Arithmetic structure of integers

Set

 definiendum $\langle \mathbb Z,+_\mathbb{Z},\cdot_\mathbb{Z} \rangle$
 postulate $[\langle a,b\rangle]+_\mathbb{Z}[\langle m,n\rangle]=[\langle a+m,b+n\rangle]$ postulate $[\langle a,b\rangle]\cdot_\mathbb{Z}[\langle m,n\rangle]=[\langle a\ m+b\ n,a\ n+b\ m\rangle]$

The operations $+$ and $\cdot$ on the right hand sides are these of arithmetic structure of natural numbers.

Discussion

We'll generally use the notation introduced in integer. We'll also often omit the multiplication sign.

Reference

Wikipedia: Integer