Copyright	Guillaume Sabbagh 2022
License	GPL-3
Maintainer	guillaumesabbagh@protonmail.com
Stability	experimental
Portability	portable
Safe Haskell	Safe-Inferred
Language	Haskell2010

Math.Categories.TotalOrder

Description

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)

Synopsis

data IsSmallerThan a = IsSmallerThan a a
data TotalOrder a = TotalOrder

Documentation

data IsSmallerThan a Source #

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

Instances details

PrettyPrint a => PrettyPrint (IsSmallerThan a) Source #
Instance details Defined in Math.Categories.TotalOrder Methods pprint :: IsSmallerThan a -> String Source #
Show a => Show (IsSmallerThan a) Source #
Instance details Defined in Math.Categories.TotalOrder Methods showsPrec :: Int -> IsSmallerThan a -> ShowS show :: IsSmallerThan a -> String showList :: [IsSmallerThan a] -> ShowS
Eq a => Eq (IsSmallerThan a) Source #
Instance details Defined in Math.Categories.TotalOrder Methods (==) :: IsSmallerThan a -> IsSmallerThan a -> Bool (/=) :: IsSmallerThan a -> IsSmallerThan a -> Bool
Eq a => Morphism (IsSmallerThan a) a Source #
Instance details Defined in Math.Categories.TotalOrder Methods (@?) :: IsSmallerThan a -> IsSmallerThan a -> Maybe (IsSmallerThan a) Source # source :: IsSmallerThan a -> a Source # target :: IsSmallerThan a -> a Source #
(Enum a, Ord a) => Category (OrdinalCategory a) (IsSmallerThan a) a Source #
Instance details Defined in Math.Categories.OrdinalCategory Methods identity :: OrdinalCategory a -> a -> IsSmallerThan a Source # ar :: OrdinalCategory a -> a -> a -> Set (IsSmallerThan a) Source # genAr :: OrdinalCategory a -> a -> a -> Set (IsSmallerThan a) Source # decompose :: OrdinalCategory a -> IsSmallerThan a -> [IsSmallerThan a] Source #
(Eq a, Ord a) => Category (TotalOrder a) (IsSmallerThan a) a Source #
Instance details Defined in Math.Categories.TotalOrder Methods identity :: TotalOrder a -> a -> IsSmallerThan a Source # ar :: TotalOrder a -> a -> a -> Set (IsSmallerThan a) Source # genAr :: TotalOrder a -> a -> a -> Set (IsSmallerThan a) Source # decompose :: TotalOrder a -> IsSmallerThan a -> [IsSmallerThan a] Source #

data TotalOrder a Source #

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

Instances details

PrettyPrint (TotalOrder a) Source #
Instance details Defined in Math.Categories.TotalOrder Methods pprint :: TotalOrder a -> String Source #
Show (TotalOrder a) Source #
Instance details Defined in Math.Categories.TotalOrder Methods showsPrec :: Int -> TotalOrder a -> ShowS show :: TotalOrder a -> String showList :: [TotalOrder a] -> ShowS
Eq (TotalOrder a) Source #
Instance details Defined in Math.Categories.TotalOrder Methods (==) :: TotalOrder a -> TotalOrder a -> Bool (/=) :: TotalOrder a -> TotalOrder a -> Bool
(Eq a, Ord a) => Category (TotalOrder a) (IsSmallerThan a) a Source #
Instance details Defined in Math.Categories.TotalOrder Methods identity :: TotalOrder a -> a -> IsSmallerThan a Source # ar :: TotalOrder a -> a -> a -> Set (IsSmallerThan a) Source # genAr :: TotalOrder a -> a -> a -> Set (IsSmallerThan a) Source # decompose :: TotalOrder a -> IsSmallerThan a -> [IsSmallerThan a] Source #