Idempotent semiring

Remarks

In an idempotent semiring, the relation $a \le b \iff a + b = b$ is a partial order. For this order, $a + b$ is the least upper bound of $a$ and $b$, the element $0$ is the least element, and multiplication is monotone on both sides. Example: the Boolean semiring $\set{0, 1}$ with $a + b = \max(a, b)$ and $a \cdot b = \min(a, b)$ is idempotent, since $\max(a, a) = a$. Example: the subsets of the free monoid $\Sigma^*$ on an alphabet $\Sigma$, with union as addition and concatenation as multiplication, form an idempotent semiring, since $L \cup L = L$. Non-example: $\N$ with the usual operations is a semiring but is not idempotent, since $1 + 1 = 2 \ne 1$. A ring is idempotent only if it is the zero ring, since $a + a = a$ gives $a = 0$. Idempotent semirings are also called dioids.