```{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE UndecidableInstances #-}

-- | These algebraic structures have sacrificed generality in favor of being easily used with the standard Haskell Prelude.  The fact that monoids are not guaranteed to be semigroups makes this difficult.

module HLearn.Algebra.Structures
(
-- * Type classes
RegularSemigroup (..)
, Group(..)

-- * Free Structures
, RegSG2Group (..)

, module Data.Semigroup
)
where

import Data.Semigroup

-------------------------------------------------------------------------------
-- Inverses

-- | Semigroups that also have an inverse.  See <https://en.wikipedia.org/wiki/Regular_semigroup>
class (Semigroup g) => RegularSemigroup g where
inverse :: g -> g

-- -- | Semigroups where a unique inverse exists for each element.  See <https://en.wikipedia.org/wiki/Inverse_semigroup>
-- class (RegularSemigroup g) => InverseSemigroup g

-- | Regular semigroups that also have an identity; alternatively, monoids where every element has a unique inverse.  See <https://en.wikipedia.org/wiki/Group_(mathematics)>
class (RegularSemigroup g, Monoid g) => Group g

-------------------------------------------------------------------------------
-- RegSG2Group

-- | Convert any regular semigroup into a group (and thus also a monoid) by adding a unique identity element
data (RegularSemigroup sg) => RegSG2Group sg = SGNothing | SGJust sg

instance (RegularSemigroup sg) => Semigroup (RegSG2Group sg) where
SGNothing <> m = m
m <> SGNothing = m
(SGJust sg1) <> (SGJust sg2) = SGJust \$ sg1<>sg2

instance (RegularSemigroup sg) => RegularSemigroup (RegSG2Group sg) where
inverse SGNothing = SGNothing
inverse (SGJust x) = SGJust \$ inverse x

instance (RegularSemigroup sg) => Monoid (RegSG2Group sg) where
mempty = SGNothing
mappend = (<>)

instance (RegularSemigroup sg) => Group (RegSG2Group sg)

-- -------------------------------------------------------------------------------
-- -- SG2Monoid
--
-- data (Semigroup sg) => SG2Monoid sg = SGNothing | SGJust sg