Data.Monoid.Linear.Internal.Semigroup

A linear semigroup a is a type with an associative binary operation <> that linearly consumes two as.

Laws (same as Semigroup): * ∀ x ∈ G, y ∈ G, z ∈ G, x <> (y <> z) = (x <> y) <> z

An Endo a is just a linear function of type a %1-> a. This has a classic monoid definition with id and (.).

A linear application of an Endo.

DerivingVia combinator for Semigroup (resp. Prelude.Monoid) given linear Semigroup (resp. Monoid).

newtype Endo a = Endo (a %1-> a)
  deriving (Prelude.Semigroup) via NonLinear (Endo a)

Boolean monoid under conjunction (&&).

All x <> All y = All (x && y)

Examples

>>> All True <> mempty <> All False)All {getAll = False}

>>> mconcat (map (\x -> All (even x)) [2,4,6,7,8])All {getAll = False}

>>> All True <> memptyAll {getAll = True}

Boolean monoid under disjunction (||).

Any x <> Any y = Any (x || y)

Examples

>>> Any True <> mempty <> Any FalseAny {getAny = True}

>>> mconcat (map (\x -> Any (even x)) [2,4,6,7,8])Any {getAny = True}

>>> Any False <> memptyAny {getAny = False}

Beware that Data.Semigroup.First is different from Data.Monoid.First. The former simply returns the first value, so Data.Semigroup.First Nothing <> x = Data.Semigroup.First Nothing. The latter returns the first non-Nothing, thus Data.Monoid.First Nothing <> x = x.

Examples

>>> First 0 <> First 10First 0

>>> sconcat $ First 1 :| [ First n | n <- [2 ..] ]First 1

Beware that Data.Semigroup.Last is different from Data.Monoid.Last. The former simply returns the last value, so x <> Data.Semigroup.Last Nothing = Data.Semigroup.Last Nothing. The latter returns the last non-Nothing, thus x <> Data.Monoid.Last Nothing = x.

Examples

>>> Last 0 <> Last 10Last {getLast = 10}

>>> sconcat $ Last 1 :| [ Last n | n <- [2..]]Last {getLast = * hangs forever *

The dual of a Monoid, obtained by swapping the arguments of (<>).

Dual a <> Dual b == Dual (b <> a)

Examples

>>> Dual "Hello" <> Dual "World"Dual {getDual = "WorldHello"}

>>> Dual (Dual "Hello") <> Dual (Dual "World")Dual {getDual = Dual {getDual = "HelloWorld"}}

Monoid under addition.

Sum a <> Sum b = Sum (a + b)

Examples

>>> Sum 1 <> Sum 2 <> memptySum {getSum = 3}

>>> mconcat [ Sum n | n <- [3 .. 9]]Sum {getSum = 42}

Monoid under multiplication.

Product x <> Product y == Product (x * y)

Examples

>>> Product 3 <> Product 4 <> memptyProduct {getProduct = 12}

>>> mconcat [ Product n | n <- [2 .. 10]]Product {getProduct = 3628800}