HomogeneousDirectProduct(dim, S)

gdirprod.spad line 94 [edit on github]

• dim: NonNegativeInteger

• S: OrderedAbelianMonoid

This type represents the finite direct or cartesian product of an underlying ordered component type. The vectors are ordered first by the sum of their components, and then refined using a reverse lexicographic ordering. This type is a suitable third argument for GeneralDistributedMultivariatePolynomial.

0: %

from AbelianMonoid

1: % if S has Monoid

from MagmaWithUnit

#: % -> NonNegativeInteger

from Aggregate

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

from Magma

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

from RightModule Integer

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

from DirectProductCategory(dim, S)

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

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

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

from DirectProductCategory(dim, S)

+: (%, %) -> %

from AbelianSemiGroup

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

from AbelianGroup

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

from AbelianGroup

<=: (%, %) -> Boolean

from PartialOrder

<: (%, %) -> Boolean

from PartialOrder

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean

from PartialOrder

>: (%, %) -> Boolean

from PartialOrder

^: (%, NonNegativeInteger) -> % if S has Monoid

from MagmaWithUnit

^: (%, PositiveInteger) -> % if S has SemiGroup

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean if S has Ring

from Rng

antiCommutator: (%, %) -> % if S has SemiRng

from NonAssociativeSemiRng

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

from HomogeneousAggregate S

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

from NonAssociativeRng

characteristic: () -> NonNegativeInteger if S has Ring

from NonAssociativeRing

coerce: % -> % if S has CommutativeRing

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: % -> Vector S

from CoercibleTo Vector S

coerce: Fraction Integer -> % if S has RetractableTo Fraction Integer

from CoercibleFrom Fraction Integer

coerce: Integer -> % if S has RetractableTo Integer or S has Ring

from NonAssociativeRing

coerce: S -> %

from Algebra S

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

from NonAssociativeRng

convert: % -> InputForm if S has Finite

from ConvertibleTo InputForm

copy: % -> %

from Aggregate

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

from HomogeneousAggregate S

count: (S, %) -> NonNegativeInteger

from HomogeneousAggregate S

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

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

D: (%, S -> S) -> % if S has Ring

from DifferentialExtension S

D: (%, S -> S, NonNegativeInteger) -> % if S has Ring

from DifferentialExtension S

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

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

from DifferentialRing

differentiate: (%, S -> S) -> % if S has Ring

from DifferentialExtension S

differentiate: (%, S -> S, NonNegativeInteger) -> % if S has Ring

from DifferentialExtension S

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

from PartialDifferentialRing Symbol

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

from PartialDifferentialRing Symbol

directProduct: Vector S -> %

from DirectProductCategory(dim, S)

dot: (%, %) -> S if S has SemiRng

from DirectProductCategory(dim, S)

elt: (%, Integer) -> S

from Eltable(Integer, S)

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

from EltableAggregate(Integer, S)

empty?: % -> Boolean

from Aggregate

empty: () -> %

from Aggregate

entries: % -> List S

from IndexedAggregate(Integer, S)

entry?: (S, %) -> Boolean

from IndexedAggregate(Integer, S)

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

from Finite

eq?: (%, %) -> Boolean

from Aggregate

eval: (%, Equation S) -> % if S has Evalable S

from Evalable S

eval: (%, List Equation S) -> % if S has Evalable S

from Evalable S

eval: (%, List S, List S) -> % if S has Evalable S

from InnerEvalable(S, S)

eval: (%, S, S) -> % if S has Evalable S

from InnerEvalable(S, S)

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

from HomogeneousAggregate S

first: % -> S

from IndexedAggregate(Integer, S)

hash: % -> SingleInteger if S has Finite

from Hashable

hashUpdate!: (HashState, %) -> HashState if S has Finite

from Hashable

index?: (Integer, %) -> Boolean

from IndexedAggregate(Integer, S)

index: PositiveInteger -> % if S has Finite

from Finite

indices: % -> List Integer

from IndexedAggregate(Integer, S)

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

from OrderedAbelianMonoidSup

latex: % -> String

from SetCategory

leftPower: (%, NonNegativeInteger) -> % if S has Monoid

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> % if S has SemiGroup

from Magma

leftRecip: % -> Union(%, failed) if S has Monoid

from MagmaWithUnit

less?: (%, NonNegativeInteger) -> Boolean

from Aggregate

lookup: % -> PositiveInteger if S has Finite

from Finite

map: (S -> S, %) -> %

from HomogeneousAggregate S

max: % -> S

from HomogeneousAggregate S

max: (%, %) -> %

from OrderedSet

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

from HomogeneousAggregate S

maxIndex: % -> Integer

from IndexedAggregate(Integer, S)

member?: (S, %) -> Boolean

from HomogeneousAggregate S

members: % -> List S

from HomogeneousAggregate S

min: % -> S

from HomogeneousAggregate S

min: (%, %) -> %

from OrderedSet

minIndex: % -> Integer

from IndexedAggregate(Integer, S)

more?: (%, NonNegativeInteger) -> Boolean

from Aggregate

one?: % -> Boolean if S has Monoid

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

parts: % -> List S

from HomogeneousAggregate S

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

from NonAssociativeAlgebra S

qelt: (%, Integer) -> S

from EltableAggregate(Integer, S)

random: () -> % if S has Finite

from Finite

recip: % -> Union(%, failed) if S has Monoid

from MagmaWithUnit

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

from LinearlyExplicitOver Integer

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix S, vec: Vector S) if S has Ring

from LinearlyExplicitOver S

reducedSystem: Matrix % -> Matrix Integer if S has Ring and S has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix S if S has Ring

from LinearlyExplicitOver S

retract: % -> Fraction Integer if S has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retract: % -> Integer if S has RetractableTo Integer

from RetractableTo Integer

retract: % -> S

from RetractableTo S

retractIfCan: % -> Union(Fraction Integer, failed) if S has RetractableTo Fraction Integer

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Integer, failed) if S has RetractableTo Integer

from RetractableTo Integer

retractIfCan: % -> Union(S, failed)

from RetractableTo S

rightPower: (%, NonNegativeInteger) -> % if S has Monoid

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> % if S has SemiGroup

from Magma

rightRecip: % -> Union(%, failed) if S has Monoid

from MagmaWithUnit

sample: %

from AbelianMonoid

size?: (%, NonNegativeInteger) -> Boolean

from Aggregate

size: () -> NonNegativeInteger if S has Finite

from Finite

smaller?: (%, %) -> Boolean

from Comparable

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

from CancellationAbelianMonoid

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

from OrderedAbelianMonoidSup

unitVector: PositiveInteger -> % if S has Monoid

from DirectProductCategory(dim, S)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup if S has AbelianGroup

AbelianMonoid

AbelianProductCategory S

AbelianSemiGroup

Aggregate

Algebra % if S has CommutativeRing

Algebra S if S has CommutativeRing

BasicType

BiModule(%, %) if S has SemiRng

BiModule(S, S) if S has SemiRng

CancellationAbelianMonoid if S has CancellationAbelianMonoid

CoercibleFrom Fraction Integer if S has RetractableTo Fraction Integer

CoercibleFrom Integer if S has RetractableTo Integer

CoercibleFrom S

CoercibleTo OutputForm

CoercibleTo Vector S

CommutativeRing if S has CommutativeRing

CommutativeStar if S has CommutativeRing

Comparable

ConvertibleTo InputForm if S has Finite

DifferentialExtension S if S has Ring

DifferentialRing if S has Ring and S has DifferentialRing

DirectProductCategory(dim, S)

Eltable(Integer, S)

EltableAggregate(Integer, S)

Evalable S if S has Evalable S

Finite if S has Finite

finiteAggregate

FullyLinearlyExplicitOver S if S has Ring

FullyRetractableTo S

Hashable if S has Finite

HomogeneousAggregate S

IndexedAggregate(Integer, S)

InnerEvalable(S, S) if S has Evalable S

LeftModule % if S has SemiRng

LeftModule S if S has SemiRng

LinearlyExplicitOver Integer if S has Ring and S has LinearlyExplicitOver Integer

LinearlyExplicitOver S if S has Ring

Magma if S has SemiGroup

MagmaWithUnit if S has Monoid

Module % if S has CommutativeRing

Module S if S has CommutativeRing

Monoid if S has Monoid

NonAssociativeAlgebra % if S has CommutativeRing

NonAssociativeAlgebra S if S has CommutativeRing

NonAssociativeRing if S has Ring

NonAssociativeRng if S has Ring

NonAssociativeSemiRing if S has Ring

NonAssociativeSemiRng if S has SemiRng

OrderedAbelianMonoid

OrderedAbelianMonoidSup if S has OrderedAbelianMonoidSup

OrderedAbelianSemiGroup

OrderedCancellationAbelianMonoid if S has OrderedAbelianMonoidSup

OrderedSet

PartialDifferentialRing Symbol if S has PartialDifferentialRing Symbol and S has Ring

PartialOrder

RetractableTo Fraction Integer if S has RetractableTo Fraction Integer

RetractableTo Integer if S has RetractableTo Integer

RetractableTo S

RightModule % if S has SemiRng

RightModule Integer if S has Ring and S has LinearlyExplicitOver Integer

RightModule S if S has SemiRng

Ring if S has Ring

Rng if S has Ring

SemiGroup if S has SemiGroup

SemiRing if S has Ring

SemiRng if S has SemiRng

SetCategory

TwoSidedRecip if S has CommutativeRing

unitsKnown if S has unitsKnown