A representation of a free group over an alphabet a.

The intuition here is that Left a represents a "negative" a, whereas Right a represents "positive" a.

Note: This does not perform simplification upon multiplication or construction. To do this, one should use simplify.

O(n) Simplifies a word in a free group.

Examples:

>>> simplify $ FreeGroup [Right 'a', Left 'b', Right 'c', Left 'c', Right 'b', Right 'a']
FreeGroup {runFreeGroup = [Right 'a',Right 'a']}

O(n) Interpret a word in a free group over some group g as an element in a group g.

O(n) Strict variant of interpret.

Present a Group as a FreeGroup modulo relations.

A representation of a free abelian group over an alphabet a.

The intuition here is group elements correspond with their positive or negative multiplicities, and as such are simplified by construction.

Examples:

>>> let single a = MkFreeAbelianGroup $ Map.singleton a 1
>>> a = single 'a'
>>> b = single 'b'
>>> a
FreeAbelianGroup $ fromList [('a',1)]
>>> a <> b
FreeAbelianGroup $ fromList [('a',1),('b',1)]
>>> a <> b == b <> a
True
>>> invert a
FreeAbelianGroup $ fromList [('a',-1)]
>>> a <> b <> invert a
FreeAbelianGroup $ fromList [('b',1)]
>>> gtimes 5 (a <> b)
FreeAbelianGroup $ fromList [('a',5),('b',5)]

Bidirectional pattern synonym for the construction of FreeAbelianGroups.

O(n) Constructs a FreeAbelianGroup from a finite Map from the set of generators (a) to its multiplicities.

O(1) Gets a representation of FreeAbelianGroup as Map. The returned map contains no records with multiplicity 0 i.e. lookup a on the returned map never returns Just 0.

Given a function from generators to an abelian group g, lift that function to a group homomorphism from FreeAbelianGroup to g.

In other words, it's a function analogus to foldMap for Monoid or goldMap for Group.

Functorial fmap for a FreeAbelianGroup.

Examples:

>>> singleton 'a' <> singleton 'A'
FreeAbelianGroup $ fromList [('A',1),('a',1)]
>>> import Data.Char (toUpper)
>>> abmap toUpper $ singleton 'a' <> singleton 'A'
FreeAbelianGroup $ fromList [('A',2)]

Monadic join for a FreeAbelianGroup.

Lift a singular value into a FreeAbelianGroup. Analogous to pure.

Examples:

>>> singleton "foo"
FreeAbelianGroup $ fromList [("foo",1)]

Interpret a free group as a word in the underlying group g.