BSD-3-Clause licensed by Mario Blažević
Maintained by Mario Blažević
This version can be pinned in stack with:monoid-subclasses-,2647


Subclasses of Semigroup and Monoid with a solid theoretical foundation and practical purposes

The monoid-subclasses package has been released on Hackage. The package defines several classes that are richer than semigroups and monoids but less demanding than groups:

  • Reductive provides the operator </> which acts as a partial inverse of the semigroup <> operator.

  • Cancellative is a subclass of Reductive that provides additional guarantees about the </> operation result:

      (a <> b) </> a == Just b
      (a <> b) </> b == Just a

    Every group (i.e., every Monoid a with the operation inverse :: a -> a) is a cancellative monoid where a </> b = Just (a <> inverse b) but not every Cancellative monoid is a group.

  • GCDMonoid is a subclass of Reductive and Monoid that provides the gcd operation for getting the greatest common denominator for two given monoid values.

  • LCMMonoid is a subclass of Reductive and Monoid that provides the lcm operation for getting the least common multiple for two given monoid values.

  • Monus provides the <\> monus operation. The set difference is one familiar instance of this operation.

  • MonoidNull class provides the Boolean null operation that checks if the argument monoid is mempty.

  • Factorial and FactorialMonoid classes represent semigroups and monoids that can be split up into irreducible factors.

That’s the theoretical point of view. From the practical point of view, the main purpose of the monoid-subclasses package is similar to that of ListLike - to provide unifying abstractions for various monoidal data types in Haskell, primarily String, ByteString, and Text. All three types are already instances of the Monoid class. While that abstraction is useful for building sequences of data, it doesn’t help with deconstructing them.

That being said, there are two major differences in the goals of ListLike and monoid-subclasses:

  • ListLike strives to reproduce the standard Data.List interface, whereas monoid-subclasses builds from deeper theoretical foundations; and
  • The monoid-subclasses implementation uses standard Haskell 2010, with the exception of two minor extensions which can be worked around if necessary.

The incremental-parser package can serve as a compact example of a parser library that can be applied to different input types thanks to monoid-subclasses. There is also picoparsec, a fork of attoparsec, and the heavy-duty grammatical-parsers library.

A more thorough description of the library design can be found in the Haskell Symposium 2013 paper Adding Structure to Monoids



  • Bumped the containers dependency upper bound

Version 1.2.5

  • Data.Monoid.Null.MonoidNull now has a default implementation
  • Add instances of MonoidNull and (where appropriate) PositiveMonoid for:
    • (a, b, c, d, e)
    • Data.Functor.Compose f g a
    • Data.Functor.Const r a
    • Data.Functor.Identity a
    • Data.Functor.Product f g a
    • Data.Ord.Down a
    • Data.Proxy.Proxy a
    • Data.Semigroup.Max a
    • Data.Semigroup.Min a
    • Data.Semigroup.WrappedMonoid a
    • GHC.Generics.(:*:) f g a
    • GHC.Generics.(:.:) f g a
    • GHC.Generics.K1 i c p
    • GHC.Generics.M1 i c f p
    • GHC.Generics.Par1 p
    • GHC.Generics.Rec1 f p
    • GHC.Generics.U1 p
  • Added instances of Reductive for Max a, Min a, Any, and All
  • Bumped the commutative-semigroups dependency upper bound


  • Bumped the text dependency upper bound

Version 1.2.4

  • Added Data.Monoid.Instances.PrefixMemory.Shadowed monoid transformer

Version 1.2.3

  • Added DistributiveGCDMonoid and DistributiveLCMMonoid type class by Jonathan Knowles

Version 1.2.2

  • Added Data.Monoid.LCM module with LCMMonoid type class by Jonathan Knowles
  • Repaired links to Hackage within by Jonathan Knowles

Version 1.2.1

  • Fix for the Monus instance for Maybe by Jonathan Knowles

Version 1.2

  • Dropped support for GHC < 8.4
  • Depending on new commutative-semigroups package
  • Modified the instance OverlappingMonoid/Monus Map/IntMap instances to conform with the class laws
  • Bumped the vector dependency upper bounds

Version 1.1.4

  • Canonicalized all mappend definitions
  • Added deriving (Data, Typeable) to all data types

Version 1.1.3

  • Support for text-2.0 by Bodigrim

Version 1.1.2

  • CI tests
  • Fallback implementation of stripCommonSuffix @Text for GHCjs by Jack Kelly
  • Fixed documentation bug #31, Factorial laws too strong

Version 1.1.1

  • Fixed compilation with GHC 8.0.2
  • Positioned doesn’t use a column for zero-width characters any more

Version 1.1

  • Added a new TextualMonoid method toText for performance
  • Fixed the calculations of column of LinePositioned
  • Changed the column of LinePositioned to be one-based
  • LinePositioned now treats FF, CR, and TAB characters as special, in accordance with Haskell language specification.

Version 1.0.1

  • Incremented the lower bound of the base dependency
  • Fixed Haddock links

Version 1.0

  • Fixed issue #24, unlawful LeftReductiveMonoid instance for Map
  • Tightened the laws of GCD classes, dropped instances for Sum and Product
  • Introduced the Monus class
  • Introduced the OverlappingGCDMonoid class
  • Added the instances of type Sum Natural and Product Natural
  • Using the language extensions FlexibleInstances and OverlappingInstances
  • Removed the linear complexity requirement
  • Added and documented less efficient instances
  • Moved various GCD classes into the new module Data.Monoid.GCD
  • Added module Data.Semigroup.Cancellative with Semigroup subclasses
  • Added module Data.Semigroup.Factorial with Semigroup subclasses
  • Deprecated several Monoid subclasses and made them constraint synonyms instead:
    • type CommutativeMonoid m = (Monoid m, Commutative m)
    • type ReductiveMonoid m = (Monoid m, Reductive m)
    • type LeftReductiveMonoid m = (Monoid m, LeftReductive m)
    • type RightReductiveMonoid m = (Monoid m, RightReductive m)
    • type CancellativeMonoid m = (Monoid m, Cancellative m)
    • type LeftCancellativeMonoid m = (Monoid m, LeftCancellative m)
    • type RightCancellativeMonoid m = (Monoid m, RightCancellative m)


Bumped the containers dependency upper bounds

Version 0.4.6

Added the Semigroup instances to fix the compilation errors with base-4.11

Version 0.4.4

Fixed boundary condition bugs in ByteStringUTF8 uncovered by a new version of QuickCheck


Fixed compilation errors with GHC 7.8.4 and older


Bumped the vector dependency upper bounds

Version 0.4.3

  • Added instances for 3- and 4-tuples
  • Re-implemented Concat as an own data type, dropping Seq


  • Fixed compilation problems with GHC 8 and containers-0.5.7
  • Fixed compilation problems with GHC 8 and containers-0.5.7
  • Merge pull request #10 from mgiles: minor typo in FactorialMonoid laws

Version 0.4.2

  • Fixed a bug in splitAt implementation for ByteStringUTF8
  • Merge pull request #9 from phadej: use newest quickcheck-instances
  • Removed the overzealous assertions from ByteStringUTF8


Removing accidental reference to Instances.Markup module


  • Bumped the vector dependency upper bounds
  • Removed GHC-prof-options from the cabal file

Version 0.4.1

  • Changed the Prelude imports to enable compilation with GHC 7.4
  • Added INLINE pragmas
  • Added the toString method to TextualMonoid class
  • Importing Text.Show.Functions to avoid overlapping instances
  • Eliminated the redundant import warnings from GHC 7.10.1


  • Added -Wall GHC option and eliminated almost all the warnings
  • Fixed a bug in the Textual instance of ByteStringUTF8


  • Excluding the imports of foldMap from Prelude


  • Added more tests and fixed a bug in Stateful
  • Fixed a bug in Positioned.span_
  • Optimized the Stateful data type


  • Added a bunch of pragmas

Version 0.3.6

  • Deprecated all the inject functions
  • Registered the new Stateful module


Accommodating the text-1.0 release

  • Introduced the function ByteStringUTF8.decode
  • Removed the utf-string dependency
  • Replaced the utf-string import by a more efficient UTF-8 encoding

Version 0.3.1

  • Added the Data.Monoid.Instances.Concat module and tests
  • Added the PositiveMonoid class
  • Added the StableFactorialMonoid subclass of FactorialMonoid
  • Added more instances for ()

Version 0.3

Added the CommutativeMonoid class at the root of the Cancellative classes

Version 0.2

  • Added TextualMonoid instances for Seq Char and Vector Char
  • Renamed the FactorialMonoid method map to foldMap in keeping with Foldable

Version 0.1.2

Optimizations of the default Factorial methods and of the ButeStringUTF8 instances

Version 0.1

Initial release