Do non-associative objects have a natural notion of representation?

Since magmas in general don't have much structure, we can't reasonably expect a representation to preserve much structure. We can therefore define a left representation of a magma $M$ to be a set $V$ equipped with a map $M \times V \to V$. We do the analogous thing for general nonassociative algebras. Serge Lang liked to describe a notion of left regular representation of an algebra $A$, which is just the linear map $A \to \operatorname{End} (A)$ that takes an element to the linear transformation it induces by left multiplication. As expected, this map is a homomorphism if and only if the algebra is associative.

There are special cases of nonassociative algebras that admit good notions of representation, and in the cases I know, these arise from operads that have "good relationships" with the associative operad. The standard example is the natural map from the Lie operad to the Associative operad that yields the forgetful functor from associative algebras to Lie algebras. This functor admits the universal enveloping algebra functor as left adjoint. There is a formalism of enveloping operads, which generalizes this case. The upshot is that these special cases have a lot more structure than a simple composition law, so we can demand more from a representation (namely, that it respect the operad structure as manifested through the universal enveloping algebra).

Let me concentrate on your first question (frankly speaking, the way you formulate your second question slightly lacks motivation).

The case where there is a reasonable suggestion, assumes that you work with some type of nonassociative algebras over a field, and all identities follow from the multilinear ones. In other words, the category of algebras you are studying is the category of algebras over some operad O. In this case, there is a nice way to describe a module over such an algebra. For an algebra A, a module structure on V is given by a collection of operations defined by all possible operations from O, where you are allowed to plug an element from V into one slot of operations, and plug elements of A into other slots. To write down the module axioms, take the defining identities of O, and form new identities, marking one element there in all possible ways; now treat the unmarked elements as elements of A, and marked elements as belonging to V.

For example, for associative algebras the original identity is (ab)c=a(bc), which leads to the following definition. A module structure is defined by two operations, a,v\mapsto av and a,v\mapsto va satisfying the identities (ab)v=a(bv), (av)b=a(vb), (va)b=v(ab). This means that in the case of associative algebras we defined bimodules. Also, for Lie algebras we get the module structure which, as it is immediate to check, coincides with the usual module/representation structure. In general, this construction provides a reasonable "enveloping algebra" for your nonassociative algebra. Thus, one way to approach your question is to study representations of the enveloping algebra, and sometimes it's the best you can get.

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:

1. For all $a \in S$ and all $x,y \in S_*$, we have $ax = ay \rightarrow x=y.$
2. For all $a \in S$ and all $x \in S_*$, we have $ax = 1 \rightarrow \bot$
3. For all $a,b \in S$ and all $x \in S_*$, we have $ax = by \rightarrow a=b.$
4. (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}.

1. For all $x,y \in \mathbb{N}$, we have $sx = sy \rightarrow x=y.$
2. For all $x \in \mathbb{N}$, we have $sx = 0 \rightarrow \bot$
3. (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 $S_*$.