How to (or why cannot) define complex conjugate in the structure $(\mathbb{Z}+i\mathbb{Z},+)$?

Question

Let $(\mathbb{Z}+i\mathbb{Z},+)$, where $i$ is the imaginary unit, be a structure with an only operation $+$, the ordinary addition in $\mathbb{Z}$, and with no constant symbols. In this structure, the number zero and the inverse of any number are definable (I think), for they can be defined as $\forall x(x=0 \leftrightarrow\forall y(x+y=y))$ and $\forall x\forall y(x=-y\leftrightarrow x+y=0)$. Then is there any way to define the complex conjugate of any number or define some specific complex numbers, like $i$, $1+i$, etc., within the given structure?

Answer 1

There is no way to do this.

Remember that for a function $f: \mathcal{M}\rightarrow\mathcal{M}$ (or indeed any function or relation in general) to be definable in $\mathcal{M}$, it must be fixed by automorphisms: if $\alpha$ is an automorphism of $\mathcal{M}$, we must have that $f(\alpha(m))=\alpha(f(m))$ for all $m\in\mathcal{M}$. Note that this is not a sufficient condition - it's a good exercise to prove this.

Now observe that since we don't have the multiplicative structure on $\mathbb{Z}+i\mathbb{Z}$ in this case, the "switching" map $\alpha: a+bi\mapsto b+ai$ is an automorphism of the structure; but it doesn't preserve conjugation, since e.g. $$conj(\alpha(1-2i))=conj(-2+i)=-2-i\quad\mbox{ but }\quad\alpha(conj(1-2i))=\alpha(1+2i)=2+i.$$