Definition:Addition

Let $$\left({S, \circ; \preceq}\right)$$ be a naturally ordered semigroup.

Then the operation $$\circ$$ in $$\left({S, \circ; \preceq}\right)$$ is called addition.

Natural Numbers
The addition operation in the domain of natural numbers is written $$+$$.

As the Natural Numbers are a Naturally Ordered Semigroup, it is appropriate to write this set as $$\left({\mathbb{N}, +; \le}\right)$$.

From this, it can be seen that $$+$$ corresponds with the operation $$\circ$$ in the structure $$\left({S, \circ; \preceq}\right)$$.