OrderedDirectProduct(dim, S, f)ΒΆ

gdirprod.spad line 69

This type represents the finite direct or cartesian product of an underlying ordered component type. The ordering on the type is determined by its third argument which represents the less than function on vectors. 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
*: (%, 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
/: (%, S) -> % if S has Field
from VectorSpace S
<: (%, %) -> 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 RetractableTo Fraction Integer
coerce: Integer -> % if S has Ring or S has RetractableTo Integer
from NonAssociativeRing
coerce: S -> %
from RetractableTo 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 DifferentialRing and S has Ring
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 DifferentialRing and S has Ring
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 DifferentialRing and S has Ring
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 DifferentialRing and S has Ring
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
dimension: () -> CardinalNumber if S has Field
from VectorSpace S
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: () -> %
from Aggregate
empty?: % -> Boolean
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
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
index: PositiveInteger -> % if S has Finite
from Finite
index?: (Integer, %) -> Boolean
from IndexedAggregate(Integer, S)
indices: % -> List Integer
from IndexedAggregate(Integer, S)
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: (%, %) -> %
from OrderedSet
maxIndex: % -> Integer
from IndexedAggregate(Integer, S)
member?: (S, %) -> Boolean
from HomogeneousAggregate S
members: % -> List 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
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 LinearlyExplicitOver Integer and S has Ring
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 LinearlyExplicitOver Integer and S has Ring
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 if S has Finite
from Finite
size?: (%, NonNegativeInteger) -> Boolean
from Aggregate
smaller?: (%, %) -> Boolean
from Comparable
subtractIfCan: (%, %) -> Union(%, failed)
from CancellationAbelianMonoid
sup: (%, %) -> %
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

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 DifferentialRing and S has Ring

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

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 LinearlyExplicitOver Integer and S has Ring

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

NonAssociativeRing if S has Ring

NonAssociativeRng if S has Ring

NonAssociativeSemiRing if S has Ring

NonAssociativeSemiRng if S has SemiRng

OrderedAbelianMonoid

OrderedAbelianMonoidSup

OrderedAbelianSemiGroup

OrderedCancellationAbelianMonoid

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 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

unitsKnown if S has unitsKnown

VectorSpace S if S has Field