semirings-0.6: two monoids as one, in holy haskimony

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.