Let $X$ denote a magma. Then $\mathrm{List}(X)$ is a monoid equipped with both a left and a right action on $X$, where the actions are defined in the obvious way. To illustrate these actions, suppose the operation of $X$ is denote $\diamond$, then we can act on $x \in X$ by the list $[u,v]$ like so: $$[u,v]x = u \diamond (v \diamond x), \qquad x[u,v] = (x \diamond u) \diamond v.$$
With that in mind, we can use $X$ to construct two interesting-looking sets like so: $$\mathcal{L}(X) = \frac{\mathrm{List}(X)}{\{(a,b) \in \mathrm{List}(X)^2 : \forall x \in X : ax=bx\}}, \quad \mathcal{R}(X) = \frac{\mathrm{List}(X)}{\{(a,b) \in \mathrm{List}(X)^2 : \forall x \in X : xa=xb\}}$$
This works because the denominators above are easily shown to be equivalence relations on $\mathrm{List}(X)$. In fact these are easily seen to be monoid congruences, so we find that $\mathcal{L}(X)$ and $\mathcal{R}(X)$ are quotient monoids of $\mathrm{List}(X)$.
Like $\mathrm{List}(X)$, the monoid $\mathcal{L}(X)$ is equipped with a natural left action on $X$, but unlike $\mathrm{List}(X),$ the monoid $\mathcal{L}(X)$ has no extra distinctions beyond those that are necessary to understand this action. Similar comments apply to $\mathcal{R}(X).$
These monoids seem to be pretty interesting. Among other things, they seem to give us a systematic way of "leftifying" identities in the language of monoids in order to obtain conditions on magmas. For example, we can define that a magma $X$ is left-commutative if and only if the monoid $\mathcal{L}(X)$ is commutative. It follows that $X$ is left-commutative if and only if it satisfies the identity $x \diamond (y \diamond z) = y \diamond (x \diamond z).$ Similarly, we can define that $X$ is left-idempotent if and only if $\mathcal{L}(X)$ is idempotent. I guess it's probably true that $X$ is left-idempotent if and only if it satisfies the identity $x \diamond (x \diamond y) = x \diamond y$. Anyway:
Question. Do these monoids have an accepted name, and are they studied anywhere?
I'm also interested in the module-theoretic version of this stuff, in which $X$ is an $R$-module together with a bilinear function $X \times X \rightarrow X$, and $\mathrm{List}(X)$ is replaced with the tensor algebra $T_R(X).$