Haskellers are usually familiar with monoids and semigroups. A monoid has
an appending operation <> (or mappend), and an identity element, mempty. A
semigroup has an appending <> operation, but does not require a mempty
element.
A Semiring has two appending operations, plus and times, and two respective
identity elements, zero and one.
More formally, a Semiring R is a set equipped with two binary relations +
and *, such that:
- (R,+) is a commutative monoid with identity element 0,
- (R,*) is a monoid with identity element 1,
- (*) left and right distributes over addition, and multiplication by '0'
annihilates R.
