Suppose we have a structure $U$ = ($\mathbb{R}^{+}\cup\{0\}; 0, 1, *)$ where * is multiplication. Equality is also present. What are the definable elements of $U$?
My thoughts:
I think you can define inequality here (although it's not an element of $U$). And I also think now that you can only define 0,1.
and on a similar note... is $<$ definable on $(\mathbb{R};0,+)$? I suspect not. I think there is a way to do it in the naturals ($\exists x \in \mathbb{N}(y=z+x \wedge x+y \neq y))$ but I don't see how you can do it in R given the existence of negative numbers. Any help would be appreciated.
Hint: note that $x\mapsto x^\alpha$ is an automorphism for each $\alpha\neq 0$. This is enough to answer both questions.
(More generally, if you ignore $0$, it is a divisible, torsion-free abelian group. Those have quantifier elimination, so definable sets are quantifier-free definable. The zero doesn't change the picture much: there are essentially no interactions between it and the rest of the structure.)