# OrderedDirectProduct(dim, S, f)ΒΆ

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 LeftModule %

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

from DirectProductCategory(dim, S)

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

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

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

from DirectProductCategory(dim, S)

+: (%, %) -> %

from AbelianSemiGroup

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

from AbelianGroup

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

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

from HomogeneousAggregate S

associator: (%, %, %) -> % if S has Ring
characteristic: () -> NonNegativeInteger if S has Ring
coerce: % -> % if S has CommutativeRing

from Algebra %

coerce: % -> OutputForm
coerce: % -> Vector S

from CoercibleTo Vector S

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

from Algebra S

commutator: (%, %) -> % if S has Ring
convert: % -> InputForm if S has Finite
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
D: (%, List Symbol, List NonNegativeInteger) -> % if S has PartialDifferentialRing Symbol and S has Ring
D: (%, NonNegativeInteger) -> % if S has DifferentialRing and S has Ring

from DifferentialRing

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

from DifferentialRing

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

from DifferentialRing

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

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

index?: (Integer, %) -> Boolean

from IndexedAggregate(Integer, S)

index: PositiveInteger -> % if S has Finite

from Finite

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: % -> 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

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

from LinearlyExplicitOver S

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

from RetractableTo S

retractIfCan: % -> Union(Fraction Integer, failed) if S has RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if S has 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
sup: (%, %) -> % if S has OrderedAbelianMonoidSup
unitVector: PositiveInteger -> % if S has Monoid

from DirectProductCategory(dim, S)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup if S has AbelianGroup

AbelianMonoid

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

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)

Evalable S if S has Evalable S

Finite if S has Finite

finiteAggregate

FullyLinearlyExplicitOver S if S has Ring

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

NonAssociativeRing if S has Ring

NonAssociativeRng if S has Ring

NonAssociativeSemiRing if S has Ring

NonAssociativeSemiRng if S has SemiRng

OrderedAbelianMonoid

OrderedAbelianSemiGroup

OrderedSet

PartialDifferentialRing Symbol if S has PartialDifferentialRing Symbol and S has Ring

PartialOrder

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

TwoSidedRecip if S has CommutativeRing

unitsKnown if S has unitsKnown