Idempotent semiring

Summary

An idempotent semiring, also called a dioid, is a semiring in which $a + a = a$ for every element $a$. In such a semiring the addition defines a natural partial order, with $a \le b$ when $a + b = b$, and addition becomes the join of this order. The Boolean semiring, the tropical semirings and the semiring of formal languages are examples. Idempotent semirings are the algebra of shortest paths, of tropical geometry and of Kleene algebras.