Let $F$ be the set of $\alpha\subset \mathbb{Q}$ with following properties.
(I) $\alpha ≠ \emptyset$ and $\alpha ≠ \mathbb{Q}$
(II) $p\in \alpha$ and $q ⇒ $q\in \alpha$ (Notice that it's slight different from usual dedekind cut) Define $\alpha < \beta$ iff $\alpha \subsetneq \beta$. Then $F$ is fully-ordered. Plus, $F$ has least-upperbound property. For $\alpha,\beta \in F$, define $\alpha + \beta$ = {$r+s\in \mathbb{Q}$|$r\in \alpha$ and $s\in \beta$} Then $\alpha + \beta$ also satisfies properties (I)&(II). Then operation $+$ is associative and commutative and there exists an additive identity $0^*$ that is {$q\in \mathbb{Q}$|$q≦0$}. Here, i don't know how to prove that 'There doesn't exist additive inverse'. Help