Lets begin by observing that to define a "non-associative action", we actually don't need a magma; a mere set will do.
Definition 0. Whenever $S$ is a set, a representation of $S$, also known as an $S$-unary algebra, consists of a set, call it $X$, together with a function $S \times X \rightarrow X,$ denoted $a,x \mapsto ax$.
A convention: the notation $abx$ means $a(bx)$, the notation $abcx$ means $a(b(cx))$, etc.
For example:
- a $0$-unary algebra is basically just a set.
- a $1$-unary algebra is basically a set $X$ equipped with a function $f:X \rightarrow X$. In the literature, these are called monounary algebras.
- a $2$-unary algebra is basically a set $X$ equipped with a pair of functions $f,g:X \rightarrow X$. In the literature, these are called biunary algebras.
- etc.
It should be clear that $S$-unary algebras are ubiquitous; for example, any time we have a set $X$ together with a self-map $f:X \rightarrow X$, the pair $(X,f)$ is a $1$-unary algebra. A famous example: the Collatz Conjecture is a statement about the $1$-unary algebra $(\mathbb{N},\xi)$, where $\xi$ is the Collatz function.
Definition 1. Whenever $S$ is a set, write $S_*$ for the $S$-unary algebra freely generated by $\{1\}$. The elements of $S_*$ are called words in $S$.
We're all familiar with the "symmetrical" viewpoint on $S_*$ in which we view it as a monoid denoted $S^*$. In particular, its the free monoid on $S$. In fact, the "asymmetrical" viewpoint in which its viewed as an $S$-unary algebra is pretty important, too; this basically amount to taking a treelike viewpoint. For example, here's a depiction of $\{f,g\}_*$:
Lets now collect together the main facts about $S_*$:
Theorem 0. Generalized Peano Postulates. Let $S$ denote a set. Then:
- For all $a \in S$ and all $x,y \in S_*$, we have $ax = ay \rightarrow x=y.$
- For all $a \in S$ and all $x \in S_*$, we have $ax = 1 \rightarrow \bot$
- For all $a,b \in S$ and all $x \in S_*$, we have $ax = by \rightarrow a=b.$
- (Axiom of Induction.) The only $S$-unary subalgebra of $S_*$ that contains $1$ is $S_*$ itself.
Furthermore, the above facts characterize $S_*$ among all $S$-unary algebras.
Okay. Define that $\mathbb{N}$ is the $\{s\}$-unary algebra freely generated by $\{0\}$. So basically, $\mathbb{N}$ is just $\{s\}_*$ with a slight change in notation. $$\mathbb{N} = \{0,s0,ss0,sss0,\ldots\}$$
By noting that Condition 3 trivializes when $S =\{s\}$ has only one element, we essentially rediscover Peano's original axioms for $\mathbb{N}$.
Theorem 1. Peano Postulates for \mathbb{N}.
- For all $x,y \in \mathbb{N}$, we have $sx = sy \rightarrow x=y.$
- For all $x \in \mathbb{N}$, we have $sx = 0 \rightarrow \bot$
- (Axiom of Induction.) The only monounary subalgebra of $\mathbb{N}$ that contains $0$ is $\mathbb{N}$ itself.
Furthermore, the above facts characterize $\mathbb{N}$ among all monounary algebras.
So I'd say non-associative actions are pretty important!
On the other hand, they're (in some sense) unnecessary:
Theorem 2. An $S$-unary algebra is the same thing as an $S^*$-set, i.e. a set $X$ equipped with an (associative) action of the monoid $S^*$.
One application is that these ideas give a really slick definition of the term "planar tree." First, note that if $M$ is a monoid and $X$ is an $M$-set, then the closed subsets of $X$ always form an Alexandroff topology (as opposed to a mere closure system). Ergo, since an $S$-unary algebra is just an $S^*$-set, hence its closed subsets automatically form an Alexandroff topology. This means, in particular, that $S_*$ is automatically an Alexandroff space.
Definition 3. An $S$-planar tree is an open subset of $S_*$.
Of course, Alexandroff spaces are the same thing as preorders; under this translation, "open subset" means the same thing as "lowerset". So we could equally well define that an $S$-planar tree is a lowerset of $S_*$.