
from Finite
recip: % -> Union(%, failed) if R has Monoid
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
retract: % -> Fraction Integer if R has RetractableTo Fraction Integer and R has SetCategory
from RetractableTo Fraction Integer
retract: % -> Integer if R has RetractableTo Integer and R has SetCategory
from RetractableTo Integer
retract: % -> R if R has SetCategory
from RetractableTo R
retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer and R has SetCategory
from RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer and R has SetCategory
from RetractableTo Integer
retractIfCan: % -> Union(R, failed) if R has SetCategory
from RetractableTo R
rightPower: (%, NonNegativeInteger) -> % if R has Monoid
from MagmaWithUnit
rightPower: (%, PositiveInteger) -> % if R has SemiGroup
from Magma
rightRecip: % -> Union(%, failed) if R has Monoid
from MagmaWithUnit
sample: %
from AbelianMonoid
size: () -> NonNegativeInteger if R has Finite
from Finite
size?: (%, NonNegativeInteger) -> Boolean
from Aggregate
smaller?: (%, %) -> Boolean if R has OrderedSet or R has Finite
from Comparable
subtractIfCan: (%, %) -> Union(%, failed) if R has CancellationAbelianMonoid
from CancellationAbelianMonoid
sup: (%, %) -> % if R has OrderedAbelianMonoidSup
from OrderedAbelianMonoidSup
unitVector: PositiveInteger -> % if R has Monoid and R has AbelianMonoid
unitVector(n) produces a vector with 1 in position n and zero elsewhere.
zero?: % -> Boolean if R has AbelianMonoid
from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoid if R has AbelianMonoid

AbelianProductCategory R

AbelianSemiGroup if R has AbelianMonoid or R has SemiRng

Aggregate

Algebra R if R has CommutativeRing

BasicType if R has BasicType

BiModule(%, %) if R has SemiRng

BiModule(R, R) if R has SemiRng

CancellationAbelianMonoid if R has CancellationAbelianMonoid

CoercibleTo OutputForm if R has CoercibleTo OutputForm

CoercibleTo Vector R

CommutativeRing if R has CommutativeRing

CommutativeStar if R has CommutativeRing

Comparable if R has OrderedSet or R has Finite

ConvertibleTo InputForm if R has Finite

DifferentialExtension R if R has Ring

DifferentialRing if R has Ring and R has DifferentialRing

Eltable(Integer, R)

EltableAggregate(Integer, R)

Evalable R if R has SetCategory and R has Evalable R

Finite if R has Finite

finiteAggregate

FullyLinearlyExplicitOver R if R has Ring

FullyRetractableTo R if R has SetCategory

HomogeneousAggregate R

IndexedAggregate(Integer, R)

InnerEvalable(R, R) if R has SetCategory and R has Evalable R

LeftModule % if R has SemiRng

LeftModule R if R has SemiRng

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 Monoid

Module R if R has CommutativeRing

Monoid if R has Monoid

NonAssociativeRing if R has Ring

NonAssociativeRng if R has Ring

NonAssociativeSemiRing if R has Ring

NonAssociativeSemiRng if R has SemiRng

OrderedAbelianMonoid if R has OrderedAbelianMonoidSup

OrderedAbelianMonoidSup if R has OrderedAbelianMonoidSup

OrderedAbelianSemiGroup if R has OrderedAbelianMonoidSup

OrderedCancellationAbelianMonoid if R has OrderedAbelianMonoidSup

OrderedSet if R has OrderedSet

PartialDifferentialRing Symbol if R has Ring and R has PartialDifferentialRing Symbol

PartialOrder if R has OrderedSet

RetractableTo Fraction Integer if R has RetractableTo Fraction Integer and R has SetCategory

RetractableTo Integer if R has RetractableTo Integer and R has SetCategory

RetractableTo R if R has SetCategory

RightModule % if R has SemiRng

RightModule R if R has SemiRng

Ring if R has Ring

Rng if R has Ring

SemiGroup if R has SemiGroup

SemiRing if R has Ring

SemiRng if R has SemiRng

SetCategory if R has SetCategory

unitsKnown if R has unitsKnown

VectorSpace R if R has Field