{-# LANGUAGE NoImplicitPrelude #-} {- | Copyright : (c) Mikael Johansson 2006 Maintainer : mik@math.uni-jena.de Stability : provisional Portability : requires multi-parameter type classes The generic case of a k-algebra generated by a monoid. -} module MathObj.Algebra where import qualified Algebra.Vector as Vector import qualified Algebra.Ring as Ring import qualified Algebra.Additive as Additive import qualified Algebra.Monoid as Monoid import Algebra.Ring((*)) import Algebra.Additive((+),negate,zero) import Algebra.Monoid((<*>)) import Control.Monad(liftM2,Functor,fmap) import Data.Map(Map) import qualified Data.Map as Map import Data.List(intersperse) import PreludeBase(Ord,Eq,{-Read,-}Show,(++),(\$), concat,map,show) newtype {- (Ord a, Monoid.C a, Ring.C b) => -} T a b = Cons (Map a b) deriving (Eq {- ,Read -} ) instance Functor (T a) where fmap f (Cons x) = Cons (fmap f x) -- is an Indexable instance better than an Ord instance here? instance (Ord a, Additive.C b) => Additive.C (T a b) where (+) = zipWith (+) {- This implementation is attracting but wrong. It fails if terms are present in b that are missing in a. Default implementation is better here. (-) = zipWith (-) -} negate = fmap negate zero = Cons Map.empty zipWith :: (Ord a) => (b -> b -> b) -> (T a b -> T a b -> T a b) zipWith op (Cons ma) (Cons mb) = Cons (Map.unionWith op ma mb) instance Ord a => Vector.C (T a) where zero = zero (<+>) = (+) (*>) = Vector.functorScale instance (Ord a, Monoid.C a, Ring.C b) => Ring.C (T a b) where one = Cons \$ Map.singleton Monoid.idt Ring.one (Cons ma) * (Cons mb) = Cons \$ Map.fromListWith (+) \$ liftM2 mulMonomial (Map.toList ma) (Map.toList mb) mulMonomial :: (Monoid.C a, Ring.C b) => (a,b) -> (a,b) -> (a,b) mulMonomial (c1,m1) (c2,m2) = (c1<*>c2,m1*m2) instance (Show a, Show b) => Show (T a b) where show (Cons ma) = concat \$ intersperse "+" \$ map (\(m,c) -> show c ++ "." ++ show m) (Map.toList ma) monomial :: a -> b -> (T a b) monomial index coefficient = Cons (Map.singleton index coefficient)