This library provides a type class for semirings.

A semiring is an additive commutative monoid with identity zero:

Most Haskellers are familiar with the Monoid typeclass:

     zero <+> a ≡ a
(a <+> b) <+> c ≡ a <+> (b <+> c)

In this case, we've aliased

zero = mempty
(<+>) = mappend

A commutative monoid adds the requirement of symmetry:

        a <+> b ≡ b <+> a

A semiring adds the requirement of a multiplication-like operator. However, it does not require the existence of multiplicative inverses, i.e. division. Moreover, multiplication does not need to be commutative.

       one <.> a ≡ a
       a <.> one ≡ a
 (a <.> b) <.> c ≡ a <.> (b <.> c)

Multiplication distributes over addition:

a <.> (b <+> c) ≡ (a <.> b) <+> (a <.> b)
(a <+> b) <>. c ≡ (a <.> c) <+> (b <.> c)

zero annihilates a semiring with respect to multiplication:

zero <.> a ≡ zero
a <.> zero ≡ zero

The classic example of a semiring is the "Tropical numbers". The Tropical numbers, or T, are just real numbers with different operators.

zero = ∞
a <+> b = minimum {a, b}
one = 0
a <.> b = a + b

We can easily verify that these satisfy the semiring axioms:

First, the requirements for a commutative monoid

            minimum {∞, a} ≡ minimum {a, ∞} ≡ a
            minimum {a, ∞} ≡ a
            minimum {a, b} ≡ minimum {b, a}
minimum {a, minimum{b, c}} ≡ minimum {minimum {a, b}, c}

             0 + a ≡ a
             a + 0 ≡ a
       a + (b + c) ≡ (a + b) + c
a + minimum {b, c} ≡ minimum {a + b, a + c}
minimum {a, b} + c ≡ minimum {a + c, b + c}
             a + ∞ ≡ ∞
             ∞ + a ≡ ∞