| 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