In the previous post, we noted that natural arithmetic $(\mathbb{N},+,*)$ is aperiodic in both addition and multiplication. As aperiodic operations they are of course directly related to partial ordering relations. The addition operation is directly related to the standard total ordering and the multiplication operation is directly related to the divisibility partial ordering with zero as the maximal element. Both of these are essentially distributive lattices as partial orders, which are also multiset inclusion lattices. The operations are multiset combination because natural addition and multiplication are both free. That is the situation with natural arithmetic. In the case of the integers $(\mathbb{Z},+,*)$ neither addition or multiplication is aperiodic, so we have to consider the related property of being torsion-free.
Torsion-free addition: the addition operation on the integers $(\mathbb{Z},+)$ is torsion-free. The reason for this is that all the negative integers are simply mirror images (or algebraic inverses) of their positive sides and so they still have the same infinite torsion-free behavior. So this means that the origin of periodic arithmetic does not lie in addition.
Partially torsion multiplication: the multiplication operation on the integers $(\mathbb{Z},*)$ is essentially the same as the on the natural numbers except that there are two elements ${-1,1}$ which are in a product with the ordinary integers. Furthermore, these two elements ${-1,1}$ form a cyclic group $C_2$ which means that $-1$ alone has periodic properties among the integers and only with respect to multiplication.
