Algebra.Heyting.BoolRing

Documentation

newtype BoolRing a

Newtype wraper which captures Boolean ring structure, which holds for every Heyting algebra. A Boolean ring is a ring which satisfies:

a <.> a = a

Some other properties:

a <+> a = mempty                  -- thus it is a ring of characteristic 2

a <.> b = b <.> a                 -- hence it is a commutative ring

a <+> (b <+> c) = (a <+> b) <+> c -- multiplicative associativity

Constructors

BoolRing

Fields

• getBoolRing :: a

Instances

Heyting a => Semigroup (BoolRing a)

Sum is symmetric differnce.

Defined in Algebra.Heyting.BoolRing

Methods

(<>) :: BoolRing a -> BoolRing a -> BoolRing a

sconcat :: NonEmpty (BoolRing a) -> BoolRing a

stimes :: Integral b => b -> BoolRing a -> BoolRing a

Heyting a => Monoid (BoolRing a)

In a Boolean ring a + a = 0, hence negate = id.

Defined in Algebra.Heyting.BoolRing

Methods

mempty :: BoolRing a

mappend :: BoolRing a -> BoolRing a -> BoolRing a

mconcat :: [BoolRing a] -> BoolRing a

Heyting a => Semiring (BoolRing a)

Multiplication is given by /\

Defined in Algebra.Heyting.BoolRing

Methods

one :: BoolRing a

(<.>) :: BoolRing a -> BoolRing a -> BoolRing a

class Monoid m => Semiring m where

Methods

one :: m

(<.>) :: m -> m -> m

Instances

Heyting a => Semiring (BoolRing a)

Multiplication is given by /\

Defined in Algebra.Heyting.BoolRing

Methods

one :: BoolRing a

(<.>) :: BoolRing a -> BoolRing a -> BoolRing a

(<+>) :: Monoid m => m -> m -> m infixl 5

Alias for mappend.