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Lets begin by observing that that to define a "non-associative action", we actually don't need a magma; a mere set will do.

Lets now collect to together the main facts about $S_*$:

Now 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. $$\mathbb{N} = \{0,s0,ss0,sss0,\ldots\}$$

Lets begin by observing that to define a "non-associative action", we actually don't need a magma; a mere set will do.

Lets now collect together the main facts about $S_*$:

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. $$\mathbb{N} = \{0,s0,ss0,sss0,\ldots\}$$

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.

Here's a few facts about $S_*$.

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 to together the main facts about $S_*$:

Definition 0. Whenever $S$ is a set, 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.

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.