https://github.com/ppetr/semigroups-actions
Semigroup/monoid actions on sets
https://github.com/ppetr/semigroups-actions
Last synced: 11 months ago
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Semigroup/monoid actions on sets
- Host: GitHub
- URL: https://github.com/ppetr/semigroups-actions
- Owner: ppetr
- License: other
- Created: 2012-12-17T20:47:45.000Z (over 13 years ago)
- Default Branch: master
- Last Pushed: 2012-12-20T21:59:58.000Z (over 13 years ago)
- Last Synced: 2024-11-17T17:54:26.840Z (over 1 year ago)
- Language: Haskell
- Size: 137 KB
- Stars: 2
- Watchers: 4
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.markdown
- License: LICENSE
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README
# 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](http://en.wikipedia.org/wiki/Semigroup_action) and
[group action](http://en.wikipedia.org/wiki/Group_action).
This package complements and depends on [semigroups](http://hackage.haskell.org/package/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:
```haskell
n `act` x = x <> ... <> x -- x appears n-times
```
So ``0 `act` x == mempty``, ``3 `act` x == x <> x <> x`` etc.
Because `<>` is associative, such an action can be computed very efficiently
with only _O(log n)_ operations using
[binary multiplication](https://en.wikipedia.org/wiki/Peasant_multiplication).
This is expressed by `newtype Repeat` and instance
```haskell
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:
```haskell
(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:
```haskell
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](http://petr.pudlak.name/) or through github.
License: BSD3