Copyright	(c) 2020-2021 Emily Pillmore
License	BSD-style
Maintainer	Emily Pillmore <emilypi@cohomolo.gy>, Reed Mullanix <reedmullanix@gmail.com>
Stability	stable
Portability	non-portable
Safe Haskell	Safe
Language	Haskell2010

Data.Group.Cyclic

Contents

Cyclic groups
- Combinators

Description

This module contains definitions for Cyclic groups, along with the relevant combinators.

Synopsis

class Group a => Cyclic a where
- generator :: a
classify :: (Eq a, Cyclic a) => (a -> Bool) -> [a]
generated :: Cyclic a => [a]

Cyclic groups

Cyclic is a Group that is generated by a single element. This element is called a generator of the group. There can be many generators for a group, e.g., any representative of an equivalence class of prime numbers of the integers modulo n, but to make things easy, we ask for only one generator.

class Group a => Cyclic a where #

A Group G is Cyclic if there exists an element x of G such that for all y in G, there exists an n, such that

y = pow x n

Methods

generator :: a #

The generator of the Cyclic group.

Instances

Instances details

Cyclic ()
Instance details Defined in Data.Group Methods generator :: () #
Cyclic a => Cyclic (Identity a)
Instance details Defined in Data.Group Methods generator :: Identity a #
Integral a => Cyclic (Sum a)
Instance details Defined in Data.Group Methods generator :: Sum a #
Cyclic a => Cyclic (Down a)
Instance details Defined in Data.Group Methods generator :: Down a #
Cyclic (Proxy x)
Instance details Defined in Data.Group Methods generator :: Proxy x #
Cyclic a => Cyclic (Const a x)
Instance details Defined in Data.Group Methods generator :: Const a x #

Combinators

classify :: (Eq a, Cyclic a) => (a -> Bool) -> [a] Source #

Classify elements of a Cyclic group.

Apply a classifying function a -> Bool to the elements of a Cyclic group as generated by its designated generator.

Examples:

>>> take 3 $ classify (< (3 :: Sum Word8))
[Sum {getSum = 1},Sum {getSum = 2}]

generated :: Cyclic a => [a] #

Generate all elements of a Cyclic group using its generator.

Note: Fuses, does not terminate even for finite groups.