Natural arithmetic is nonperiodic

An aperiodic semigroup is a semigroup that contains no non-trivial subgroups. This means that for all elements $x$ there is a point at which further iteration of the element produces no effect or in other words $x^{n+1}=x^n$. Elements of this form a clearly aperiodic, and that is already established in the literature. A separate issue, however, is rather or not elements that can be iterated infinitely are considered periodic. It is clear that since these elements continue to infinity without any repetition they cannot be considered periodic either. As a result, a separate concept is nonperiodic commutative semigroups.

We can immediately see that $(\mathbb{N},+)$ and $(\mathbb{N},*)$ are nonperiodic. In $(\mathbb{N},+)$ the first element 0 is idempotent, and the remaining elements continue infinitely. In $(\mathbb{N},*)$ the first elements are 0 and 1 both of which are idempotent and then the rest continue infinitely. The only difference is that multiplication has more idempotents. As a result, natural arithmetic has no unorderly aperiodic behavior.

Non-periodic semigroups have the most order-theoretic behavior among the class of semigroups. Non-periodic semigroups can be defined by the composition of extensive monotone functions of a partial order. The partial order of addition is the natural partial order, and the partial order of multiplication is the divisibility partial order. The divisibility partial order is a suborder of the natural ordering except that zero is considered maximal. With respect to these orderings, we can see that addition strictly increases the natural ordering and multiplication strictly increases the divisibility ordering. Multiplication by zero simply transforms an element to the maximal element in the partial order.

It is also noticeable that the addition and multiplication semigroups are therefore related to semilattices. In particular, the maximum semilattice and the least common multiple semilattice. Considering these as upper bounds we see that $max(a,b) <= (a+b)$ and $lcm(a,b)| (a*b)$ as maximum is the least upper bound of its ordering and addition is a much greater bound and likewise lcm is the least upper bound of divisibility and multiplication is an upper bound that is non-minimal.

The nonperiodic and orderly behavior of the arithmetic of the natural numbers is the basis of the connection between arithmetic and logic. Both the natural arithmetic operations and the lattice operations are commutative, associative, and nonperiodic. This is why when we have two disjoint sets $A$ and $B$ and we take their union the cardinality is equal to the sum of the cardinality of the two of them. Addition is an abstraction of the operation of joining two sets in set theory and classical logic. It abstracts away the elements of a set and it tells us about their cardinalities. In the same way, multiplication is an abstraction of the joining of partitions in the co-partition lattice defined in partition logic. In this sense, arithmetic exists to benefit the understanding of logic.