Commutative aperiodic semigroups

Certain partial orders have semilattices associated with them. Semilattices are commutative idempotent semigroups that arise entirely from order theory. Commutative aperiodic semigroups are a generalization of semilattices, which can be used in a wider variety of cases. It is known that every commutative semigroup has an algebraic preorder associated with it, as described below. It can easily be seen that this is not a partial order, unless the semigroup is aperiodic.

\[ \exists z : x + z = y \implies x \le y \]
Set theory is based upon set union, which forms an idempotent commutative monoid. Well idempotence works well for set theory, it won't do for arithmetic. The natural numbers under addition $(\mathbb{N},+)$ can instead be modeled as an aperiodic commutative semigroup.This automatically generates the partial ordering on the natural numbers $(\mathbb{N},\le)$ because the addition operation only ever builds up a number by making it bigger, which is obviously not the case in a non-aperiodic semigroup like the integers. So set theory can be founded by an idempotent commutative monoid and arithmetic can be founded instead by an aperiodic commutative monoid.

It is interesting to consider the atoms in a commutative aperiodic semigroup like this. Atoms are defined as in order theory, as the elements that are only greater then the identity. The atoms in set theory are the individual members or singleton sets, which can be used to build up every element through set union. The atom in arithmetic is the singular unit, one which can be used to build up every natural number through addition. These are atomically generated semigroups. Atomically generated semigroups don't need to be atomistic lattices, as seen by the fact that the natural numbers are atomically generated. It is in this context that multisets can arise as values built up from a commutative aperiodic monoid, like sets are built up, except that the combiner operation is aperiodic rather then idempotent.