Arithmetic structure of integers

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.

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

Wikipedia: Integer