Safe Haskell | Trustworthy |
---|---|
Language | Haskell2010 |
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
- gcd :: m -> m -> m
- class Monoid m => LeftReductiveMonoid m where
- isPrefixOf :: m -> m -> Bool
- stripPrefix :: m -> m -> Maybe m
- class Monoid m => RightReductiveMonoid m where
- isSuffixOf :: m -> m -> Bool
- stripSuffix :: m -> m -> Maybe m
- class LeftReductiveMonoid m => LeftCancellativeMonoid m
- class RightReductiveMonoid m => RightCancellativeMonoid m
- class LeftReductiveMonoid m => LeftGCDMonoid m where
- commonPrefix :: m -> m -> m
- stripCommonPrefix :: m -> m -> (m, m, m)
- class RightReductiveMonoid m => RightGCDMonoid m where
- commonSuffix :: m -> m -> m
- stripCommonSuffix :: m -> m -> (m, m, m)
Symmetric, commutative monoid classes
class Monoid m => CommutativeMonoid m Source
Class of all Abelian ({i.e.}, commutative) monoids that satisfy the commutativity property:
a <> b == b <> a
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) |
class (CommutativeMonoid m, LeftReductiveMonoid m, RightReductiveMonoid m) => ReductiveMonoid m where Source
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
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) |
class (LeftCancellativeMonoid m, RightCancellativeMonoid m, ReductiveMonoid m) => CancellativeMonoid m Source
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
CancellativeMonoid () | |
CancellativeMonoid a => CancellativeMonoid (Dual a) | |
Integral a => CancellativeMonoid (Sum a) | |
(CancellativeMonoid a, CancellativeMonoid b) => CancellativeMonoid (a, b) |
class (ReductiveMonoid m, LeftGCDMonoid m, RightGCDMonoid m) => GCDMonoid m where Source
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
Asymmetric monoid classes
class Monoid m => LeftReductiveMonoid m where Source
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.
isPrefixOf :: m -> m -> Bool Source
stripPrefix :: m -> m -> Maybe m Source
class Monoid m => RightReductiveMonoid m where Source
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.
isSuffixOf :: m -> m -> Bool Source
stripSuffix :: m -> m -> Maybe m Source
class LeftReductiveMonoid m => LeftCancellativeMonoid m Source
Subclass of LeftReductiveMonoid
where stripPrefix
is a complete inverse of <>
, satisfying the following
additional law:
stripPrefix a (a <> b) == Just b
class RightReductiveMonoid m => RightCancellativeMonoid m Source
Subclass of LeftReductiveMonoid
where stripPrefix
is a complete inverse of <>
, satisfying the following
additional law:
stripSuffix b (a <> b) == Just a
class LeftReductiveMonoid m => LeftGCDMonoid m where Source
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 b
commonPrefix :: m -> m -> m Source
stripCommonPrefix :: m -> m -> (m, m, m) Source
LeftGCDMonoid () | |
LeftGCDMonoid ByteString | |
LeftGCDMonoid ByteString | |
LeftGCDMonoid IntSet | |
LeftGCDMonoid Text | |
LeftGCDMonoid Text | |
LeftGCDMonoid ByteStringUTF8 | |
Eq x => LeftGCDMonoid [x] | |
RightGCDMonoid a => LeftGCDMonoid (Dual a) | |
(Integral a, Ord a) => LeftGCDMonoid (Sum a) | |
Integral a => LeftGCDMonoid (Product a) | |
LeftGCDMonoid x => LeftGCDMonoid (Maybe x) | |
Eq a => LeftGCDMonoid (IntMap a) | |
Ord a => LeftGCDMonoid (Set a) | |
Eq a => LeftGCDMonoid (Seq a) | |
Eq a => LeftGCDMonoid (Vector a) | |
(Eq a, LeftGCDMonoid a, MonoidNull a, StableFactorialMonoid a) => LeftGCDMonoid (Concat a) | |
(LeftGCDMonoid a, StableFactorialMonoid a) => LeftGCDMonoid (Measured a) | |
(StableFactorialMonoid m, TextualMonoid m, LeftGCDMonoid m) => LeftGCDMonoid (LinePositioned m) | |
(StableFactorialMonoid m, LeftGCDMonoid m) => LeftGCDMonoid (OffsetPositioned m) | |
(LeftGCDMonoid a, LeftGCDMonoid b) => LeftGCDMonoid (a, b) | |
(Ord k, Eq a) => LeftGCDMonoid (Map k a) | |
(LeftGCDMonoid a, LeftGCDMonoid b) => LeftGCDMonoid (Stateful a b) |
class RightReductiveMonoid m => RightGCDMonoid m where Source
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 b
commonSuffix :: m -> m -> m Source
stripCommonSuffix :: m -> m -> (m, m, m) Source
RightGCDMonoid () | |
RightGCDMonoid ByteString | |
RightGCDMonoid ByteString | |
RightGCDMonoid IntSet | |
LeftGCDMonoid a => RightGCDMonoid (Dual a) | |
(Integral a, Ord a) => RightGCDMonoid (Sum a) | |
Integral a => RightGCDMonoid (Product a) | |
RightGCDMonoid x => RightGCDMonoid (Maybe x) | |
Ord a => RightGCDMonoid (Set a) | |
Eq a => RightGCDMonoid (Seq a) | |
Eq a => RightGCDMonoid (Vector a) | |
(Eq a, RightGCDMonoid a, MonoidNull a, StableFactorialMonoid a) => RightGCDMonoid (Concat a) | |
(RightGCDMonoid a, StableFactorialMonoid a) => RightGCDMonoid (Measured a) | |
(StableFactorialMonoid m, TextualMonoid m, RightGCDMonoid m) => RightGCDMonoid (LinePositioned m) | |
(StableFactorialMonoid m, RightGCDMonoid m) => RightGCDMonoid (OffsetPositioned m) | |
(RightGCDMonoid a, RightGCDMonoid b) => RightGCDMonoid (a, b) | |
(RightGCDMonoid a, RightGCDMonoid b) => RightGCDMonoid (Stateful a b) |