DirectProduct(dim, R)ΒΆ

vector.spad line 350 [edit on github]

This type represents the finite direct or cartesian product of an underlying component type. This contrasts with simple vectors in that the members can be viewed as having constant length. Thus many categorical properties can by lifted from the underlying component type. Component extraction operations are provided but no updating operations. Thus new direct product elements can either be created by converting vector elements using the directProduct function or by taking appropriate linear combinations of basis vectors provided by the unitVector operation.

0: % if R has AbelianMonoid or % has AbelianMonoid and R has SemiRng

from AbelianMonoid

1: % if R has Monoid

from MagmaWithUnit

#: % -> NonNegativeInteger

from Aggregate

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

from Magma

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

from RightModule Integer

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

from DirectProductCategory(dim, R)

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

from AbelianGroup

*: (NonNegativeInteger, %) -> % if R has AbelianMonoid or % has AbelianMonoid and R has SemiRng

from AbelianMonoid

*: (PositiveInteger, %) -> % if R has AbelianMonoid or R has SemiRng

from AbelianSemiGroup

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

from DirectProductCategory(dim, R)

+: (%, %) -> % if R has AbelianMonoid or R has SemiRng

from AbelianSemiGroup

-: % -> % if R has SemiRng and % has AbelianGroup or R has AbelianGroup

from AbelianGroup

-: (%, %) -> % if R has SemiRng and % has AbelianGroup or R has AbelianGroup

from AbelianGroup

<=: (%, %) -> 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

any?: (R -> Boolean, %) -> Boolean

from HomogeneousAggregate R

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

from NonAssociativeRng

characteristic: () -> NonNegativeInteger if R has Ring

from NonAssociativeRing

coerce: % -> % if R has CommutativeRing

from Algebra %

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 CoercibleFrom 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 Algebra R

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

from NonAssociativeRng

convert: % -> InputForm if R has Finite

from ConvertibleTo InputForm

copy: % -> %

from Aggregate

count: (R -> Boolean, %) -> NonNegativeInteger

from HomogeneousAggregate R

count: (R, %) -> NonNegativeInteger if R has BasicType

from HomogeneousAggregate R

D: % -> % if R has DifferentialRing and R has Ring

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

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 PartialDifferentialRing Symbol and R has Ring

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

differentiate: % -> % if R has DifferentialRing and R has Ring

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

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 PartialDifferentialRing Symbol and R has Ring

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

directProduct: Vector R -> %

from DirectProductCategory(dim, R)

dot: (%, %) -> R if R has AbelianMonoid and R has SemiRng

from DirectProductCategory(dim, R)

elt: (%, Integer) -> R

from Eltable(Integer, R)

elt: (%, Integer, R) -> R

from EltableAggregate(Integer, R)

empty?: % -> Boolean

from Aggregate

empty: () -> %

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 Evalable R and R has SetCategory

from Evalable R

eval: (%, List Equation R) -> % if R has Evalable R and R has SetCategory

from Evalable R

eval: (%, List R, List R) -> % if R has Evalable R and R has SetCategory

from InnerEvalable(R, R)

eval: (%, R, R) -> % if R has Evalable R and R has SetCategory

from InnerEvalable(R, R)

every?: (R -> Boolean, %) -> Boolean

from HomogeneousAggregate R

first: % -> R

from IndexedAggregate(Integer, R)

hash: % -> SingleInteger if R has Hashable

from Hashable

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

from Hashable

index?: (Integer, %) -> Boolean

from IndexedAggregate(Integer, R)

index: PositiveInteger -> % if R has Finite

from Finite

indices: % -> List Integer

from IndexedAggregate(Integer, R)

inf: (%, %) -> % if R has OrderedAbelianMonoidSup

from OrderedAbelianMonoidSup

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: % -> R if R has OrderedSet

from HomogeneousAggregate R

max: (%, %) -> % if R has OrderedSet

from OrderedSet

max: ((R, R) -> Boolean, %) -> R

from HomogeneousAggregate R

maxIndex: % -> Integer

from IndexedAggregate(Integer, R)

member?: (R, %) -> Boolean if R has BasicType

from HomogeneousAggregate R

members: % -> List R

from HomogeneousAggregate R

min: % -> R if R has OrderedSet

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 or % has AbelianMonoid and R has SemiRng

from AbelianMonoid

parts: % -> List R

from HomogeneousAggregate R

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

from NonAssociativeAlgebra R

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 LinearlyExplicitOver Integer and R has Ring

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 LinearlyExplicitOver Integer and R has Ring

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 MagmaWithUnit

size?: (%, NonNegativeInteger) -> Boolean

from Aggregate

size: () -> NonNegativeInteger if R has Finite

from Finite

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

from Comparable

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

from CancellationAbelianMonoid

sup: (%, %) -> % if R has OrderedAbelianMonoidSup

from OrderedAbelianMonoidSup

unitVector: PositiveInteger -> % if R has AbelianMonoid and R has Monoid

from DirectProductCategory(dim, R)

zero?: % -> Boolean if R has AbelianMonoid or % has AbelianMonoid and R has SemiRng

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 % if R has CommutativeRing

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

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

CoercibleFrom Integer if R has RetractableTo Integer and R has SetCategory

CoercibleFrom R if R has SetCategory

CoercibleTo OutputForm if R has CoercibleTo OutputForm

CoercibleTo Vector R

CommutativeRing if R has CommutativeRing

CommutativeStar if R has CommutativeRing

Comparable if R has Finite or R has OrderedSet

ConvertibleTo InputForm if R has Finite

DifferentialExtension R if R has Ring

DifferentialRing if R has DifferentialRing and R has Ring

DirectProductCategory(dim, R)

Eltable(Integer, R)

EltableAggregate(Integer, R)

Evalable R if R has Evalable R and R has SetCategory

Finite if R has Finite

finiteAggregate

FullyLinearlyExplicitOver R if R has Ring

FullyRetractableTo R if R has SetCategory

Hashable if R has Hashable

HomogeneousAggregate R

IndexedAggregate(Integer, R)

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

LeftModule % if R has SemiRng

LeftModule R if R has SemiRng

LinearlyExplicitOver Integer if R has LinearlyExplicitOver Integer and R has Ring

LinearlyExplicitOver R if R has Ring

Magma if R has SemiGroup

MagmaWithUnit if R has Monoid

Module % if R has CommutativeRing

Module R if R has CommutativeRing

Monoid if R has Monoid

NonAssociativeAlgebra % if R has CommutativeRing

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 SemiRng

OrderedAbelianMonoid if R has OrderedAbelianMonoid

OrderedAbelianMonoidSup if R has OrderedAbelianMonoidSup

OrderedAbelianSemiGroup if R has OrderedAbelianMonoid

OrderedCancellationAbelianMonoid if R has OrderedAbelianMonoidSup

OrderedSet if R has OrderedSet

PartialDifferentialRing Symbol if R has PartialDifferentialRing Symbol and R has Ring

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 Integer if R has LinearlyExplicitOver Integer and R has Ring

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

TwoSidedRecip if R has CommutativeRing

unitsKnown if R has unitsKnown