| Safe Haskell | Trustworthy |
|---|---|
| Language | Haskell2010 |
Data.Monoid.Cancellative
Description
This module defines the Monoid => ReductiveMonoid => (CancellativeMonoid, GCDMonoid) class hierarchy.
The ReductiveMonoid class introduces operation </> which is the inverse of <>. For the Sum monoid, this
operation is subtraction; for Product it is division and for Set it's the set difference. A ReductiveMonoid is
not a full group because </> may return Nothing.
The CancellativeMonoid subclass does not add any operation but it provides the additional guarantee that <> can
always be undone with </>. Thus Sum is a CancellativeMonoid but Product is not because (0*n)/0 is not
defined.
The GCDMonoid subclass adds the gcd operation which takes two monoidal arguments and finds their greatest common
divisor, or (more generally) the greatest monoid that can be extracted with the </> operation from both.
All monoid subclasses listed above are for Abelian, i.e., commutative or symmetric monoids. Since most practical monoids in Haskell are not Abelian, each of the these classes has two symmetric superclasses:
- class Monoid m => CommutativeMonoid m
- class (CommutativeMonoid m, LeftReductiveMonoid m, RightReductiveMonoid m) => ReductiveMonoid m where
- class (LeftCancellativeMonoid m, RightCancellativeMonoid m, ReductiveMonoid m) => CancellativeMonoid m
- class (ReductiveMonoid m, LeftGCDMonoid m, RightGCDMonoid m) => GCDMonoid m where
- class Monoid m => LeftReductiveMonoid m where
- class Monoid m => RightReductiveMonoid m where
- class LeftReductiveMonoid m => LeftCancellativeMonoid m
- class RightReductiveMonoid m => RightCancellativeMonoid m
- class LeftReductiveMonoid m => LeftGCDMonoid m where
- class RightReductiveMonoid m => RightGCDMonoid m where
Symmetric, commutative monoid classes
class Monoid m => CommutativeMonoid m #
Class of all Abelian ({i.e.}, commutative) monoids that satisfy the commutativity property:
a <> b == b <> a
Instances
| CommutativeMonoid () # | |
| CommutativeMonoid IntSet # | |
| CommutativeMonoid a => CommutativeMonoid (Dual a) # | |
| Num a => CommutativeMonoid (Sum a) # | |
| Num a => CommutativeMonoid (Product a) # | |
| Ord a => CommutativeMonoid (Set a) # | |
| (CommutativeMonoid a, CommutativeMonoid b) => CommutativeMonoid (a, b) # | |
| (CommutativeMonoid a, CommutativeMonoid b, CommutativeMonoid c) => CommutativeMonoid (a, b, c) # | |
| (CommutativeMonoid a, CommutativeMonoid b, CommutativeMonoid c, CommutativeMonoid d) => CommutativeMonoid (a, b, c, d) # | |
class (CommutativeMonoid m, LeftReductiveMonoid m, RightReductiveMonoid m) => ReductiveMonoid m where #
Class of Abelian monoids with a partial inverse for the Monoid <> operation. The inverse operation </> must
satisfy the following laws:
maybe a (b <>) (a </> b) == a maybe a (<> b) (a </> b) == a
Minimal complete definition
Instances
| ReductiveMonoid () # | |
| ReductiveMonoid IntSet # | |
| ReductiveMonoid a => ReductiveMonoid (Dual a) # | |
| Integral a => ReductiveMonoid (Sum a) # | |
| Integral a => ReductiveMonoid (Product a) # | |
| Ord a => ReductiveMonoid (Set a) # | |
| (ReductiveMonoid a, ReductiveMonoid b) => ReductiveMonoid (a, b) # | |
| (ReductiveMonoid a, ReductiveMonoid b, ReductiveMonoid c) => ReductiveMonoid (a, b, c) # | |
| (ReductiveMonoid a, ReductiveMonoid b, ReductiveMonoid c, ReductiveMonoid d) => ReductiveMonoid (a, b, c, d) # | |
class (LeftCancellativeMonoid m, RightCancellativeMonoid m, ReductiveMonoid m) => CancellativeMonoid m #
Subclass of ReductiveMonoid where </> is a complete inverse of the Monoid <> operation. The class instances
must satisfy the following additional laws:
(a <> b) </> a == Just b (a <> b) </> b == Just a
Instances
| CancellativeMonoid () # | |
| CancellativeMonoid a => CancellativeMonoid (Dual a) # | |
| Integral a => CancellativeMonoid (Sum a) # | |
| (CancellativeMonoid a, CancellativeMonoid b) => CancellativeMonoid (a, b) # | |
| (CancellativeMonoid a, CancellativeMonoid b, CancellativeMonoid c) => CancellativeMonoid (a, b, c) # | |
| (CancellativeMonoid a, CancellativeMonoid b, CancellativeMonoid c, CancellativeMonoid d) => CancellativeMonoid (a, b, c, d) # | |
class (ReductiveMonoid m, LeftGCDMonoid m, RightGCDMonoid m) => GCDMonoid m where #
Class of Abelian monoids that allow the greatest common denominator to be found for any two given values. The operations must satisfy the following laws:
gcd a b == commonPrefix a b == commonSuffix a b Just a' = a </> p && Just b' = b </> p where p = gcd a b
If a GCDMonoid happens to also be a CancellativeMonoid, it should additionally satisfy the following laws:
gcd (a <> b) (a <> c) == a <> gcd b c gcd (a <> c) (b <> c) == gcd a b <> c
Minimal complete definition
Instances
| GCDMonoid () # | |
| GCDMonoid IntSet # | |
| GCDMonoid a => GCDMonoid (Dual a) # | |
| (Integral a, Ord a) => GCDMonoid (Sum a) # | |
| Integral a => GCDMonoid (Product a) # | |
| Ord a => GCDMonoid (Set a) # | |
| (GCDMonoid a, GCDMonoid b) => GCDMonoid (a, b) # | |
| (GCDMonoid a, GCDMonoid b, GCDMonoid c) => GCDMonoid (a, b, c) # | |
| (GCDMonoid a, GCDMonoid b, GCDMonoid c, GCDMonoid d) => GCDMonoid (a, b, c, d) # | |
Asymmetric monoid classes
class Monoid m => LeftReductiveMonoid m where #
Class of monoids with a left inverse of mappend, satisfying the following law:
isPrefixOf a b == isJust (stripPrefix a b) maybe b (a <>) (stripPrefix a b) == b a `isPrefixOf` (a <> b)
| Every instance definition has to implement at least the stripPrefix method. Its complexity should be no worse
than linear in the length of the prefix argument.
Minimal complete definition
Instances
class Monoid m => RightReductiveMonoid m where #
Class of monoids with a right inverse of mappend, satisfying the following law:
isSuffixOf a b == isJust (stripSuffix a b) maybe b (<> a) (stripSuffix a b) == b b `isSuffixOf` (a <> b)
| Every instance definition has to implement at least the stripSuffix method. Its complexity should be no worse
than linear in the length of the suffix argument.
Minimal complete definition
Instances
class LeftReductiveMonoid m => LeftCancellativeMonoid m #
Subclass of LeftReductiveMonoid where stripPrefix is a complete inverse of <>, satisfying the following
additional law:
stripPrefix a (a <> b) == Just b
Instances
| LeftCancellativeMonoid () # | |
| LeftCancellativeMonoid ByteString # | |
| LeftCancellativeMonoid ByteString # | |
| LeftCancellativeMonoid Text # | |
| LeftCancellativeMonoid Text # | |
| LeftCancellativeMonoid ByteStringUTF8 # | |
| Eq x => LeftCancellativeMonoid [x] # | |
| RightCancellativeMonoid a => LeftCancellativeMonoid (Dual a) # | |
| Integral a => LeftCancellativeMonoid (Sum a) # | |
| Eq a => LeftCancellativeMonoid (Seq a) # | |
| Eq a => LeftCancellativeMonoid (Vector a) # | |
| (LeftCancellativeMonoid a, LeftCancellativeMonoid b) => LeftCancellativeMonoid (a, b) # | |
| (LeftCancellativeMonoid a, LeftCancellativeMonoid b, LeftCancellativeMonoid c) => LeftCancellativeMonoid (a, b, c) # | |
| (LeftCancellativeMonoid a, LeftCancellativeMonoid b, LeftCancellativeMonoid c, LeftCancellativeMonoid d) => LeftCancellativeMonoid (a, b, c, d) # | |
class RightReductiveMonoid m => RightCancellativeMonoid m #
Subclass of LeftReductiveMonoid where stripPrefix is a complete inverse of <>, satisfying the following
additional law:
stripSuffix b (a <> b) == Just a
Instances
| RightCancellativeMonoid () # | |
| RightCancellativeMonoid ByteString # | |
| RightCancellativeMonoid ByteString # | |
| RightCancellativeMonoid Text # | |
| RightCancellativeMonoid Text # | |
| LeftCancellativeMonoid a => RightCancellativeMonoid (Dual a) # | |
| Integral a => RightCancellativeMonoid (Sum a) # | |
| Eq a => RightCancellativeMonoid (Seq a) # | |
| Eq a => RightCancellativeMonoid (Vector a) # | |
| (RightCancellativeMonoid a, RightCancellativeMonoid b) => RightCancellativeMonoid (a, b) # | |
| (RightCancellativeMonoid a, RightCancellativeMonoid b, RightCancellativeMonoid c) => RightCancellativeMonoid (a, b, c) # | |
| (RightCancellativeMonoid a, RightCancellativeMonoid b, RightCancellativeMonoid c, RightCancellativeMonoid d) => RightCancellativeMonoid (a, b, c, d) # | |
class LeftReductiveMonoid m => LeftGCDMonoid m where #
Class of monoids capable of finding the equivalent of greatest common divisor on the left side of two monoidal values. The methods' complexity should be no worse than linear in the length of the common prefix. The following laws must be respected:
stripCommonPrefix a b == (p, a', b')
where p = commonPrefix a b
Just a' = stripPrefix p a
Just b' = stripPrefix p b
p == commonPrefix a b && p <> a' == a && p <> b' == b
where (p, a', b') = stripCommonPrefix a bMinimal complete definition
Instances
class RightReductiveMonoid m => RightGCDMonoid m where #
Class of monoids capable of finding the equivalent of greatest common divisor on the right side of two monoidal values. The methods' complexity must be no worse than linear in the length of the common suffix. The following laws must be respected:
stripCommonSuffix a b == (a', b', s)
where s = commonSuffix a b
Just a' = stripSuffix p a
Just b' = stripSuffix p b
s == commonSuffix a b && a' <> s == a && b' <> s == b
where (a', b', s) = stripCommonSuffix a bMinimal complete definition
Instances