Data.Semiring

Contents

• Endofunctors for construction
  • first/right like functions
• Laws (Axioms) for building algebraic structures
• Categorical Algebraic Structures
• Arrow like functions for semiring categories
• A groupoid class that is a category. Maybe this is a bad idea?
• Alegraic laws as isomorphism for groupoid instances

Description

Synopsis

• class Category k => Ctor k constr | constr -> k where
  • selfmap :: k a b -> k c d -> k (constr a c) (constr b d)
• promote :: Ctor k op => k a b -> k (op a c) (op b c)
• swap_promote :: Ctor k op => k a b -> k (op c a) (op c b)
• class Ctor k op => Absorbs k op id | op -> k, op -> id where
  • absorb :: k (op id a) a
  • unabsorb :: k a (op id a)
• class Ctor k op => Assocative k op | op -> k where
  • assoc :: k (op (op a b) c) (op a (op b c))
  • unassoc :: k (op a (op b c)) (op (op a b) c)
• class Ctor k op => Commutative k op | op -> k where
  • commute :: k (op a b) (op b a)
• class Ctor k op => Annihilates k op zero | op zero -> k where
  • annihilates :: k (op zero a) zero
• class (Ctor k add, Ctor k multi) => Distributes k add multi | add multi -> k where
  • distribute :: k (multi (add a b) c) (add (multi a c) (multi b c))
  • undistribute :: k (add (multi a c) (multi b c)) (multi (add a b) c)
• class (Assocative k dot, Absorbs k dot id) => Monoidial k dot id | dot id -> k
• class (Monoidial k dot id, Commutative k dot) => CommutativeMonoidial k dot id | dot id -> k
• class (CommutativeMonoidial k add zero, CommutativeMonoidial k multi one, Annihilates k multi zero, Distributes k add multi) => Semiring k add zero multi one | add zero multi one -> k
• first :: Semiring a add zero multi one => a b c -> a (multi b d) (multi c d)
• second :: Semiring a add zero multi one => a b c -> a (multi d b) (multi d c)
• left :: Semiring a add zero multi one => a b c -> a (add b d) (add c d)
• right :: Semiring a add zero multi one => a b c -> a (add d b) (add d c)
• class Category g => Groupoid g where
  • inv :: g a b -> g b a
• data Iso k a b = Iso {
  • embed :: k a b
  • project :: k b a
  }
• biject_sum_absorb :: Biject (BSum BZero a) a
• biject_sum_assoc :: Biject (BSum (BSum a b) c) (BSum a (BSum b c))
• biject_product_absorb :: Biject (BProduct BOne a) a
• biject_product_assoc :: Biject (BProduct (BProduct a b) c) (BProduct a (BProduct b c))
• biject_distributes :: Biject (BProduct (BSum a b) c) (BSum (BProduct a c) (BProduct b c))
• kbiject_sum_absorb :: Monad m => KBiject m (KBSum KBZero a) a
• kbiject_sum_assoc :: Monad m => KBiject m (KBSum (KBSum a b) c) (KBSum a (KBSum b c))
• kbiject_product_absorb :: Monad m => KBiject m (KBProduct KBOne a) a
• kbiject_product_assoc :: Monad m => KBiject m (KBProduct (KBProduct a b) c) (KBProduct a (KBProduct b c))
• kbiject_distributes :: Monad m => KBiject m (KBProduct (KBSum a b) c) (KBSum (KBProduct a c) (KBProduct b c))

Endofunctors for construction

first/right like functions

Laws (Axioms) for building algebraic structures

Categorical Algebraic Structures

Arrow like functions for semiring categories

A groupoid class that is a category. Maybe this is a bad idea?

Alegraic laws as isomorphism for groupoid instances