Copyright	(c) Alice Rixte 2025
License	BSD 3
Maintainer	alice.rixte@u-bordeaux.fr
Stability	unstable
Portability	non-portable (GHC extensions)
Safe Haskell	None
Language	Haskell2010

Data.Act.Torsor

Description

Presentation

Synopsis

class LActGp x g => LTorsor x g where
- ldiff :: x -> x -> g
- (.-.) :: x -> x -> g
class RActGp x g => RTorsor x g where
- rdiff :: x -> x -> g
- (.~.) :: x -> x -> g

Documentation

class LActGp x g => LTorsor x g where Source #

A left group torsor.

The most well known example of a torsor is the particular case of an affine space where the group is the additive group of the vector space and the set is a set of points. Torsors are more general than affine spaces since they don't enforce linearity. Notice that LActDistrib may correspond to a linearity condition if you need one.

See this nLab article for more information : https://ncatlab.org/nlab/show/torsor

In algebraic terms :

A left group action is a torsor if and only if for every pair (x,y) :: (x, x), there exists a unique group element g :: g such that g <>$ x = y.

In Haskell terms :

Instances must satisfy the following law :

y .-. x <>$ x == y
if g <>$ x == y then g == y .-. x

Minimal complete definition

ldiff | (.-.)

Methods

ldiff :: x -> x -> g infix 6 Source #

ldiff y x is the only group element such that ldiff y x <>$ x = y.

(.-.) :: x -> x -> g infix 6 Source #

Infix synonym for ldiff.

This represents a point minus a point.

Instances

Instances details

LTorsor x () Source #
Instance details Defined in Data.Act.Torsor Methods ldiff :: x -> x -> () Source # (.-.) :: x -> x -> () Source #
Group g => LTorsor g (ActSelf g) Source #
Instance details Defined in Data.Act.Torsor Methods ldiff :: g -> g -> ActSelf g Source # (.-.) :: g -> g -> ActSelf g Source #
LTorsor x g => LTorsor x (Identity g) Source #
Instance details Defined in Data.Act.Torsor Methods ldiff :: x -> x -> Identity g Source # (.-.) :: x -> x -> Identity g Source #
RTorsor x g => LTorsor x (Dual g) Source #
Instance details Defined in Data.Act.Torsor Methods ldiff :: x -> x -> Dual g Source # (.-.) :: x -> x -> Dual g Source #
Fractional x => LTorsor x (Product x) Source #
Instance details Defined in Data.Act.Torsor Methods ldiff :: x -> x -> Product x Source # (.-.) :: x -> x -> Product x Source #
Num x => LTorsor x (Sum x) Source #
Instance details Defined in Data.Act.Torsor Methods ldiff :: x -> x -> Sum x Source # (.-.) :: x -> x -> Sum x Source #
(Group g, Coercible x g) => LTorsor x (ActSelf' g) Source #
Instance details Defined in Data.Act.Torsor Methods ldiff :: x -> x -> ActSelf' g Source # (.-.) :: x -> x -> ActSelf' g Source #
LTorsor x g => LTorsor (Identity x) (Identity g) Source #
Instance details Defined in Data.Act.Torsor Methods ldiff :: Identity x -> Identity x -> Identity g Source # (.-.) :: Identity x -> Identity x -> Identity g Source #
(LTorsor x g, LTorsor y h) => LTorsor (x, y) (g, h) Source #
Instance details Defined in Data.Act.Torsor Methods ldiff :: (x, y) -> (x, y) -> (g, h) Source # (.-.) :: (x, y) -> (x, y) -> (g, h) Source #

class RActGp x g => RTorsor x g where Source #

A right group torsor.

In algebraic terms :

A left group action is a torsor if and only if for every pair (x,y) :: (x, x), there exists a unique group element g :: g such that g <>$ x = y.

In Haskell terms :

Instances must satisfy the following law :

x $<> y .~. x == y
if x $<> g == y then g == y .~. x

Minimal complete definition

rdiff | (.~.)

Methods

rdiff :: x -> x -> g infix 6 Source #

rdiff y x is the only group element such that rdiff y x $<> x = y.

(.~.) :: x -> x -> g infix 6 Source #

Infix synonym for rdiff.

This represents a point minus a point.

Instances

Instances details

RTorsor x () Source #
Instance details Defined in Data.Act.Torsor Methods rdiff :: x -> x -> () Source # (.~.) :: x -> x -> () Source #
Group g => RTorsor g (ActSelf g) Source #
Instance details Defined in Data.Act.Torsor Methods rdiff :: g -> g -> ActSelf g Source # (.~.) :: g -> g -> ActSelf g Source #
RTorsor x g => RTorsor x (Identity g) Source #
Instance details Defined in Data.Act.Torsor Methods rdiff :: x -> x -> Identity g Source # (.~.) :: x -> x -> Identity g Source #
LTorsor x g => RTorsor x (Dual g) Source #
Instance details Defined in Data.Act.Torsor Methods rdiff :: x -> x -> Dual g Source # (.~.) :: x -> x -> Dual g Source #
Fractional x => RTorsor x (Product x) Source #
Instance details Defined in Data.Act.Torsor Methods rdiff :: x -> x -> Product x Source # (.~.) :: x -> x -> Product x Source #
Num x => RTorsor x (Sum x) Source #
Instance details Defined in Data.Act.Torsor Methods rdiff :: x -> x -> Sum x Source # (.~.) :: x -> x -> Sum x Source #
(Group g, Coercible x g) => RTorsor x (ActSelf' g) Source #
Instance details Defined in Data.Act.Torsor Methods rdiff :: x -> x -> ActSelf' g Source # (.~.) :: x -> x -> ActSelf' g Source #
RTorsor x g => RTorsor (Identity x) (Identity g) Source #
Instance details Defined in Data.Act.Torsor Methods rdiff :: Identity x -> Identity x -> Identity g Source # (.~.) :: Identity x -> Identity x -> Identity g Source #
(RTorsor x g, RTorsor y h) => RTorsor (x, y) (g, h) Source #
Instance details Defined in Data.Act.Torsor Methods rdiff :: (x, y) -> (x, y) -> (g, h) Source # (.~.) :: (x, y) -> (x, y) -> (g, h) Source #