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Struct/The max-plus algebra of natural numbers
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Title: 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 $\oplus$.
