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Questions tagged [monoids]

A monoid is a set that is closed under an associative binary operation and has an identity element I ∈ Such that for all a ∈ S, Ia = aI = a. Note that unlike a group, its elements need not have inverses. It can also be thought of as a semigroup with an identity element.

A monoid is a set that is closed under an associative binary operation and has an identity element I ∈ Such that for all a ∈ S, Ia = aI = a. Note that unlike a group, its elements need not have inverses. It can also be thought of as a semigroup with an identity element. Put simply, a monoid is an algebraic structure with an associative binary operation that has an identity element. Examples include:

  • lists under concatenation
  • numbers under addition or multiplication
  • booleans under conjunction or disjunction
  • sets under union or intersection
  • functions from a type to itself, under composition

Note that in most of these cases the operation is also commutative, but it need not be; concatenation and function composition are not commutative.

Useful links:

Wikipedia - Monoid

Wikipedia - Monoid (category theory)

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non-lawful Monoid instances for building up AST not considered harmful?

I've seen a data type defined like the following with a corresponding Monoid instance: data Foo where FooEmpty :: String -> Foo FooAppend :: Foo -> Foo -> Foo -- | Create a 'Foo' with a specific 'String'. foo :: String -> Foo foo =…
illabout
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Haskell: How to write a `Monoid` instance for something that depends on parameters

I am working on a small library for the university that does integer calculations in a cyclic group; Things like: (3 (% 11)) + (10 (% 11)) --> (2 (% 11)) 'Integers (% n)' clearly form a monoid under addition with '0 (% n)' as identity element.…
Qqwy
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What can we do with Alternative but cannot do with Monoid?

I read Why MonadPlus and not Monad + Monoid? and I understand a theoretical difference, but I cannot figure out a practical difference, because for List it looks the same. mappend [1] [2] == [1] <|> [2] Yes. Maybe has different…
ais
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Applicative instance for a tuple with monoid and function inside

I was trying to convert a haskell example, I came across earlier, to scalaz. The original example was this: ("Answer to the ", (*)) <*> ("Ultimate Question of ", 6) <*> ("Life, the Universe, and Everything", 7) Which, as far as I am able to…
George
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How to write monoid protocol in Clojure?

The following does not work, for obvious reasons. (defprotocol Monoid (mappend [a b]) (mzero [])) mzero has zero arguments, and zero argument methods are not allowed (or do not make sense) in protocols. In Haskell or Scala, where the dispatch…
missingfaktor
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Why does mconcat require a list rather than a Foldable?

While looking at the definition of Monoid I noticed that mconcat has the following definition (source): mconcat :: Monoid a => [a] -> a mconcat = foldr mappend mempty Why would the signature limit this to [a] rather than the more generic Foldable…
fphilipe
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How can foldMap do the same as foldr when the latter does not have anything to do with monoids?

foldr and foldMap can be used to define each other as I understand. But how is that possible, as the latter uses monoids, while the former does not? Do we have any guarantees that the stuff the foldr works on can have a monoid?
The Unfun Cat
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Is my understanding of monoid valid?

So, I'm learning Haskell at the moment, and I would like to confirm or debunk my understanding of monoid. What I figured out from reading CIS194 course is that monoid is basically "API" for defining custom binary operation on custom set. Than I…
Reygoch
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Can you formulate a monoid or semigroup for the radix sort?

This is the pseudocode for the radix sort: Pseudocode for Radix Sort: Radix-Sort(A, d) // Each key in A[1..n] is a d-digit integer. (Digits are // numbered 1 to d from right to left.) 1. for i = 1 to d do Use a stable sorting algorithm to sort A on…
hawkeye
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Folding without Monoid instance

I have a simple tree structure: data Tree a = Leaf | Node a (Tree a) (Tree a) And a Foldable implementation: import qualified Data.Foldable as F instance F.Foldable Tree where foldMap f Leaf = mempty foldMap f (Node x l r) = F.foldMap…
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Where do the foldl/foldr implementations of Foldable come from for binary trees in haskell?

I'm working through Learn You a Haskell, and I'm on the section on monoids. In this section, the author defines the foldMap method for a tree as follows: instance F.Foldable Tree where foldMap f Empty = mempty foldMap f (Node x l r) =…
MYV
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Monoid mempty in pattern matching

I tried to write a generalized maximum function similar to the one in Prelude. My first naiv approach looked like this: maximum' :: (F.Foldable a, Ord b) => a b -> Maybe b maximum' mempty = Nothing maximum' xs = Just $ F.foldl1 max xs However, when…
mmh
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What are those class extensions for the Cartesian class for?

The Cartesian class from the constrained-category project is for categories, products of objects in which are objects in the same category yet again. I wonder about the classes Cartesian extends: class ( Category k …
Zhiltsoff Igor
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What is the purpose of the ArgMin and ArgMax type synonyms in Data.Semigroup?

The base library in Haskell has the following type synonyms in Data.Semigroup: type ArgMin a b = Min (Arg a b) type ArgMax a b = Max (Arg a b) Here are links to the haddocks: ArgMin and ArgMax What is the purpose of these two type synonyms? …
illabout
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Identity element for std::min/float monoid

In this (very interesting) talk the speaker poses a question: What is the e value for the float/std::min monoid. In other words: what is the identity element for the monoid composed of standard C++ floats and std::min operation? The speaker says…
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