Monoid package:universum

This module reexports functions to work with monoids plus adds extra useful functions.

The class of monoids (types with an associative binary operation that has an identity). Instances should satisfy the following:

• Right identity x <> mempty = x
• Left identity mempty <> x = x
• Associativity x <> (y <> z) = (x <> y) <> z (Semigroup law)
• Concatenation mconcat = foldr (<>) mempty

You can alternatively define mconcat instead of mempty, in which case the laws are:

• Unit mconcat (pure x) = x
• Multiplication mconcat (join xss) = mconcat (fmap mconcat xss)
• Subclass mconcat (toList xs) = sconcat xs

The method names refer to the monoid of lists under concatenation, but there are many other instances. Some types can be viewed as a monoid in more than one way, e.g. both addition and multiplication on numbers. In such cases we often define newtypes and make those instances of Monoid, e.g. Sum and Product. NOTE: Semigroup is a superclass of Monoid since base-4.11.0.0.

Provide a Semigroup for an arbitrary Monoid. NOTE: This is not needed anymore since Semigroup became a superclass of Monoid in base-4.11 and this newtype be deprecated at some point in the future.

Extracts Monoid value from Maybe returning mempty if Nothing.

>>> maybeToMonoid (Just [1,2,3] :: Maybe [Int])
[1,2,3]

>>> maybeToMonoid (Nothing :: Maybe [Int])
[]

This is a valid definition of stimes for an idempotent Monoid. When x <> x = x, this definition should be preferred, because it works in <math> rather than <math>

This is a valid definition of stimes for a Monoid. Unlike the default definition of stimes, it is defined for 0 and so it should be preferred where possible.