Safe Haskell | Safe |
---|---|

Language | Haskell98 |

- class C a => C a where
- isUnit :: C a => a -> Bool
- stdAssociate, stdUnit, stdUnitInv :: C a => a -> a
- stdAssociate, stdUnit, stdUnitInv :: C a => a -> a
- stdAssociate, stdUnit, stdUnitInv :: C a => a -> a
- intQuery :: (Integral a, C a) => a -> Bool
- intAssociate :: (Integral a, C a, C a) => a -> a
- intStandard :: (Integral a, C a, C a) => a -> a
- intStandardInverse :: (Integral a, C a, C a) => a -> a
- propComposition :: (Eq a, C a) => a -> Bool
- propInverseUnit :: (Eq a, C a) => a -> Bool
- propUniqueAssociate :: (Eq a, C a) => a -> a -> Property
- propAssociateProduct :: (Eq a, C a) => a -> a -> Bool

# Class

class C a => C a where Source #

This class lets us deal with the units in a ring.
`isUnit`

tells whether an element is a unit.
The other operations let us canonically
write an element as a unit times another element.
Two elements a, b of a ring R are _associates_ if a=b*u for a unit u.
For an element a, we want to write it as a=b*u where b is an associate of a.
The map (a->b) is called
StandardAssociate by Gap,
"unitCanonical" by Axiom,
and "canAssoc" by DoCon.
The map (a->u) is called
"canInv" by DoCon and
"unitNormal(x).unit" by Axiom.

The laws are

stdAssociate x * stdUnit x === x stdUnit x * stdUnitInv x === 1 isUnit u ==> stdAssociate x === stdAssociate (x*u)

Currently some algorithms assume

stdAssociate(x*y) === stdAssociate x * stdAssociate y

Minimal definition:
`isUnit`

and (`stdUnit`

or `stdUnitInv`

) and optionally `stdAssociate`

isUnit, (stdUnit | stdUnitInv)

stdAssociate, stdUnit, stdUnitInv :: a -> a Source #

stdAssociate, stdUnit, stdUnitInv :: C a => a -> a Source #

stdAssociate, stdUnit, stdUnitInv :: C a => a -> a Source #

stdAssociate, stdUnit, stdUnitInv :: C a => a -> a Source #