Numeric Typeclasses

This pacakge attempts to model the numeric hierachay found in abstract algebra.

These mathemtaical objects are defined as sets with accompanying closed binary operations that also, depending on the object, obey certain propetries such as assosiativity, commutivity, and distributative laws.

For example, the most elementary mathematical object is called a Magma and it is defined as a set with a closed binary operation. However, the binary operation associated with the Magma does Not have to conform to the assosiative property. Magma is the most elementary mathemtaical object and is used here only for illustrative purposes. It does not have much value in math or programming and so this library starts its hierarchy with the Semigroup.

Definitions

Semigroup

+ A Set  
+ Closed binary operation  
+ Binary operation conforms to the associative property  

CommutativeSemigroup

+ Semigroup
+ Semigroup binary operation conforms to the commutative property

Monoid

+ Semigroup
+ Identity value

CommutativeMonoid

+ Monoid
+ Monoid binary operation conforms to the commutative property

Group

+ Monoid
+ Inverses

AbelianGroup

+ Group
+ Group binary operation conforms to the commutative property

Semring

+ Two Binary operations
    + Addition
        + CommutativeMonoid
    + Multiplication
        + Monoid

Ring

+ Two Binary operations
    + Addition
        + AbelianGroup
    + Multiplication
        + Monoid

CommutativeRing

+ Ring
+ Ring multiplication operation conforms to the commutative property

DivisionRing

+ Ring
+ Ring multiplication is a Group

CommutativeDivisionRing

+ DivisionRing
+ Ring multiplication operation conforms to the commutative property

Field

+ CommutativeDivisionRing

Usage

This library is used extensively in jonathanfishbein1/linear-algebra.

{-| Zero vector given a Field and dimension
-}
zeros : Monoid.Monoid a -> Int -> Vector a
zeros { identity } dim =
    List.repeat dim identity
        |> Vector