Iteration as a monoid action

As I take a more action-oriented perspective, old concepts need to be reconsidered. One such concept is iteration, which we shall see produces a monoid action. Let $S$ be a semigroup and $(\mathbb{Z}_+,*)$ the monoid of positive integers under multiplication. Then we define a monoid action like so: \[ iter : (\mathbb{Z}_+,*) \times S \to S \] \[ iter_n(x) = x^n \] It is easy to see that this forms a monoid action. The identity $1$ produces an identity action so $iter_1(x) = x^1 = x$. The composition of iterations is $iter_m(iter_n(x)) = (x^n)^m = x^{nm} = iter_{nm}(x)$. These are semigroup endomorphisms when $S$ is commutative. The relevance of this monoid action is it produces the iteration preorder:

Proposition. the iteration preorder of a semigroup is the action preorder of its increasing monoid action.

The iteration preorder is familiar as a subpreorder of the commutativity preorder of the semigroup, but it may not be familiar that is the action preorder of a monoid action. We also saw that Green's preorders: L,R,J can be defined by certain monoid actions: left actions, right actions, and two sided actions by the monoid established from the semigroup by adjoining an identity. In this way, we have reconsidered old preorders in the light of monoid actions.

In general, we should as much as possible for any given preorder establish the structure of either a category or a monoid action on it which determines it. In the case of a category, this means the morphic preordering and in the case of a monoid action this means the action preorder. For example, the input/output action preorders of a category are better described by associated comma categories.

The additional data of a category or a monoid action, provides extra context to a preordering relation. Categories and monoid actions describe the processes of motion of elements. It may not be possible, however, to describe every preorder by some interesting monoid action or category. For example, it is unlikely that there is any sort of action that describes the commutativity preordering.

Monoid iteration:
In the special case of a monoid, we can use $(\mathbb{Z}_{\ge 0},*)$ as a base monoid. This assigns the zeroth power of each element to the identity, but it produce a different iteration preorder because then the identity is a successor of every element. This larger iteration preorder is still a subpreorder of the commutativity preorder though, because the identity is always a central element in any monoid.

Group iterations:
In the case of groups, we have an even bigger base monoid in the form of $(\mathbb{Z},*)$. Negative iteration then corresponds to inverses, which only exist in the special case of groups. As with the case of semigroups, iteration actions need not form endomorphisms unless $S$ is commutative. In that case, when we have a commutative group $G$ then it also forms a $\mathbb{Z}$-module, which has an underlying multiplicative monoid action.

Radical subsemigroups:
Every subsemigroup is iteration closed, which means it is an upper set of the iteration preorder. We can define the radical of a subsemigroup $\sqrt{S}$ to be the lower set closure of $S$ with respect to the iteration preorder. Then $S$ is both a lower set and an upper set of the iteration preorder, which necessarily means it is a disjoint union of connected components. Therefore, when considering radical subsemigroups, it is sufficient to consider the connected components of the iteration preorder. The same is true for radical ideals of semigroups.