Data.Monoid.Coproduct

The coproduct of two monoids.

Documentation

data m :+: n

m :+: n is the coproduct of monoids m and n. Values of type m :+: n consist of alternating lists of m and n values. The empty list is the identity, and composition is list concatenation, with appropriate combining of adjacent elements when possible.

Instances

Monoid (:+: m n)

The coproduct of two monoids is itself a monoid.

Semigroup (:+: m n)

(Action m r, Action n r) => Action (:+: m n) r

Coproducts act on other things by having each of the components act individually.

inL :: m -> m :+: n

Injection from the left monoid into a coproduct.

inR :: n -> m :+: n

Injection from the right monoid into a coproduct.

mappendL :: m -> (m :+: n) -> m :+: n

Prepend a value from the left monoid.

mappendR :: n -> (m :+: n) -> m :+: n

Prepend a value from the right monoid.

killL :: Monoid n => (m :+: n) -> n

killL takes a value in a coproduct monoid and sends all the values from the left monoid to the identity.

killR :: Monoid m => (m :+: n) -> m

killR takes a value in a coproduct monoid and sends all the values from the right monoid to the identity.

untangle :: (Action m n, Monoid m, Monoid n) => (m :+: n) -> (m, n)

Take a value from a coproduct monoid where the left monoid has an action on the right, and "untangle" it into a pair of values. In particular,

 m1 <> n1 <> m2 <> n2 <> m3 <> n3 <> ...

is sent to

 (m1 <> m2 <> m3 <> ..., (act m1 n1) <> (act (m1 <> m2) n2) <> (act (m1 <> m2 <> m3) n3) <> ...)

That is, before combining n values, every n value is acted on by all the m values to its left.