Semigroup/monoid actions on sets
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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.


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. Because <> is associative, such an action can be computed very efficiently with only O(log n) operations using binary multiplication.

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))


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 2012, Petr Pudlák

Contact: or through github.

License: BSD3