I described previously how $(\mathbb{N},+,*)$ has aperiodic addition and multiplication. Addition is completely free and multiplication is free on non-zero elements, so that addition and multiplication are essentially multiset combination. As aperiodic commutative operations these operations produce non-minimal upper bounds well the least upper bounds are produced by the max and lcm operations instead. This keeps the natural operations in the category of aperiodic commutative operations.
The operations of the integers $(\mathbb{Z},+,*)$ are certainly not aperiodic. Neither of them are, but I noticed there is something special about addition different from multiplication. Well every element has an inverse, there are no elements that invert themselves under iteration. Elements exist separately from their inverses. But since the inverses exist they are not aperiodic. To avoid confusion, we can call the first type of semigroup aperiodic and the other type torsion free.
So well neither $(\mathbb{Z},+)$ nor $(\mathbb{Z},*)$ are aperiodic we have that $(\mathbb{Z},+)$ is torsion-free well $(\mathbb{Z},*)$ is neither torsion-free nor aperiodic. The importance of this comes in the construction of the complex numbers $\mathbb{C}$ from the integers $\mathbb{Z}$ because the property of having torsion-free addition is preserved from the integers all the way to the complex numbers well the fact that multiplication is not torsion free is key to the construction of complex multiplication which includes elements of all orders as the unit circle group extends the torsion group of integer multiplication.
We have a certain intuitive sense of "groupness" of semigroups if you allow me a figure of expression. Groups are operations that always have inverses for all of their elements. But since infinite torsion free groups allow you to separate elements from their inverses they have subsemigroups that are not subgroups. Indeed we can get an infinite width distributive lattice ordered subsemigroup of $\mathbb{Q}_{\ne 0}$ as seen by the natural numbers. We can get ordered subsets out of unordered structures when they are not torsion. So in a sense torsion groups are the most strongly group-like of operations among the groups.
