The semigroups-actions package

[Tags:bsd3, library]

In mathematics, an action of a semigroup on a set is an operation that associates each element of the semigroup is with a transformation on the set.


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Properties

Versions 0.1
Dependencies base (>=2 && <5), containers (>=0.3 && <0.6), semigroups (==0.8.*) [details]
License BSD3
Copyright Copyright (C) 2012 Petr Pudlák
Author Petr Pudlák
Maintainer Petr Pudlák <petr.mvd@gmail.com>
Stability provisional
Category Algebra, Data, Data Structures, Math
Home page http://github.com/ppetr/semigroups-actions/
Bug tracker http://github.com/ppetr/semigroups-actions/issues
Source repository head: git clone git://github.com/ppetr/semigroups-actions.git
Uploaded Thu Dec 20 21:48:01 UTC 2012 by PetrPudlak
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Readme for semigroups-actions

Readme for semigroups-actions-0.1

semigroups-actions

In mathematics, an action of a semigroup on a set is an operation that associates each element of the semigroup is with a transformation on the set. See Wikipedia articles on semigroup action and group action.

This package complements and depends on semigroups.

It requires MultiParamTypeClasses extension.

Examples

Similarly to semigroups, semigroup (or monoid) actions arise almost everywhere (if you look for them).

Natural numbers acting on monoids

The multiplication monoid of natural numbers acts on any other monoid:

n `act` x = x <> ... <> x -- x appears n-times

So 0 `act` x == mempty, 3 `act` x == x <> x <> x etc. TODO efficiency

This is expressed by newtype Repeat and instance

instance (Monoid w, Whole n) => SemigroupAct (Product n) (Repeat w) where

Many different concepts can be expressed using such an action, including

  • repeating a list n-times (replicate),

  • compose a function f n-times, commonly denoted as fⁿ in mathematics:

    (Repeat (Endo fⁿ)) = (Product n) `act` (Repeat (Endo f))
    

Self-application

Any semigroup (or monoid) can be viewed as acting on itself. In this case, act simply becomes <>. This is expressed by the SelfAct type:

newtype SelfAct a = SelfAct a
instance Semigroup g => Semigroup (SelfAct g) where
    (SelfAct x) <> (SelfAct y) = SelfAct $ x <> y

Matrices acting on vectors

Matrices with multiplication can be viewed as a group acting on vectors.

Copyright

Copyright 2012, Petr Pudlák

Contact: petr.pudlak.name or through github.

License: BSD3