# Bialgebra(R, MxM)ΒΆ

- 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.

- 0: %
- from AbelianMonoid
- 1: %
- from MagmaWithUnit
- *: (%, %) -> %
- from Magma
- *: (%, R) -> %
- from RightModule R
- *: (Integer, %) -> %
- from AbelianGroup
- *: (NonNegativeInteger, %) -> %
- from AbelianMonoid
- *: (PositiveInteger, %) -> %
- from AbelianSemiGroup
- *: (R, %) -> %
- from LeftModule R
- +: (%, %) -> %
- from AbelianSemiGroup
- -: % -> %
- from AbelianGroup
- -: (%, %) -> %
- from AbelianGroup
- =: (%, %) -> Boolean
- from BasicType
- ^: (%, NonNegativeInteger) -> %
- from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
- from Magma
- ~=: (%, %) -> Boolean
- from BasicType
- annihilate?: (%, %) -> Boolean
- from Rng
- antiCommutator: (%, %) -> %
- from NonAssociativeSemiRng
- associator: (%, %, %) -> %
- from NonAssociativeRng
- characteristic: () -> NonNegativeInteger
- from NonAssociativeRing
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- coerce: Integer -> %
- from NonAssociativeRing
- coerce: R -> %
- from Algebra R
- commutator: (%, %) -> %
- from NonAssociativeRng
- coproduct: % -> MxM
- from Coalgebra(R, MxM)
- counit: % -> R
- from Coalgebra(R, MxM)
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- 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
- 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

Algebra R

BiModule(%, %)

BiModule(R, R)

Coalgebra(R, MxM)

Module R