Struct/The max-plus algebra of natural numbers

Consider the ordered triple $(\NN, \oplus, \otimes)$, where $\NN$ denotes the set of natural numbers, and $\oplus$ and $\otimes$ stand for the following operations:
 * If $x,y \in \NN$, then $x \oplus y = max(x,y)$ (the maximum of $x$ and $y$).
 * If $x,y \in \NN$, then $x \otimes y = x + y$ (ordinary addition of integers).

The operation $\oplus$ is commutative, and associative, and the ordered pair $(\NN, \oplus)$ is a monoid, but is not a group. The identity element in this monoid is $0$.

The operation $\otimes$ is commutative and associative, and the ordered pair $(\NN, \otimes)$ is a monoid, but not a group. The identity element for $(\NN, \otimes)$ is $0$.

The operation $\otimes$ distributes over $\oplus$.