nLab
idempotent semiring

Idea

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 $L+L\prime =L\cup L\prime$ and multiplication is given by concatentation.

• The tropical algebra and the max-plus algebra are idempotent semirings.