Math.Categories.TotalOrder

Any total (or linear) order induces a preorder category where elements are objects, there is an arrow between two objects iff the relation is satisfied.

(See Categories for the working mathematican. Saunders Mac Lane. p.11)

Documentation

data IsSmallerThan a

IsSmallerThan is the type of morphisms in a linear order, it reminds the fact that there is a morphism from a source to a target iff the source is smaller than the target.

Constructors

IsSmallerThan a a

Instances

Eq a => Eq (IsSmallerThan a)

Defined in Math.Categories.TotalOrder

Methods

(==) :: IsSmallerThan a -> IsSmallerThan a -> Bool

(/=) :: IsSmallerThan a -> IsSmallerThan a -> Bool

Show a => Show (IsSmallerThan a)

Defined in Math.Categories.TotalOrder

Methods

showsPrec :: Int -> IsSmallerThan a -> ShowS

show :: IsSmallerThan a -> String

showList :: [IsSmallerThan a] -> ShowS

PrettyPrint a => PrettyPrint (IsSmallerThan a)

Defined in Math.Categories.TotalOrder

Methods

pprint :: IsSmallerThan a -> String

Eq a => Morphism (IsSmallerThan a) a

Defined in Math.Categories.TotalOrder

Methods

(@?) :: IsSmallerThan a -> IsSmallerThan a -> Maybe (IsSmallerThan a)

source :: IsSmallerThan a -> a

target :: IsSmallerThan a -> a

(Eq a, Ord a) => Category (TotalOrder a) (IsSmallerThan a) a

Defined in Math.Categories.TotalOrder

Methods

identity :: TotalOrder a -> a -> IsSmallerThan a

ar :: TotalOrder a -> a -> a -> Set (IsSmallerThan a)

genAr :: TotalOrder a -> a -> a -> Set (IsSmallerThan a)

decompose :: TotalOrder a -> IsSmallerThan a -> [IsSmallerThan a]

(Enum a, Ord a) => Category (OrdinalCategory a) (IsSmallerThan a) a

Defined in Math.Categories.OrdinalCategory

Methods

identity :: OrdinalCategory a -> a -> IsSmallerThan a

ar :: OrdinalCategory a -> a -> a -> Set (IsSmallerThan a)

genAr :: OrdinalCategory a -> a -> a -> Set (IsSmallerThan a)

decompose :: OrdinalCategory a -> IsSmallerThan a -> [IsSmallerThan a]

data TotalOrder a

A TotalOrder category is the category induced by a total order.

(See Categories for the working mathematican. Saunders Mac Lane. p.11)

Constructors

TotalOrder

Instances

Eq (TotalOrder a)

Defined in Math.Categories.TotalOrder

Methods

(==) :: TotalOrder a -> TotalOrder a -> Bool

(/=) :: TotalOrder a -> TotalOrder a -> Bool

Show (TotalOrder a)

Defined in Math.Categories.TotalOrder

Methods

showsPrec :: Int -> TotalOrder a -> ShowS

show :: TotalOrder a -> String

showList :: [TotalOrder a] -> ShowS

PrettyPrint (TotalOrder a)

Defined in Math.Categories.TotalOrder

Methods

pprint :: TotalOrder a -> String

(Eq a, Ord a) => Category (TotalOrder a) (IsSmallerThan a) a

Defined in Math.Categories.TotalOrder

Methods

identity :: TotalOrder a -> a -> IsSmallerThan a

ar :: TotalOrder a -> a -> a -> Set (IsSmallerThan a)

genAr :: TotalOrder a -> a -> a -> Set (IsSmallerThan a)

decompose :: TotalOrder a -> IsSmallerThan a -> [IsSmallerThan a]