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.