Numeric.Algebra

# Additive

## additive semigroups

class Additive r

sum1

## additive Abelian semigroups

class Abelian r

## additive idempotent semigroups

class Idempotent r

sinnum1pIdempotent

sinnumIdempotent

## partitionable additive semigroups

class Partitionable m

## additive monoids

class Monoidal m

sum

## additive groups

class Group r

# Multiplicative

## multiplicative semigroups

class Multiplicative r

product1

## commutative multiplicative semigroups

class Commutative r

## multiplicative monoids

class Unital r

product

## idempotent multiplicative semigroups

class Band r

pow1pBand

powBand

## multiplicative groups

class Division r

## factorable multiplicative semigroups

class Factorable m

## involutive multiplicative semigroups

class InvolutiveMultiplication r

class TriviallyInvolutive r

# Ring-Structures

## Semirings

class Semiring r

class InvolutiveSemiring r

class Dioid r

## Rngs

class Rng r

## Rigs

class Rig r

# Rings

class Ring r

## Division Rings

class LocalRing r

class DivisionRing r

class Field r

# Modules

class LeftModule r m

class RightModule r m

class Module r m

# Algebras

## associative algebras over (non-commutative) semirings

class Algebra r a

class Coalgebra r c

## unital algebras

class UnitalAlgebra r a

class CounitalCoalgebra r c

class Bialgebra r a

## involutive algebras

class InvolutiveAlgebra r a

class InvolutiveCoalgebra r c

class InvolutiveBialgebra r h

class TriviallyInvolutiveAlgebra r a

class TriviallyInvolutiveCoalgebra r a

class TriviallyInvolutiveBialgebra r h

## idempotent algebras

class IdempotentAlgebra r a

class IdempotentBialgebra r h

## commutative algebras

class CommutativeAlgebra r a

class CommutativeBialgebra r h

class CocommutativeCoalgebra r c

## division algebras

class DivisionAlgebra r a

## Hopf alegebras

class HopfAlgebra r h

# Ring Properties

## Characteristic

class Characteristic r

charInt

charWord

## Order

class Order a

class OrderedRig r

class AdditiveOrder r

class LocallyFiniteOrder a

class DecidableZero r

class DecidableUnits r

class DecidableAssociates r

# Natural numbers

data Natural

class Whole n

# Representable Additive

addRep

sinnum1pRep

# Representable Monoidal

zeroRep

sinnumRep

# Representable Group

negateRep

minusRep

subtractRep

timesRep

# Representable Multiplicative (via Algebra)

mulRep

# Representable Unital (via UnitalAlgebra)

oneRep

# Representable Rig (via Algebra)

fromNaturalRep

# Representable Ring (via Algebra)

fromIntegerRep

# Norm

class Quadrance r m

# Covectors

data Covector r a

## Covectors as linear functionals

counitM

unitM

comultM

multM

invM

coinvM

antipodeM

convolveM

memoM