Data.Algebra

Algebras

type FreeAlgebra a f = (FreeSemimodule a f, Algebra a (Rep f))

An algebra over a free module f.

Note that this is distinct from a free algebra.

class Semiring a => Algebra a b where

An algebra algebra over a semiring.

Note that the algebra needn't be associative.

Methods

append :: (b -> b -> a) -> b -> a

Instances

Semiring r => Algebra r IntSet

Defined in Data.Algebra

Methods

append :: (IntSet -> IntSet -> r) -> IntSet -> r

Semiring a => Algebra a ()

Defined in Data.Algebra

Methods

append :: (() -> () -> a) -> () -> a

Semiring r => Algebra r E4

Defined in Data.Semimodule.Basis

Methods

append :: (E4 -> E4 -> r) -> E4 -> r

Semiring r => Algebra r E3

Defined in Data.Semimodule.Basis

Methods

append :: (E3 -> E3 -> r) -> E3 -> r

Semiring r => Algebra r E2

Defined in Data.Semimodule.Basis

Methods

append :: (E2 -> E2 -> r) -> E2 -> r

Semiring r => Algebra r E1

Defined in Data.Semimodule.Basis

Methods

append :: (E1 -> E1 -> r) -> E1 -> r

(Semiring r, Ord a) => Algebra r (Set a)

Defined in Data.Algebra

Methods

append :: (Set a -> Set a -> r) -> Set a -> r

Semiring r => Algebra r (Seq a)

The tensor algebra

Defined in Data.Algebra

Methods

append :: (Seq a -> Seq a -> r) -> Seq a -> r

Semiring a => Algebra a [a]

Tensor algebra

>>> append (<>) [1..3 :: Int]
[1,2,3,1,2,3,1,2,3,1,2,3]

>>> append (\f g -> fold (f ++ g)) [1..3] :: Int
24

Defined in Data.Algebra

Methods

append :: ([a] -> [a] -> a) -> [a] -> a

(Algebra a b, Algebra a c) => Algebra a (b, c)

Defined in Data.Algebra

Methods

append :: ((b, c) -> (b, c) -> a) -> (b, c) -> a

(Algebra a b, Algebra a c, Algebra a d) => Algebra a (b, c, d)

Defined in Data.Algebra

Methods

append :: ((b, c, d) -> (b, c, d) -> a) -> (b, c, d) -> a

(.*.) :: FreeAlgebra a f => f a -> f a -> f a infixl 7

Multiplication operator on an algebra over a free semimodule.

Caution in general (.*.) needn't be commutative, nor associative.

type FreeUnital a f = (FreeAlgebra a f, Unital a (Rep f))

A unital algebra over a free semimodule f.

class Algebra a b => Unital a b where

A unital algebra over a semiring.

Methods

aempty :: a -> b -> a

Instances

Semiring r => Unital r IntSet

Defined in Data.Algebra

Methods

aempty :: r -> IntSet -> r

Semiring a => Unital a ()

Defined in Data.Algebra

Methods

aempty :: a -> () -> a

Semiring r => Unital r E4

Defined in Data.Semimodule.Basis

Methods

aempty :: r -> E4 -> r

Semiring r => Unital r E3

Defined in Data.Semimodule.Basis

Methods

aempty :: r -> E3 -> r

Semiring r => Unital r E2

Defined in Data.Semimodule.Basis

Methods

aempty :: r -> E2 -> r

Semiring r => Unital r E1

Defined in Data.Semimodule.Basis

Methods

aempty :: r -> E1 -> r

(Semiring r, Ord a) => Unital r (Set a)

Defined in Data.Algebra

Methods

aempty :: r -> Set a -> r

Semiring r => Unital r (Seq a)

Defined in Data.Algebra

Methods

aempty :: r -> Seq a -> r

Semiring a => Unital a [a]

Defined in Data.Algebra

Methods

aempty :: a -> [a] -> a

(Unital a b, Unital a c) => Unital a (b, c)

Defined in Data.Algebra

Methods

aempty :: a -> (b, c) -> a

(Unital a b, Unital a c, Unital a d) => Unital a (b, c, d)

Defined in Data.Algebra

Methods

aempty :: a -> (b, c, d) -> a

unital :: FreeUnital a f => a -> f a

Insert an element into an algebra.

>>> V4 1 2 3 4 .*. unital two :: V4 Int
V4 2 4 6 8

unit :: FreeUnital a f => f a

Unital element of a unital algebra over a free semimodule.

>>> unit :: Complex Int
1 :+ 0
>>> unit :: QuatD
Quaternion 1.0 (V3 0.0 0.0 0.0)

Coalgebras

type FreeCoalgebra a f = (FreeSemimodule a f, Coalgebra a (Rep f))

A coalgebra over a free semimodule f.

class Semiring a => Coalgebra a c where

A coalgebra over a semiring.

( id *** coempty ) . coappend = id = ( coempty *** id ) . coappend

Methods

coappend :: (c -> a) -> c -> c -> a

Instances

Semiring r => Coalgebra r IntSet

the free commutative band coalgebra over Int

Defined in Data.Algebra

Methods

coappend :: (IntSet -> r) -> IntSet -> IntSet -> r

Semiring r => Coalgebra r ()

Defined in Data.Algebra

Methods

coappend :: (() -> r) -> () -> () -> r

Semiring r => Coalgebra r E4

Defined in Data.Semimodule.Basis

Methods

coappend :: (E4 -> r) -> E4 -> E4 -> r

Semiring r => Coalgebra r E3

Defined in Data.Semimodule.Basis

Methods

coappend :: (E3 -> r) -> E3 -> E3 -> r

Semiring r => Coalgebra r E2

Defined in Data.Semimodule.Basis

Methods

coappend :: (E2 -> r) -> E2 -> E2 -> r

Semiring r => Coalgebra r E1

Defined in Data.Semimodule.Basis

Methods

coappend :: (E1 -> r) -> E1 -> E1 -> r

(Semiring r, Ord a) => Coalgebra r (Set a)

the free commutative band coalgebra

Defined in Data.Algebra

Methods

coappend :: (Set a -> r) -> Set a -> Set a -> r

Semiring r => Coalgebra r (Seq a)

The tensor Hopf algebra

Defined in Data.Algebra

Methods

coappend :: (Seq a -> r) -> Seq a -> Seq a -> r

Semiring r => Coalgebra r [a]

The tensor Hopf algebra Δ(x) = x ⊗ 1 + 1 ⊗ x, x in V, Δ(1) = 1 ⊗ 1

Defined in Data.Algebra

Methods

coappend :: ([a] -> r) -> [a] -> [a] -> r

Algebra r a => Coalgebra r (a -> r)

Defined in Data.Algebra

Methods

coappend :: ((a -> r) -> r) -> (a -> r) -> (a -> r) -> r

(Coalgebra r a, Coalgebra r b) => Coalgebra r (a, b)

Defined in Data.Algebra

Methods

coappend :: ((a, b) -> r) -> (a, b) -> (a, b) -> r

(Coalgebra r a, Coalgebra r b, Coalgebra r c) => Coalgebra r (a, b, c)

Defined in Data.Algebra

Methods

coappend :: ((a, b, c) -> r) -> (a, b, c) -> (a, b, c) -> r

type FreeCounital a f = (FreeCoalgebra a f, Counital a (Rep f))

A counital coalgebra over a free semimodule f.

class Coalgebra a c => Counital a c where

A counital coalgebra over a semiring.

Methods

coempty :: (c -> a) -> a

Instances

Semiring r => Counital r IntSet

Defined in Data.Algebra

Methods

coempty :: (IntSet -> r) -> r

Semiring r => Counital r ()

Defined in Data.Algebra

Methods

coempty :: (() -> r) -> r

Semiring r => Counital r E4

Defined in Data.Semimodule.Basis

Methods

coempty :: (E4 -> r) -> r

Semiring r => Counital r E3

Defined in Data.Semimodule.Basis

Methods

coempty :: (E3 -> r) -> r

Semiring r => Counital r E2

Defined in Data.Semimodule.Basis

Methods

coempty :: (E2 -> r) -> r

Semiring r => Counital r E1

Defined in Data.Semimodule.Basis

Methods

coempty :: (E1 -> r) -> r

(Semiring r, Ord a) => Counital r (Set a)

Defined in Data.Algebra

Methods

coempty :: (Set a -> r) -> r

Semiring r => Counital r (Seq a)

Defined in Data.Algebra

Methods

coempty :: (Seq a -> r) -> r

Semiring r => Counital r [a]

Defined in Data.Algebra

Methods

coempty :: ([a] -> r) -> r

Algebra r a => Counital r (a -> r)

Defined in Data.Algebra

Methods

coempty :: ((a -> r) -> r) -> r

(Counital r a, Counital r b) => Counital r (a, b)

Defined in Data.Algebra

Methods

coempty :: ((a, b) -> r) -> r

(Counital r a, Counital r b, Counital r c) => Counital r (a, b, c)

Defined in Data.Algebra

Methods

coempty :: ((a, b, c) -> r) -> r

counital :: FreeCounital a f => f a -> a

Obtain an element from a coalgebra over a free semimodule.

Bialgebras

type FreeBialgebra a f = (FreeAlgebra a f, FreeCoalgebra a f, Bialgebra a (Rep f))

A bialgebra over a free semimodule f.

class (Unital a b, Counital a b) => Bialgebra a b

A bialgebra over a semiring.

Instances

Semiring r => Bialgebra r E4

Defined in Data.Semimodule.Basis

Semiring r => Bialgebra r E3

Defined in Data.Semimodule.Basis

Semiring r => Bialgebra r E2

Defined in Data.Semimodule.Basis

Semiring r => Bialgebra r E1

Defined in Data.Semimodule.Basis