Product(A, B)

product.spad line 1 [edit on github]

• A : SetCategory
• B : SetCategory

This domain implements cartesian product. If the underlying domains are both Finite then the resulting Product is also Finite and can be enumerated via size(), index(), location(), etc. The index of the second component (B) varies most quickly.

* : (%, %) -> % if A has Group and B has Group or A has Monoid and B has Monoid

from Magma

* : (Integer, %) -> % if B has AbelianGroup and A has AbelianGroup

from AbelianGroup

* : (NonNegativeInteger, %) -> % if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid or A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has AbelianGroup and A has AbelianGroup or B has AbelianMonoid and A has AbelianMonoid

from AbelianMonoid

* : (PositiveInteger, %) -> % if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid or A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has AbelianGroup and A has AbelianGroup or B has AbelianMonoid and A has AbelianMonoid

from AbelianSemiGroup

+ : (%, %) -> % if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid or A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has AbelianGroup and A has AbelianGroup or B has AbelianMonoid and A has AbelianMonoid

from AbelianSemiGroup

- : % -> % if B has AbelianGroup and A has AbelianGroup

from AbelianGroup

- : (%, %) -> % if B has AbelianGroup and A has AbelianGroup

from AbelianGroup

/ : (%, %) -> % if A has Group and B has Group

from Group

0 : () -> % if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid or A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has AbelianGroup and A has AbelianGroup or B has AbelianMonoid and A has AbelianMonoid

from AbelianMonoid

1 : () -> % if A has Group and B has Group or A has Monoid and B has Monoid

from MagmaWithUnit

< : (%, %) -> Boolean if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has OrderedSet and A has OrderedSet

from PartialOrder

<= : (%, %) -> Boolean if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has OrderedSet and A has OrderedSet

from PartialOrder

= : (%, %) -> Boolean

from BasicType

> : (%, %) -> Boolean if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has OrderedSet and A has OrderedSet

from PartialOrder

>= : (%, %) -> Boolean if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has OrderedSet and A has OrderedSet

from PartialOrder

^ : (%, Integer) -> % if A has Group and B has Group

from Group

^ : (%, NonNegativeInteger) -> % if A has Group and B has Group or A has Monoid and B has Monoid

from MagmaWithUnit

^ : (%, PositiveInteger) -> % if A has Group and B has Group or A has Monoid and B has Monoid

from Magma

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> % if A has Group and B has Group

from Group

conjugate : (%, %) -> % if A has Group and B has Group

from Group

construct : (A, B) -> %

construct(a, b) creates element of the product with components a and b.

convert : % -> InputForm if A has Finite and B has Finite

from ConvertibleTo(InputForm)

enumerate : () -> List(%) if A has Finite and B has Finite

from Finite

first : % -> A

first(x) selects first component of the product

hash : % -> SingleInteger if A has Hashable and B has Hashable or A has Finite and B has Finite

from Hashable

hashUpdate! : (HashState, %) -> HashState if A has Hashable and B has Hashable or A has Finite and B has Finite

from Hashable

index : PositiveInteger -> % if A has Finite and B has Finite

from Finite

inf : (%, %) -> % if B has OrderedAbelianMonoidSup and A has OrderedAbelianMonoidSup

from OrderedAbelianMonoidSup

inv : % -> % if A has Group and B has Group

from Group

latex : % -> String

from SetCategory

leftPower : (%, NonNegativeInteger) -> % if A has Group and B has Group or A has Monoid and B has Monoid

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> % if A has Group and B has Group or A has Monoid and B has Monoid

from Magma

leftRecip : % -> Union(%, "failed") if A has Group and B has Group or A has Monoid and B has Monoid

from MagmaWithUnit

lookup : % -> PositiveInteger if A has Finite and B has Finite

from Finite

max : (%, %) -> % if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has OrderedSet and A has OrderedSet

from OrderedSet

min : (%, %) -> % if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has OrderedSet and A has OrderedSet

from OrderedSet

one? : % -> Boolean if A has Group and B has Group or A has Monoid and B has Monoid

from MagmaWithUnit

opposite? : (%, %) -> Boolean if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid or A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has AbelianGroup and A has AbelianGroup or B has AbelianMonoid and A has AbelianMonoid

from AbelianMonoid

random : () -> % if A has Finite and B has Finite

from Finite

recip : % -> Union(%, "failed") if A has Group and B has Group or A has Monoid and B has Monoid

from MagmaWithUnit

rightPower : (%, NonNegativeInteger) -> % if A has Group and B has Group or A has Monoid and B has Monoid

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> % if A has Group and B has Group or A has Monoid and B has Monoid

from Magma

rightRecip : % -> Union(%, "failed") if A has Group and B has Group or A has Monoid and B has Monoid

from MagmaWithUnit

sample : () -> % if A has Group and B has Group or B has AbelianGroup and A has AbelianGroup or A has Monoid and B has Monoid or A has CancellationAbelianMonoid and B has CancellationAbelianMonoid or A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has AbelianMonoid and A has AbelianMonoid

from MagmaWithUnit

second : % -> B

second(x) selects second component of the product

size : () -> NonNegativeInteger if A has Finite and B has Finite

from Finite

smaller? : (%, %) -> Boolean if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has OrderedSet and A has OrderedSet or A has Finite and B has Finite or B has Comparable and A has Comparable

from Comparable

subtractIfCan : (%, %) -> Union(%, "failed") if B has AbelianGroup and A has AbelianGroup or A has CancellationAbelianMonoid and B has CancellationAbelianMonoid or A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup

from CancellationAbelianMonoid

sup : (%, %) -> % if B has OrderedAbelianMonoidSup and A has OrderedAbelianMonoidSup

from OrderedAbelianMonoidSup

zero? : % -> Boolean if A has CancellationAbelianMonoid and B has CancellationAbelianMonoid or A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup or B has AbelianGroup and A has AbelianGroup or B has AbelianMonoid and A has AbelianMonoid

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

OrderedAbelianMonoidSup

Monoid

OrderedCancellationAbelianMonoid

SemiGroup

Finite

CancellationAbelianMonoid

BasicType

OrderedAbelianMonoid

CoercibleTo(OutputForm)

OrderedSet

Hashable

Group

AbelianGroup

Magma

AbelianSemiGroup

SetCategory

Comparable

AbelianMonoid

unitsKnown

OrderedAbelianSemiGroup

PartialOrder

TwoSidedRecip

MagmaWithUnit

ConvertibleTo(InputForm)