Let $N$ be a set of non-negative integers. Of course we know that $a+b=0$ implies that $a=b=0$ for $a, b \in N$.
How do (or can) we prove this fact if we don't know the subtraction or order?
In other words, we can only use the addition and multiplication.
Please give me advise.
The addition law mean that for $a, b \in N$, there is an element $a+b$ in $N$ and this operation is associative. The multiplication law means that for $a, b \in N$, there is an element $ab$ in $N$ and this operation is associative. Also the distribution laws hold.
Let me rephrase the question since I don't want arguments on orders.
Let $N$ be a set with operation $+$ and $\times$.
$N$ is a monoid with the operation $+$ and $\times$ respectively. There is an unit element $0\in N$.
The distribution laws hold as in the case of the set of integers.
Can we prove the fact above with this assumption?