Bialgebra(R, MxM)

tensor.spad line 471 [edit on github]

• R : CommutativeRing
• MxM : Module(R)

A bialgebra is a coalgebra which at the same time is an algebra such that the comultiplication is also an algebra homomorphism. MxM: Module(R) should be replaced by a more restricted category, but it is not clear at this point which one.

* : (%, %) -> %

from Magma

* : (%, R) -> %

from RightModule(R)

* : (R, %) -> %

from LeftModule(R)

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

= : (%, %) -> Boolean

from BasicType

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associator : (%, %, %) -> %

from NonAssociativeRng

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : R -> %

from Algebra(R)

coerce : Integer -> %

from NonAssociativeRing

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

coproduct : % -> MxM

from Coalgebra(R, MxM)

counit : % -> R

from Coalgebra(R, MxM)

latex : % -> String

from SetCategory

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(R)

recip : % -> Union(%, "failed")

from MagmaWithUnit

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

subtractIfCan : (%, %) -> Union(%, "failed")

from CancellationAbelianMonoid

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

RightModule(%)

Rng

Coalgebra(R, MxM)

Monoid

NonAssociativeAlgebra(R)

Ring

SemiGroup

CancellationAbelianMonoid

LeftModule(%)

Algebra(R)

LeftModule(R)

unitsKnown

NonAssociativeRing

Magma

SemiRing

Module(R)

AbelianGroup

NonAssociativeSemiRing

SetCategory

AbelianSemiGroup

BiModule(R, R)

AbelianMonoid

BiModule(%, %)

NonAssociativeRng

NonAssociativeSemiRng

BasicType

MagmaWithUnit

RightModule(R)

CoercibleTo(OutputForm)

SemiRng