-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Algebraic structures
--   
--   Attempt to define algebraic structures in more robust and useful way.
@package morphisms-objects
@version 0.1.3

module Control.Object.Semigroup

-- | <pre>
--   When providing a new instance, you should ensure it satisfies the one law:
--   * Associativity: x &lt;&gt; (y &lt;&gt; z) ≡ (x &lt;&gt; y) &lt;&gt; z
--   </pre>
class Semigroup a

-- | Infix version of <tt>append</tt>
(<>) :: Semigroup a => a -> a -> a

module Control.Object.Monoid

-- | <pre>
--   When providing a new instance, you should ensure it satisfies the two law:
--   * Right absorption: unit &lt;&gt; x ≡ x
--   * Left absorption: x &lt;&gt; unit ≡ x
--   </pre>
class Semigroup a => Monoid a
unit :: Monoid a => a

module Control.Object.Group

-- | <pre>
--   When providing a new instance, you should ensure it satisfies the two law:
--   * Right absorption: x &lt;&gt; inverse x ≡ unit
--   * Left absorption: inverse x &lt;&gt; x ≡ unit
--   </pre>
class Monoid a => Group a
inverse :: Group a => a -> a

module Control.Object.Semilattice

-- | <pre>
--   When providing a new instance, you should ensure it satisfies the three laws:
--   * Associativity: x /\ (y /\ z) ≡ (x /\ y) /\ z
--   * Commutativity: x /\ y ≡ y /\ x
--   * Idempotency: x /\ x ≡ x
--   </pre>
class Infimum a
(/\) :: Infimum a => a -> a -> a

-- | <pre>
--   When providing a new instance, you should ensure it satisfies the three laws:
--   * Associativity: x \/ (y \/ z) ≡ (x \/ y) \/ z
--   * Commutativity: x \/ y ≡ y \/ x
--   * Idempotency: x \/ x ≡ x
--   </pre>
class Supremum a
(\/) :: Supremum a => a -> a -> a

module Control.Object.Lattice

-- | <pre>
--   When providing a new instance, you should ensure it satisfies the one law:
--   * Absorption: a \/ (a /\ b) ≡ a /\ (a \/ b) ≡ a
--   </pre>
class (Infimum a, Supremum a) => Lattice a

module Control.Object.Semiring

-- | <pre>
--   When providing a new instance, you should ensure it satisfies the two laws:
--   * Commutativity: a &gt;&lt; b = b &gt;&lt; a
--   * Distributivity: x &lt;&gt; (y &gt;&lt; z) = x &lt;&gt; y &gt;&lt; x &lt;&gt; z
--   </pre>
class Semigroup a => Semiring a
(><) :: Semiring a => a -> a -> a

module Control.Object.Setoid
data Boolean
True :: Boolean
False :: Boolean
(&&) :: Boolean -> Boolean -> Boolean
infixr 3 &&
(||) :: Boolean -> Boolean -> Boolean
infixr 9 ||
not :: Boolean -> Boolean
bool :: a -> a -> Boolean -> a

-- | <pre>
--   When providing a new instance, you should ensure it satisfies the four law:
--   * Reflexivity: x == x ≡ True
--   * Symmetry: x == y ≡ y == x
--   * Transitivity: x == y &amp;&amp; y == z ≡ True ===&gt; x == z ≡ True
--   * Negation: x /= y ≡ not (x == y)
--   </pre>
class Setoid a
(==) :: Setoid a => a -> a -> Boolean
(/=) :: Setoid a => a -> a -> Boolean

module Control.Object.Chain
data Ordering
Less :: Ordering
Equal :: Ordering
Greater :: Ordering
order :: a -> a -> a -> Ordering -> a

-- | <pre>
--   When providing a new instance, you should ensure it satisfies the three law:
--   * Reflexivity: x &lt;= x ≡ True
--   * Transitivity: x &lt;= y &amp;&amp; y &lt;= z ≡ True ===&gt; x &lt;= z ≡ True
--   * Antisymmetry: x &lt;= y &amp;&amp; y &lt;= x ≡ True ===&gt; x == y ≡ True
--   </pre>
class Setoid a => Chain a
(<=) :: Chain a => a -> a -> Ordering
(>=) :: Chain a => a -> a -> Ordering
(<) :: Chain a => a -> a -> Boolean
(>) :: Chain a => a -> a -> Boolean

module Control.Object