What all can we define with successor and division by 2?

Question

I know that multiplication is definable in any one of: $\langle \mathbb{N}; x^y \rangle$, $\langle \mathbb{N}; x^2+y^2 \rangle$, $\langle \mathbb{N}; xy+x \rangle$ $\langle \mathbb{N}; +, \div \rangle$ $\langle \mathbb{N}; <,\div \rangle$, $\langle \mathbb{N}; S, \div \rangle$

I don't think addition is definable in $\langle \mathbb{Q}; <, \times \rangle$, but it seems it is definable in $\langle \mathbb{Q}; S, \times \rangle$.

I am trying to understand how much can be defined (how close we can get to defining $+$ and $\times$) from any/each of $S$, $<$, and $+$ together with division by 2? The domain isn't particularly important to me.

Answer 1

$y=x/2$ is just $y+y=x$, so you can't get any more expressibility than addition from division by 2.