TITLE: Partition of $\mathbb{R}^+$ into two semigroups
QUESTION [10 upvotes]: Can the semigroup  $(\mathbb R^+ ,+)$ be partitioned into two semigroups? 

I have been trying but haven't found anything, please help.

REPLY [3 votes]: Let $H=\{h_i:i\in I\}$ be a Hamel basis for $\mathbb R$ over $\mathbb Q$, where the index set $I$ is well-ordered. Thus each real number $x$ has a unique expression as a finite linear combination of elements of $H$ with rational coefficients, i.e., $x=\sum_{i\in I}x_ih_i$ where $x_i\in\mathbb Q$ and $\{i\in I:x_i\ne0\}$ is finite. If $x\ne0$, define $f(x)$ to be the first nonzero coefficient in this expression, i.e., $f(x)=x_i\ne0$, and $x_j=0$ for all $j\lt i$.
Let$$A=\{x\in\mathbb R^+:f(x)\gt0\}$$and let$$B=\{x\in\mathbb R^+:f(x)\lt0\}.$$
It is easy to see that $A$ and $B$ are disjoint nonempty sets whose union is $\mathbb R^+$, and that each is closed under addition.