Tropical semiring

Example

Let $T = \R \cup \set{+\infty}$. For $a, b \in T$, define $$a \oplus b = \min(a, b), \qquad a \otimes b = a + b,$$ with the conventions $\min(a, +\infty) = a$ and $a + (+\infty) = +\infty$. Then $(T, \oplus, \otimes)$ is a commutative semiring, the tropical semiring in its min-plus form.

The check of the semiring axioms:

1. $\oplus$ is associative and commutative, and $+\infty$ is its neutral element, since $\min(a, +\infty) = a$.
2. $\otimes$ is associative and commutative, and $0$ is its neutral element, since $a + 0 = a$.
3. $\otimes$ distributes over $\oplus$: $a + \min(b, c) = \min(a + b, a + c)$, because adding $a$ preserves the order.
4. $+\infty$ is absorbing for $\otimes$: $a + (+\infty) = +\infty$.

For instance, $3 \oplus 5 = 3$ and $3 \otimes 5 = 8$, and $2 \otimes (3 \oplus 7) = 5 = (2 \otimes 3) \oplus (2 \otimes 7)$.