Recall that a semiring is a set $R$ equipped with two binary operations, denoted $+$, and $\cdot$ and called addition and multiplication, satisfying the ring (or rng) axioms except that there may not be a zero nor a negative.

Definition

An idempotent semiring (also known as a dioid) is one in which addition is idempotent: $x + x = x$, for all $x\in R$.

Terminology

The term dioid is sometimes used as an alternative name for idempotent semirings.

Examples

Any quantale is an idempotent semiring, or dioid, under join and multiplication.

The powerset of the set of languages over a given alphabet forms an idempotent semiring in which $L + L' = L \cup L'$ and multiplication is given by concatentation.