Let $S$ be an abelian semigroup generated by a nonempty set $X$. Suppose $\sim$ is an equivalence relation on $S$ satisfying the condition:
"If $x \sim x'$ and $y \sim y'$ then $x+y \sim x'+y'$ for all $x, x', y, y' \in X$."
Does it follow that $\sim~$ induces a semigroup congruence on S?
To prove this, I thought to define the height of an element s in S as the minimal number of generators it takes to express s. We could then induct on heights somehow, but I'm not sure how to set this up.
Any input would be appreciated.
It is not true. Consider the (free) Abelian semigroup on one element (say, $\Bbb Z^+=\{1,2,3,..\}$), it is generated by $1$, and consider any equivalence relation. Since $|X|=1$, ~ will satisfy the criterium. (only $1+1\sim 1+1$ is required).